MULTI-PHYSICS MODELING AND SIMULATION OF PHOTOVOLTAIC DEVICES AND

SYSTEMS

By

Timofey Golubev

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Physics – Doctor of Philosophy

2020

ABSTRACT

MULTI-PHYSICS MODELING AND SIMULATION OF PHOTOVOLTAIC DEVICES AND

SYSTEMS

By

Timofey Golubev

Physics-based computational modeling is an essential aspect of research and development of new
photovoltaic (PV) technologies. For example, PV modeling allows us to improve our understanding
of device physics, evaluate the potential of new device architectures prior to experimentation, and
predict the energy production of PV systems. This dissertation describes the development of
improvements to PV modeling at both the device and system levels, as well as the application of
these models to obtain new insights into device physics and PV system performance.

The first half of this thesis focuses on solar cell modeling using the drift-diffusion approach.
Investigations into numerical instabilities of a drift-diffusion model for bilayer organic solar cells are
described, and techniques for improving convergence behavior are presented. The drift-diffusion
approach is then adapted for planar perovskite solar cells, validated against literature results, and
utilized to understand a new experimental result involving the impact of ultrathin fullerene layers on
device performance. The second half of this thesis expands our modeling to PV systems. Here, we
focus on the effects of temperature and soiling (accumulation of particulate matter on PV module
surfaces) on PV system energy production. A general semi-physical model for predicting annual PV
soiling losses is developed and integrated with open-source PV performance models and commercial
algorithms for PV system cleaning schedule optimization. Additionally, the potentials of machine
learning approaches to soiling modeling are discussed and a proof-of-concept is demonstrated.
In order to consider the impact of temperature on PV module performance, a coupled thermal-
electrical modeling approach is developed by combining temperature-dependent equivalent circuit
models with a commercial heat transfer solver. This approach also allows for predicting energy
production of PV modules or films installed on irregular surfaces, such as vehicles. Application of
the coupled thermal-electrical approach to simulation of residential rooftop and vehicle-integrated

PV systems is demonstrated. Overall, this work has resulted in an improved understanding of the
numerical methods necessary to ensure stability of drift-diffusion codes, insight into the role of
fullerenes in perovskite solar cells, and the development of modeling approaches than can aid in
PV system engineering and improve accuracy in PV system energy production forecasting.

Copyright by
TIMOFEY GOLUBEV
2020

ACKNOWLEDGEMENTS

I am deeply grateful for the support I have received during my graduate career. First, I would like
to thank my advisor, Phillip Duxbury, for his guidance on both my thesis research projects and
general career planning. Thank you for providing me an opportunity to work on very interesting
photovoltaics research and encouraging me to pursue an internship, which later turned into a
permanent position.

I also wish to thank my Ph.D. guidance committee members: Andrew Cristlieb, Richard Lunt,
Morten Hjorth-Jensen, Hui-Chia Yu, and Pengpeng Zhang. I thank Richard Lunt for providing
me the opportunity to collaborate on perovskite solar cells research. I thank Morten Hjorth-Jensen
for his great computational physics course that helped me discover a passion for computational
physics research, as well as encouragement and support during a difficult time of my graduate
studies when I was struggling to find a thesis advisor and topic that would allow me to pursue
this passion.
I thank Pengpeng Zhang for being my advisor during my 2nd and 3rd years of
graduate school and providing an opportunity to do some interesting experimental research, as well
as her patience and support when I realized that I prefer computational research. My experience
as an experimentalist has certainly made me a better scientist, since it has given me a hands-on
understanding of experimental physics. I thank Andrew Christlieb and Hui-Chia Yu for general
thesis advice and support.

I would like to thank Non Thongprong for helping me get started with drift-diffusion modeling,
Dianyi Liu for performing the experiments for the perovskite devices that I modeled, and Reza
Loloee for his constant technical support during my time as an experimentalist. I am grateful to
Ryan Jones from Enlighten Energy for providing the opportunity to work on PV soiling research and
technical advice on the project. I also would like to thank Mark Hepokoski and Corey Packard from
ThermoAnalytics for providing advice and supporting the coupled thermal-electrical PV modeling
project, as well as their encouragement to finish my thesis. I acknowledge financial support for my
graduate studies and research from Michigan State University, the U.S. Department of Education

v

GAANN fellowship, Enlighten Energy, and ThermoAnalytics.

I would like to express gratitude to Vladimir Zelevinsky for his excellent theoretical physics
lectures and Kim Crosslan for support with all of the various paperwork and organizational aspects
of being a graduate student. I would like to thank Alexey Yamilov for being my undergraduate
research advisor and first introducing me to programming and computational physics. I am thankful
to all of my undergraduate and graduate school professors for the knowledge and skills that they
taught me, which provided a solid foundation for my research work. Also, I would like to thank
my colleagues at ThermoAnalytics for making my work experience enjoyable, and allowing me to
maintain a good work/Ph.D-research balance for the last 1.5 years.

I am grateful to all the graduate students who started graduate school with me in 2014 and
accompanied me in course work, research, and recreational activities: Andrew Tan, Bakul Agarwal,
Xingze Mao, Joe Williams, Crispin Contreras, John Bower, Bryan Isherwood, Victor Aguilar, Hao
Lin, Tenzin Rabga, Safwan Shanab, and more. I would like to extend my thanks to all the members
of the Foot-Pounds soccer team, which I was glad to be a part of. I am also very thankful to Xi
Dong, who was my labmate for a few months before I switched to computational research, and
became a great friend and fishing companion. Our fishing trips gave me much-needed breaks
from the challenges and stress of graduate school. Finally, I would like to thank my parents and
grandmother for their love and support throughout my life.

vi

TABLE OF CONTENTS

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LIST OF TABLES .
LIST OF FIGURES .
CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.1 Motivation .
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1.2 Solar Cell Devices .
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1.3 Photovoltaic Device Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Equivalent Circuit Approach . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Drift-Diffusion Approach . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix
xi
1
1
2
5
6
8
1.4 Photovoltaic System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Outline .

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2.1
2.2 Discretization .
2.3

PART I DRIFT-DIFFUSION MODELING OF SOLAR CELLS
CHAPTER 2

17
IMPLEMENTING DRIFT-DIFFUSION MODELS . . . . . . . . . . . 18
1D Drift-Diffusion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Iteration Schemes .
2.3.1 Gummel Method .
. 22
2.3.2 Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Single-Carrier Device in Space-Charge-Limited Current Regime . . . . . . 28
. 31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4.1
2.4.2 Two-Carrier Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 Model Verification .

2.5 Conclusions . .

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CHAPTER 3

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IMPROVING CONVERGENCE OF A DRIFT-DIFFUSION MODEL
FOR BILAYER ORGANIC SOLAR CELLS . . . . . . . . . . . . . . . 34
3.1 DD-BI Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Convergence Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Convergence Acceleration through Pulay Mixing . . . . . . . . . . . . . . 41
3.2.2 Newton vs. Gummel Iterative Methods
. . . . . . . . . . . . . . . . . . . 42
Slotboom Variables Approach . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Conclusions . .

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CHAPTER 4 DRIFT DIFFUSION MODELING OF PEROVSKITE SOLAR CELLS 48
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1 Numerical Model
.
4.2 Model Validation .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 Understanding the Impact of C60 at the Interface of Perovskite Solar Cells . . . . . 53
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Conclusions . .

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vii

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IMPROVING PV SYSTEM SOILING LOSS PREDICTIONS . . . .

PART II MODELING OF PHOTOVOLTAIC SYSTEMS
CHAPTER 5

66
. 68
5.1 Background .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 Semi-Physical Modeling of Soiling Accumulation . . . . . . . . . . . . . . . . . . 71
5.2.1 Background .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.2 Limitations of Existing Models . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2.3 Developing a General Soiling Accumulation Model . . . . . . . . . . . . . 85
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3 Effect of Rainfall
Incidence Angle Modifier due to Soiling . . . . . . . . . . . . . . . . . . . . . . . 94
5.4
. . . . . 97
5.5 Combining Incidence Angle Modifier with Soiling Accumulation Models
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.6 Model Validation .
5.7
Integration into Commercial Software . . . . . . . . . . . . . . . . . . . . . . . . 103
5.8 Predicting Soiling with Machine Learning . . . . . . . . . . . . . . . . . . . . . . 113
Introduction to MLR and ANN . . . . . . . . . . . . . . . . . . . . . . . . 114
Soiling Mass Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . 116
.
Soiling Loss .
. 122
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5.9 Conclusions . .

5.8.1
5.8.2
5.8.3

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CHAPTER 6 COUPLED THERMAL-ELECTRICAL MODELING OF INTEGRATED

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Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . .

PV SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.1 Empirical Models .
6.2 Physical Thermal-Electrical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2.1
. 133
6.2.2 Methods .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2.3 Model Validation .
. 148
6.2.4 Comparison with Empirical Models . . . . . . . . . . . . . . . . . . . . . 153
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
. . . . . . . . . . . . . . . . . . . . . . . . 155
6.3.1 Building Rooftop PV Systems
6.3.2 Vehicle-Integrated Solar Panels . . . . . . . . . . . . . . . . . . . . . . . . 158
. 165
6.3.3
PV-Battery Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.4 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 169
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

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6.3 Example Applications .

6.4 Conclusions . .

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CHAPTER 7 CONCLUSIONS AND OUTLOOK . . . . . . . . . . . . . . . . . . . . 171
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
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7.1 Conclusions . .
7.2 Outlook .
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APPENDICES .
APPENDIX A
APPENDIX B
APPENDIX C

BIBLIOGRAPHY .

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PV SOILSAYER SOFTWARE DESCRIPTION . . . . . . . . . 177
. 201
PV MODULE DATASHEETS . . . . . . . . . . . . . . . . . .
LIST OF PUBLICATIONS RESULTING FROM THIS WORK 205
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
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viii

LIST OF TABLES

Table 4.1: Complete set of parameters used for the model. References for parameters

obtained from literature are listed.

. . . . . . . . . . . . . . . . . . . . . . . . . 56

Table 5.1: Empirical parameters of the Guo et al. model. . . . . . . . . . . . . . . . . . . . 84

Table 5.2: Daily rainfall adjustments needed to account for measured soiling ratio time-

series for NREL Site 3. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Table 5.3: Daily rainfall adjustments needed to account for measured soiling ratio time-

series for NREL Site 11.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Table 5.4: Variability of forecast for a site in Lamont, CA (NREL Site 11 from [125])

when using different weather prediction resources.

. . . . . . . . . . . . . . . . 111

Table 5.5: Comparison of ANN and MLR performance when different numbers of fea-

tures (environmental variables) are used for model training. . . . . . . . . . . . . 120

Table 5.6: Statistical metrics comparing MLR and ANN model performance when using
a dataset generated with the semi-physical soiling model without the resuspen-
sion term.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

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Table 6.1: SAPM temperature model coefficients. . . . . . . . . . . . . . . . . . . . . . . . 130

Table 6.2: Errors of empirical models for results shown in Figure 6.1.

. . . . . . . . . . . . 131

Table 6.3: Errors of empirical models for entire year using default parameters.

. . . . . . . 132

Table 6.4: Errors of empirical models’ results shown in Figure 6.2.

. . . . . . . . . . . . . 133

Table 6.5: Errors for one year of module temperatures predicted by empirical models

using fitted parameters. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Table 6.6: Physical properties of the solar module.

. . . . . . . . . . . . . . . . . . . . . . 148

Table 6.7: Electrical parameters of the solar module.

. . . . . . . . . . . . . . . . . . . . . 149

Table 6.8: Physical properties of the CdTe solar module.

. . . . . . . . . . . . . . . . . . . 151

Table 6.9: Derived equivalent circuit parameters of the First Solar 4 Series FS-4122-3

solar module.

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ix

Table A.1: SAPM temperature model coefficients. . . . . . . . . . . . . . . . . . . . . . . . 182

Table A.2: Spectral mismatch modifier coefficients. . . . . . . . . . . . . . . . . . . . . . . 183

Table A.3: Sample albedo values. From [130].

. . . . . . . . . . . . . . . . . . . . . . . . 186

Table A.4: Default annual losses in PVWatts for reference.

. . . . . . . . . . . . . . . . . . 189

x

LIST OF FIGURES

Figure 1.1: Diagram of the energetics, and drift and diffusion currents at the p-n junction

of a solar cell. Adapted from [8].

. . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.2: General JV curve showing the main parameters used to describe a solar cell’s

performance. Adapted from [9]. . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.3: Equivalent circuit of an ideal solar cell (a) and a solar cell with losses modeled

with series and shunt resistance (b). Adapted from [13].

. . . . . . . . . . . . .

3

4

7

Figure 2.1: Two Gummel iteration approaches. Approach b) is used in this study. . . . . . . 23

Figure 2.2: Flow chart of the main steps in the Newton-Raphson method.

. . . . . . . . .

. 27

Figure 2.3: Comparison of results from drift-diffusion model of an arbitrary single-carrier
device under SCLC conditions with Mott-Gurney analytical equations. a)
Current density versus voltage. b) The electric field distribution through the
thickness of a device under high voltage.

. . . . . . . . . . . . . . . . . . . . . 30

Figure 2.4: Drift-diffusion (DD) model results for JV curves of a hole-only device with
varying sizes of injection barriers compared with the Mott-Gurney equation.

. . 31

Figure 2.5: Results from drift-diffusion models of single-layer, two-carrier devices: one
with uniform mobilities (a-d) and another with 100x larger hole mobility than
electron mobility (e-i). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Figure 3.1: Basic architecture (a) and energy level alignment (b) of a bilayer solar cell.

Adapted from [32].

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Figure 3.2: Carrier densities at short-circuit (a) and open-circuit (b) conditions for the

bilayer device.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Figure 3.3: Recombination and generation processes at the bilayer interface. Adapted

from [15].

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Figure 3.4: The hole density at a point close to the interface inside the device (which
exact point does not matter) is plotted verses the number of iterations of the
Gummel loop. a) A converging solution, which occurs when the mixing
factor is sufficiently small for the applied voltage. b) A solution that cannot
converge. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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xi

Figure 3.5: Comparison of convergence behaviors between Gummel and Newton meth-
ods. a) Mixing factor needed to achieve convergence. b) Number of iterations
required to converge.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

.

Figure 4.1: Device structure of planar heterojunction perovskite solar cell modeled by

Sherkar et al. [47]. . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Figure 4.2: Simulated JV curves for different carrier mobilities in perovskite layer from

this work (a), and Sherkar et al. (b). . . . . . . . . . . . . . . . . . . . . . . . . 51

Figure 4.3: Simulated JV curves for different trap densities at HTL/perovskite interface
(a, b) and ETL/perovskite interface (c, d). Results from this work’s model (a,
c) are compared with results from Sherkar et al. (b, d). . . . . . . . . . . . . . . 52

Figure 4.4: Simulated JV curves for different injection barriers at the anode and cathode
(φa and φc respectively) on the JV curve from this work’s model (a), and
from Sherkar et al. (b).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 4.5: Energy levels and thicknesses of the device layers with the rectangles repre-
senting the bandgap of the material. The levels were found from a combination
of literature values and fitting of the numerical model (more details in Table
4.1). The levels within the BCP bandgap are gap-states due to diffusion of
Ag into BCP. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

.

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.

.

Figure 4.6: Carrier photogeneration rate profile for the device without C60 as deter-
mined from the transfer-matrix optical model using the sum over wavelengths
350-750 nm. Position is displayed relative to the beginning of the HTL
(PEDOT:PSS) layer.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

.

Figure 4.7: Comparison of the JV curve results from the experiment and numerical model. . 58

Figure 4.8: JV curves showing the effects of perovskite/BCP interface carrier mobility
(a) with ETL trap density set to 6 × 1020 m−3 and trap density (b) on the 0
nm device with perovskite/BCP interface mobility set to 2× 10−9 m2/Vs (the
same value as for the 20 nm C60 device). All other parameters were kept the
same as in Table 4.1. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Figure 4.9: Charge carrier densities for the simulated devices with 0 nm, 1 nm, and 20 nm
C60 under illumination at zero applied bias. The perovskite/C60 and C60/BCP
interfaces are marked by dotted black lines. The parameters used are listed in
Table 4.1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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xii

Figure 4.10: The effect of using an energetic barrier for electron extraction instead of a low
effective mobility region at the perovskite/BCP interface of devices without
C60. The fit to experiment that was achieved when using a low mobility
region is shown for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Figure 4.11: Charge carrier densities for device with 0 nm C60 under illumination at 0.7 V

(a) and 0.9 V (b) applied bias. The parameters used are listed in Table 4.1.

. . . 64

Figure 4.12: The effect of variation of the HOMO/LUMO of the HTL/ETL on the JV curve

of the device with 0 nm C60. The fit to experiment is included for comparison. . 65

Figure 5.1: Soiling is a complex problem with many influencing factors over a large
range of size and time scales. Environmental parameters are shown in blue,
microscopic soiling processes are shown in green, and controllable influenc-
ing factors related to plant design and maintenance strategies are in orange.
Adapted from [86].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

.

Figure 5.2: Schematic of particle dry deposition model resistance analogy. Adapted from

[97].

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.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Figure 5.3: Schematic of deposition due to particle impaction. Adapted from [98].

. . . . . 79

Figure 5.4: One year of hourly soiling rates predicted by the Guo et al. model with (a)

and without (b) the resuspension term.

. . . . . . . . . . . . . . . . . . . . . . 85

Figure 5.5: Attachment probability as a function of a) particle diameter (with wind speed
of 1 m/s and relative humidity of 70%) and b) wind speed (with particle
diameter of 15 µm and relative humidity of 50%).

. . . . . . . . . . . . . . . . 87

Figure 5.6: Wind-induced velocity as a function of a) wind speed (for 10 µm particles in
40% relative humidity) and b) relative humidity (for 50 µm particles at 2 m/s
wind speed).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

.

.

.

.

.

.

Figure 5.7: Resuspension flux versus wind speed for RH 40%. . . . . . . . . . . . . . . . . 89
Figure 5.8: Net soiling mass accumulation flux at 40% RH, 50 µg/m3 PM10 concentra-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

tion, and 20◦ tilt angle.

Figure 5.9: Comparison of hourly soiling mass accumulation rates for Riverside, CA as
predicted by the Guo et al. model (a) and the improved model proposed in
this thesis (b).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

. .

.

Figure 5.10: Reduction in soiling losses due to rainfall. This plot is from Enlighten
Energy’s PV SoilSayer software, which contains an implementation of the
soiling model described here.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 93

xiii

Figure 5.11: Empirical angular loss factors for direct (a), diffuse (b), and albedo (c) ir-
radiance as functions of angle of incidence for two angular loss coefficients
(ar).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

. .

.

.

.

.

.

.

.

Figure 5.12: Hourly soiling loss profile over a single day for two different panels: a "no
clean" panel that was never cleaned and a "clean" panel that is periodically
cleaned according to a cleaning cycle. This plot is from Enlighten Energy’s
PV SoilSayer software, which contains an implementation of the soiling
model described here.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Figure 5.13: Comparison of model predictions with NREL Site 3 (Calipatria, California)
when using precipitation data provided by NREL (a) and when using modified
rainfall data (b).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

. .

.

Figure 5.14: Comparison of model predictions with NREL Site 11 (Lamont, California)
measured data when using precipitation data provided by NREL (a) and when
using modified rainfall data (b). . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Figure 5.15: Flow-chart of the approach used to incorporate the soiling loss models that

were developed in this work into PV SoilSayer. . . . . . . . . . . . . . . . . . . 104

Figure 5.16: Beginning of PV SoilSayer results report, showing the forecast inputs and
comparison of the predicted annual energy production and soiling losses for a
system that is never cleaned ("No Clean" case) versus a system that is cleaned
according to an optimized cleaning schedule ("Clean" case). . . . . . . . . . . . 106

Figure 5.17: Optimal manual cleaning dates and soiling rates results page from a sample

PV SoilSayer forecast. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Figure 5.18: Visualization of soiling losses and energy production from a sample PV
SoilSayer forecast. The sharp reductions in soiling loss correspond to manual
and rainfall cleanings of the panels. . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 5.19: Visualization of soiling and energy production for a single month from a
sample PV SoilSayer forecast. A manual cleaning event occurs from June 13
to June 22, reducing the soiling loss on the "Clean" panel and increasing the
difference between "Clean" and "No Clean" panel energy production. . . . . . . 109

Figure 5.20: AC soiling loss with oscillations caused by inverter clipping.

. . . . . . . . . . 110

Figure 5.21: AC soiling loss for same forecast as Figure 5.20, but without inverter clipping.

. 111

Figure 5.22: Comparison of soiling time-series predictions for a site in Lamont, CA (NREL

Site 11 from [125]) when using different weather prediction resources.

. . . . . 112

xiv

Figure 5.23: Diagram of an ANN. From [135]. . . . . . . . . . . . . . . . . . . . . . . . .

. 115

Figure 5.24: Heatmap of Pearson correlation coefficients between environmental variables

and predicted soiling rate.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Figure 5.25: Heatmap of Spearman correlation coefficients between environmental vari-

ables and predicted soiling rate. . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Figure 5.26: Comparison of hourly soiling rates predicted by MLR and ANN models to
rates predicted by the semi-physical model (with resuspension) that they were
trained on. Only data points from the testing set are shown.

. . . . . . . . . . . 121

Figure 5.27: Comparison of hourly soiling rates predicted by MLR and ANN models to
rates predicted by the semi-physical model (without resuspension) that they
were trained on. Only the test data points are shown. . . . . . . . . . . . . . . . 122

Figure 5.28: Heatmap of Spearman correlation coefficients between ten-day moving aver-
ages of soiling and environmental variables measured at a PV power plant in
California.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

.

.

.

.

.

.

Figure 5.29: Comparison of Kimber, MLR, and ANN models’ ability to fit the training

data (measured soiling and environmental parameters for 2019). . . . . . . . . . 124

Figure 5.30: Comparison of Kimber, MLR, and ANN models’ ability to predict soiling in
2020, when trained on data from 2019. Statistical metrics for two different
prediction periods are shown: February 25 through June 16, and April 23
through June 16. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

.

.

Figure 6.1: Comparison of empirical models using default parameters.

. . . . . . . . . . . 131

Figure 6.2: Comparison of empirical models with parameters fitted using least-squares

optimization. .

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Figure 6.3: Five-parameter equivalent circuit representation of a solar cell. Adapted from

[147].

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Figure 6.4: Flow chart of the Cubas et al. procedure to find the equivalent circuit parameters.140

Figure 6.5: Flow chart of the improved equivalent circuit parameters extraction method.

. . 142

Figure 6.6: Example of results from our modified parameter extraction tool. a) Conver-
gence behavior as the ideality factor guess is incremented. b, c) IV curves
generated during the iterative process, with the optimized ones plotted in
red. Fill factor decreases as ideality factor increases. d) Values of optimal
parameters for this run.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

xv

Figure 6.7: Flow chart of the calculations at each time step of the coupled thermal-electric

PV model.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Figure 6.8: TAITherm prediction of cell temperature variation with a) irradiance, b) air

temperature, and c) wind speed. . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Figure 6.9: One day simulation results for a) cell temperature, b) current, c) efficiency, d)

silicon bandgap, e) modified ideality factor, and f) diode saturation current. . . . 150

Figure 6.10: Example of the effect of cell temperature on a) IV curve and b) power production.151

Figure 6.11: Comparison of one day of modeled and measured First Solar Series 4 CdTe

module temperature and short-circuit current.

. . . . . . . . . . . . . . . . . . 153

Figure 6.12: Comparison of cell temperature versus irradiance predictions under 1 m/s

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

and 5 m/s wind.

Figure 6.13: Comparison of cell temperature versus air temperature and wind speed predictions.154

Figure 6.14: Comparison of cell temperature predictions versus wind speed and irradiance
made with a fitted PVsyst model (ηm=18%, α = 0.9, uc = 23.2, uv = 7.4)
and TAITherm.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

.

.

.

Figure 6.15: TAITherm house model with roof-mounted solar panel.

. . . . . . . . . . . . . 156

Figure 6.16: PV rooftop system simulation results for a typical day in July in Phoenix,
Arizona. Predicted power (a) and efficiency (b) with and without considering
temperature-induced losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Figure 6.17: Predicted PV energy production with and without including soiling losses. a)

Daily DC power. b) Monthly DC energy production. . . . . . . . . . . . . . . . 157

Figure 6.18: a) Soiling coefficients used as input. b) Transmission and angular losses due
to soiling calculated by the thermal-electrical model. Daily rainfall totals are
shown with blue bars.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Figure 6.19: TAITherm Jeep model with roof-mounted solar panel. . . . . . . . . . . . . . . 159

Figure 6.20: Comparison of a single day of modeled vehicle-integrated solar panel power
(a) and efficiency (b) when different factors affecting PV performance are
considered (irradiance dependence only, irradiance and angle dependence,
and irradiance, angle, and temperature dependence). . . . . . . . . . . . . . . . 160

xvi

Figure 6.21: a) TPV UV-vis spectrum for front illumination. Data from [165]. b) Incident
angle modifier for a conventional thin film PV and one that is optimized for
improved angle-dependence. Data from [149].

. . . . . . . . . . . . . . . . . . 163

Figure 6.22: a) Predicted total annual energy production in five U.S. cities for three different
vehicle-integrated PV systems: TPVs covering entire surface of the vehicle,
TPVs only used in windows, and a conventional silicon PV roof module.
b) Predicted monthly energy production for the TPV system that covers the
entire surface of the vehicle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Figure 6.23: Schematic of the thermal-electrical battery model. Q is the heat generated by
the battery, calculated in terms of Joule heating (electrical losses) within the
battery equivalent circuit and entropic heating (heat from chemical reactions
in the battery). Adapted from [167].

. . . . . . . . . . . . . . . . . . . . . . . 166

Figure 6.24: Flow diagram of the battery life prediction model. ("Eqc" stands for "equiv-

alent circuit".) .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 6.25: a) Battery charge/discharge current over a single summer day. Positive current
is defined as charging. b) State of charge of the 30 Ah battery due to the current
profile in a).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

. .

. 167

. 168

Figure 6.26: Relative resistance (a) and capacity (b) of the 30 Ah and 60 Ah batteries.

Values are relative to those of a brand-new battery. . . . . . . . . . . . . . . . . 168

Figure 6.27: PV-battery and battery management system modeling approach. . . . . . . . . . 169

Figure A.1: Weather Resource UI.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Figure A.2: System Design UI. .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Figure A.3: Inverter partial-load efficiency curve. Adapted from [183].

. . . . . . . . . . . 181

Figure A.4: Array types. Adapted from [185]. . . . . . . . . . . . . . . . . . . . . . . . . . 184

Figure A.5: Ground cover ratio. Adapted from [185].

. . . . . . . . . . . . . . . . . . . . . 185

Figure A.6: PV System Losses UI. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Figure A.7: Monthly energy production chart example.

. . . . . . . . . . . . . . . . . . . . 189

Figure A.8: Soiling loss calculation/upload options. . . . . . . . . . . . . . . . . . . . . . . 190

Figure A.9: Soiling Loss Calculator UI.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

xvii

Figure A.10:DC Loss Over a Dry Period UI. . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Figure A.11:Daily Soiling Rates UI.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 192

Figure A.12:Angular Losses UI.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Figure A.13:Rainfall Parameters UI.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 195

Figure A.14:Sample Monthly Average Soiling Losses Output. . . . . . . . . . . . . . . . . . 195

Figure A.15:Clean Cycle Configuration UI.

. . . . . . . . . . . . . . . . . . . . . . . . .

. 196

Figure A.16:Optimal Clean Cycle UI.

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 197

Figure A.17:Clean Based on Specific Thresholds UI.

. . . . . . . . . . . . . . . . . . . . . 199

Figure A.18:Clean Based on Specified Cycle UI. . . . . . . . . . . . . . . . . . . . . . . . . 200

xviii

CHAPTER 1

INTRODUCTION

1.1 Motivation

Fossil fuel energy sources have been a fundamental driver of industrialization and the resulting
technological, economic, and social progress. However, fossil fuels are finite with predictions
expecting worldwide reserves of oil, natural gas, and coal to be depleted in 50, 51, and 132 years
respectively [1]. Additionally, fossil fuels are the dominant source of greenhouse gases, which are
causing climate change [2]. The Intergovernmental Panel on Climate Change (IPCC) predicts that
without significant emissions reductions, annual carbon dioxide emissions would rise to over 100%
above 2010 levels by 2050, which is likely to cause a rise in global temperatures of 4◦C from pre-
industrial levels. Such a temperature rise is expected to cause serious groundwater shortages, more
severe weather events, decreased agricultural production, and loss of critical ecosystem functions.
The IPCC predicts that it is possible to limit the global temperature increase to 2◦C if carbon dioxide
emissions drop to 66% below 2010 levels by 2050 [3]. A rapid transition to clean renewable energy
sources is essential to maintain the standard of living in developed countries, allow for the continued
development of the rest of the world, and reduce the effects of greenhouse gas emissions.

With the increasing concerns and education about fossil fuels, renewable energy usage has
been expanding worldwide. Sunlight is the most abundant renewable resource. There is more
energy in the sunlight striking the earth’s surface in two hours than the annual worldwide energy
consumption [4]. In 2018, increases in solar power production capacity accounted for over 53% of
the total new net capacity of all renewable sources [5]. The growth in photovoltaics (PV) energy has
been supported by improved technologies that have led to increasing efficiencies and rapidly falling
costs [6]. There is still an immense opportunity for growth since PV energy currently supplies
only 2.6% of the global electricity demand [7]. In order to effectively expand PV technologies, it
is necessary not just to install more solar panels, but also to continue to improve power generation

1

efficiencies and reduce manufacturing costs. These improvements can be made in two different
categories. One area of potential improvement is to develop better solar cell materials and devices
that are more efficient and less expensive to produce. The second area where improvements can be
made is at the PV module and system level, by reducing losses due to the environmental conditions
such as high temperatures and reduction in absorbed irradiance due to soiling (i.e. particulate
matter) accumulation.

Physics and computational modeling are an essential part of the research and development of
PV devices and systems. Computational modeling allows us to explain experimental observations
and test new designs before investing time and financial resources into conducting experiments.
Developing models for solar cell devices that can explain experimental observations requires a
strong understanding of solar cell device physics. Consideration of semiconductor physics, optics,
heat transfer, weather, and soiling is necessary to realistically model solar modules that are outside
of controlled laboratory conditions. Therefore, accurate modeling of PV technologies in real-
world conditions is a complex multi-physics problem whose solution requires the combination of
electrical, thermal, and environmental models. This dissertation describes the development of
improvements to PV modeling at both the device and system levels, as well as the application of
these models to obtain new insights into device physics and PV system performance.

1.2 Solar Cell Devices

A solar cell is a two-terminal semiconductor device that generates current and voltage when
under illumination. The basic components of a solar cell are two electrodes (at least one of which
allows transmission of light) and a solar irradiance absorber (also known as the active layer), which
is made of a semiconductor. The working principle is based on the photovoltaic effect where
photons of energy greater than the bandgap of the absorber layer excite electrons from the valence
band (VB) to the conduction band (CB), leaving behind holes in the VB. Holes are the absence
of an electron and behave like a particle, which is positively charged and has an effective mass
that reflects its ability to move through a material. Due to an energy level difference between

2

the cathode and anode work functions, a solar cell has a built-in electric potential. The electric
potential causes the photogenerated free electrons and holes (also known as charge carriers, or
simply carriers) to separate and move on-average in opposite directions to the positive and negative
contacts, thus generating a current. Charge carriers also move due to diffusion, which describes the
tendency of particles to move from higher to lower concentration. The net current in the device has
contributions from carrier drift due to the electric potential and carrier diffusion due to the particle
concentration gradient (Figure 1.1). The mechanism of separation of the carriers is dependent on
the materials and architecture of the cell. The accumulation of electrons and holes at the electrodes
results in the generation of a voltage, which counteracts the built-in potential. Once the voltage due
to charge accumulation at the contacts becomes high enough to cancel both the drift and diffusion
currents, there will be zero current in the solar cell. The voltage at which this occurs is called the
open-circuit voltage (Voc). Beyond Voc, current flows in the opposite direction, and the solar cell
no longer produces electrical energy.

Figure 1.1: Diagram of the energetics, and drift and diffusion currents at the p-n junction of a
solar cell. Adapted from [8].

The performance of a solar cell can be described in terms of a generated current density (J)
versus voltage (V) curve (commonly known as JV curve), as shown in Figure 1.2. The JV curve
has several parameters that are used to characterize a device’s performance: short-circuit current

3

density (Jsc), open-circuit voltage (Voc), and fill factor (FF). The short-circuit current is the current
which flows when the voltage between the electrodes is zero, or equivalently when the device is
short-circuited. This current depends strongly on the photon absorption rate in the active layer.
In the ideal case, Jsc is equal to the photocurrent (Jph); in other words, all of the photogenerated
electrons and holes reach the electrodes. The open-circuit voltage is the maximum voltage that
a solar cell can produce or the voltage that develops inside the cell when the two electrodes are
isolated. Voc is the zero of the JV curve. A solar cell will have the highest efficiency if it is operated
at the J and V that gives maximum power output, where power is defined by P = JV. The fill
factor is defined as the ratio Pmax/JscVoc and represents the "squareness" of the JV curve. Power
conversion efficiency is the ratio between the maximum generated power and the incident power

PCE =

Pmax
Pinc

=

JscVocFF

Pinc

(1.1)

Figure 1.2: General JV curve showing the main parameters used to describe a solar cell’s
performance. Adapted from [9].

To maximize efficiency, the Jsc, Voc, and FF should be maximized. Qualitatively, this means
that an ideal solar cell will have a high absorption rate of photons with all of the generated charge
carriers reaching the electrodes without loss. In practice, we can describe four main criteria for an
efficient solar cell. First, the absorbing layer’s bandgap must be small enough so that the photons
from solar radiation have enough energy to excite the electrons. Simultaneously, the bandgap

4

should be large enough to prevent carrier recombination. Second, the absorber layer should be
thick enough to absorb a significant amount of light. However, this thickness is limited to the
finite diffusion lengths of charge carriers in the material. Therefore, an active layer that has high
carrier mobilities is beneficial. Third, the cell’s layers should have suitable energy bands so that
both the electrons and holes encounter no energy barriers as they travel to their contacts. Finally,
the losses during transport of the generated carriers to the electrodes should be minimized. These
losses can occur due to carriers recombining with defects and impurities in the active layer and at
the interfaces between layers [10].

Solar cell technologies are often classified into three generations. First-generation solar cells
are silicon-based and dominate the market. Second-generation solar cells are based on thin-film
technologies such as cadmium telluride (CdTe), copper indium gallium diselenide (CIGS), and
thin-film amorphous silicon. Third-generation solar cells are based on semiconducting organic
molecules, inorganic nanoparticles, or organic-inorganic hybrid materials, and generally manu-
factured through solution-processing. Prominent examples are organic and perovskite solar cells.
Some advantages of third-generation solar cells are reduced manufacturing costs due to low tem-
perature and vacuum-free fabrication and the possibility of roll-to-roll printing. They are also light,
flexible, and can be made semi-transparent [11]. The commercial market is currently dominated by
first-generation solar cells, but this may soon change as third-generation PV technologies advance
to commercially-viable products. The device modeling portion of this dissertation focuses on the
emerging third-generation solar cells, while the system-level modeling portion concentrates mostly
on the first and second-generation solar cells that are commonly used today commercially.

1.3 Photovoltaic Device Modeling

As briefly described above, many aspects need to be optimized in order to design efficient
solar cells. Experimentally optimizing all of these aspects is time-consuming and can be costly.
Computational modeling can help rapidly test possible combinations of materials and device
architectures for ones that have the greatest potential of producing high-performance solar cells.

5

Also, comparing experimental results to simulations based on the physics equations improves our
understanding of the mechanisms that affect the device performance. Computational modeling has
been used since the 1950s to study the physical properties of semiconductor devices, including solar
cells. Many PV device models are based on the fitting of experimental JV curves to an equivalent
circuit equation. More detailed and physics-based approaches are based on various simplifications
of the Boltzmann transport equation. One of the most popular and efficient approaches is the
drift-diffusion model, which involves numerically solving equations for the electric potential and
currents in the device in a self-consistent loop. The equivalent circuit and drift-diffusion approaches
are introduced in the next two sections.

1.3.1 Equivalent Circuit Approach

Equivalent circuit modeling is a widely used approach for describing the electrical behavior of solar
cell devices. This approach uses equivalent circuit equations based on the Shockley diode equation,
which describes the ideal recombination current from diffusion and recombination of carriers at
the p-n junction (a diode) [12]. The Shockley diode equation for current (I) in a diode as a function
of applied voltage (Va) is

(cid:20)

(cid:19)

(cid:18) eVa

kBT

(cid:21)

− 1

I(Va) = Io

exp

(1.2)

where I0 is the saturation-current (thermal generation current at zero applied bias), e is the elemen-
tary charge, kB is the Boltzmann constant, and T is the device temperature. The thermal generation
current is the current due to drift of the minority carriers (electrons in the p-type region and holes
in the n-type region) across the junction.

The simplest model of a solar cell treats it as an equivalent circuit composed of an ideal diode
and a current source that are connected in parallel, as shown in Figure 1.3a. The relationship
between current and voltage is described by adding a photocurrent term to the Shockley diode
equation

I(Va) = Io

exp

− 1

− Iph

(1.3)

(cid:21)

(cid:20)

(cid:18) eVa

(cid:19)

kBT

6

In practice, a solar cell has losses which can be described by adding a series resistance (Rs)
and a shunt (or parallel) resistance (Rp), as shown in Figure 1.3b. The series resistance describes
the losses related to non-infinite mobilities of the charge carriers and imperfect contacts between
the active layer and electrodes. The parallel resistance determines the leakage current, which is the
current that does not flow through the electrodes and can be due to pinholes between the layers in
the cell that provide an alternate path for the current. The charges that flow through the leakage
path will not contribute to the powering of the device or charging of the battery that the solar cell
is connected to.

Figure 1.3: Equivalent circuit of an ideal solar cell (a) and a solar cell with losses modeled with
series and shunt resistance (b). Adapted from [13].

A non-dimensional constant, a, called the ideality factor, is added to the Shockley diode term
to account for deviation of the diodes from Shockley diffusion theory [14]. Recognizing e/kBT as
the thermal voltage (VT), this more realistic equivalent circuit can be described by

I(V) = Io

exp

− 1

V − IRs

Rp

+

− Iph

(1.4)

(cid:20)

(cid:18)(V − IRs)

(cid:19)

(cid:21)

aVT

7

For a high-performance solar cell, Rs should be small, and Rp should be large. Other equivalent
circuit models have been developed, such as the ones for organic heterojunctions [15, 16]. The effect
of temperature on device performance can be included by considering its effect on the parameters
of the equivalent circuit equation. An advantage of equivalent circuit modeling is that it is a simple
approach with low computational expense and reasonable accuracy, as long as the equations are
fitted to experimental data. Equivalent circuit modeling of solar cells with consideration of the
effects of temperatures is described in Chapter 6.

1.3.2 Drift-Diffusion Approach

While using equivalent circuit equations allows for a quick fitting of experimental current-voltage
curves and some understanding of the physical parameters, it does not give a deep physical insight
into the mechanisms occurring inside the device. In order to understand the device physics, it is
necessary to perform simulations which explicitly treat the electrodynamics, charge generation,
separation, and recombination processes. The most general equation for transport in semiconduct-
ing devices is the semi-classical Boltzmann transport equation (BTE), which describes the time
evolution of a seven-dimensional particle distribution function f((cid:174)r,(cid:174)t, t) [17].

(cid:18) ∂ f

(cid:19)

+ (cid:174)v · ∇r f + (cid:174)F · ∇p f =

∂ f
∂t

∂t

collisions

(1.5)

The distribution function represents the probability for a particle to occupy position (cid:174)r with mo-
mentum (cid:174)p at time t. By definition, integrating f over all momentums gives the particle density at
position (cid:174)r, and further integrating over all positions gives the total number of particles N in the
system. (Note that this equation is often written in terms of wavenumber (cid:174)k, which is related to
particle momentum through the de Broglie relation (cid:174)p = (cid:174)k).

The collisions term on the right-hand side describes scattering in and out of a momentum state.
The probability of scattering of particles out of a momentum state (cid:174)p to a momentum (cid:174)p(cid:48) is defined
as S((cid:174)p, (cid:174)p(cid:48)). The total number of scattering events from (cid:174)p to (cid:174)p(cid:48) is given by the transition probability
multiplied by the probability distribution function. The net change in f((cid:174)p) due to scattering is the

8

difference between scattering of particles into state (cid:174)p and scattering of particles out of state (cid:174)p. For
each state (cid:174)p, we sum the contributions to scattering from all other states (cid:174)p(cid:48).
, (cid:174)p) − f((cid:174)p)S((cid:174)p, (cid:174)p(cid:48))

=


f((cid:174)p(cid:48))S((cid:174)p(cid:48)

(cid:18) ∂ f

(1.6)

(cid:19)

(cid:21)

(cid:20)

∂t

collisions

(cid:174)p(cid:48)

However, we should correct for the Pauli Exclusion Principle which states that only one particle
can occupy a given state at a given time (ignoring spin). To account for exclusion, the transition
rate probability is multiplied by the probability that the state is not occupied (1 − f ), giving

(1 − f((cid:174)p))S((cid:174)p(cid:48)

, (cid:174)p) − f((cid:174)k)(1 − f((cid:174)p(cid:48)))S((cid:174)p, (cid:174)p(cid:48))

(1.7)

(cid:18) ∂ f

(cid:19)

∂t

collisions

(cid:20)

=


(cid:174)p(cid:48)

(cid:21)

Þ
(cid:174)p(cid:48)[(1 − f((cid:174)p))S((cid:174)p(cid:48)

In the limit of small d (cid:174)p, the sum can be converted to an integral, making the BTE an integro-
differential equation

+ (cid:174)v · ∇r f + (cid:174)F · ∇p f =

∂ f
∂t

, (cid:174)p) − f((cid:174)k)(1 − f((cid:174)p(cid:48)))S((cid:174)p, (cid:174)p(cid:48))]d (cid:174)p(cid:48)

(1.8)

With seven partial derivatives of the distribution function on the left-hand side, and a scattering
integral on the right-hand side, the general BTE is very difficult and computationally expensive to
solve. Simplifications of the BTE can be derived through the Method of Moments, where the BTE
is multiplied by a moment generating function [18]. Moments of different orders can be taken,
with the order of the moment determined by the power of the momentum in the moment generating
function. Every moment of the BTE generates a new transport equation. A moment Mφ is defined
in terms of the moment-generating function φ((cid:174)p) as

Þ

Mφ =

1
V

φ((cid:174)p) f((cid:174)r, (cid:174)p, t)d (cid:174)p

(1.9)

where the 1/V is to cancel out the volume that arises from taking the three-dimensional integral.
The moment-generating functions for the zeroth, first, and second moments are

0 = 1, φ

1 = (cid:174)p, φ

2 = p2

φ

(1.10)

(cid:20) 1

Þ

Taking an nth order moment of the BTE, we have
1
V

+ ∇r · 1
V

(cid:174)v f φnd (cid:174)p +

φn f d (cid:174)p

∂
∂t

V

(cid:21)

Þ

Þ

φn (cid:174)F · ∇p f d (cid:174)p =

Þ

1
V

φnQ( f)d (cid:174)p

(1.11)

9

Þ

(cid:21)

where we used (cid:174)v · ∇r f = ∇r · ((cid:174)v f) and the scattering term of the BTE was denoted as Q( f) for
brevity. The third term of the left-hand side can be simplified by integration by parts

(cid:174)F
V

·

φn · ∇p f dpd2p =

(cid:174)F
V

·

φn f

d2p −

d (cid:174)p

f ∂φ
∂p

(1.12)

(cid:20)Þ

(cid:12)(cid:12)(cid:12)(cid:12)p=∞

p=−∞

The first term on the right-hand side equals zero since the distribution function must go to zero at
infinite momentum. Thus, we have the following equation for the nth moment of the BTE

(cid:21)

Þ

Þ

φn f d (cid:174)p

+ ∇r · 1
V

(cid:174)v f φnd (cid:174)p − F
V

·

d (cid:174)p =

f ∂φ
∂p

1
V

φnQ( f)d (cid:174)p

(1.13)

By taking the zeroth and first moments of the BTE, one can derive the continuity and drift-diffusion
equations (respectively) which are used for drift-diffusion modeling [17, 19].

To derive the continuity equation, we evaluate the first moment of the BTE by substituting φ = 1

Þ

Þ

Þ

(cid:20) 1

V

∂
∂t

into Equation 1.13 which yields

Þ

(cid:20) 1

V

∂
∂t

Þ

Þ

f d (cid:174)p

+ ∇r · 1
V

(cid:174)v f d (cid:174)p =

1
V

Q( f)d (cid:174)p

(1.14)

Note that this is simply taking the mean of the BTE over momentum space. In semiconductors, the
mean effect of collisions results in generation and recombination of charge carriers. Therefore, the
zeroth moment of the collision term is the net carrier density generation rate Uρ. We recognize
the first term on the left-hand side as the time derivative of the density distribution of particles in
space. We recognize the integral in the second term on the left-hand side as a particle density flux

(cid:21)

Þ

(cid:174)v f d (cid:174)p = ρ(cid:174)v ≡ (cid:174)Φρ

(1.15)

(1.16)

Applying these simplifications, we have

= Uρ − ∇r · (cid:174)Φρ

∂ ρ
∂t

This is the continuity equation which describes a general conservation law, where ρ is some quantity
per unit volume, (cid:174)Φρ is the flux of that quantity, and Uρ is the generation rate of that quantity per unit
time per unit volume. In the context of semiconductor device modeling, the continuity equation
is a carrier mass conservation law, stating that the rate of change in carrier density is equal to the

10

net carrier generation rate minus the divergence of the particle flux. Rewriting the particle flux in
terms of an electrical current density divided by the charge of the carriers (where e denotes the
elementary charge), we get the continuity equations for electrons (n) and holes (p) that are used for
semiconductor device modeling

∂p
∂t

= Up − 1
e

∇r · (cid:174)Jp

and

∂n
∂t

= Un +

∇r · (cid:174)Jn

1
e

(1.17)

To derive the drift-diffusion equation, we evaluate the first moment of the BTE by substituting

φ = (cid:174)p = m(cid:174)v into Equation 1.13 and dividing all terms by m

(cid:174)v f d (cid:174)p

+ ∇r · 1
V

(cid:174)v(cid:174)v f d (cid:174)p −

(cid:174)F
mV

·

f d (cid:174)p =

1
V

(cid:174)vQ( f)d (cid:174)p

(1.18)

Þ

(cid:20) 1

V

∂
∂t

(cid:21)

We recognize the integrals in the first term and third terms on the left-hand side as a particle density
flux (cid:174)Φρ and particle density distribution ρ, respectively. Therefore, we have

(cid:174)v(cid:174)v f d (cid:174)p −

(cid:174)F
m ρ =

1
V

(cid:174)vQ( f)d (cid:174)p

(1.19)

The second term can be simplified as follows.

∂ (cid:174)Φρ
∂t

+ ∇r · 1
V
Þ

∇r · 1
V

(cid:174)v(cid:174)v f d (cid:174)p =

Þ

Þ

Þ

Þ

Þ

(cid:21)
(cid:21)

∇r · (cid:174)v(cid:174)v f d (cid:174)p
(cid:174)v(cid:174)v · ∇r f d (cid:174)p
i ∇r f d (cid:174)p
2

(cid:21)

v

(cid:20)Þ
(cid:20)Þ
(cid:20)Þ

=

1
V
1
V
1
V
=(cid:104)v

=

i (cid:105)∇r ρ
2

(1.20)

where we have assumed an isotropic velocity distribution ((cid:104)v
use the equipartition theorem for a single component of velocity

x(cid:105) = (cid:104)v
2

y(cid:105) = (cid:104)v
2

z(cid:105) ≡ (cid:104)v
2

i (cid:105)). Next we
2

1
2mv

2
i =

1
2 kBT

where kB is the Boltzmann constant and T is the temperature. Therefore,

i (cid:105)∇r ρ =
(cid:104)v
2

kBT
m

∇r ρ

11

(1.21)

(1.22)

The right-hand side collision term of Equation 1.19 can be simplified by using the relaxation
time approximation (RTA), where one replaces the scattering integral by ( f 0− f)/τ, where f 0 is the
equilibrium particle distribution, and τ is a time-constant called the relaxation time, which defines
the time it takes the system to relax to its equilibrium state after being perturbed by collisions [17].
Thus we have

d (cid:174)p = ρ

(cid:174)vQ( f)d (cid:174)p =

(cid:174)v

(1.23)
where (cid:104)(cid:174)v(cid:105) represents the average velocity of the ensemble of particles. By definition, the average
velocity of the ensemble at equilibrium is zero, therefore
(cid:104)(cid:174)v(cid:105)
τ

(cid:174)vQ( f)d (cid:174)p = −ρ

(1.24)

= −

(cid:174)Φρ

τ

Þ

τ

( f 0 − f)

τ

(cid:104)(cid:174)v0(cid:105) − (cid:104)(cid:174)v(cid:105)

Þ

Þ

Combining Equations 1.19, 1.22, and 1.24, we have

∂ (cid:174)Φρ
∂t

+

kBT
m

∇r ρ −

(cid:174)F
m ρ = −

(cid:174)Φρ

τ

(1.25)

In the context of semiconductors, the particle density flux can be written in terms of an electrical
current density divided by the elementary charge. For holes (p) and electrons (n), we have

(cid:174)Φp =

(cid:174)Jp
e

(cid:174)Φn = −

(cid:174)Jn
e

(1.26)
The force is due to the electric field so we can use (cid:174)F = e (cid:174)E. Substituting these quantities and
rearranging Equation 1.25, we get

and

∂ (cid:174)Jp
∂t

τ

+ (cid:174)Jp =

e2
τ
m

(cid:174)E p − eτ

kBT
m

∇r p

and

∂ (cid:174)Jn
∂t

τ

+ (cid:174)Jn =

e2
τ
m

(cid:174)En + eτ

kBT
m

∇rn

(1.27)

Now we substitute the definition of carrier mobility (µ = eτ/m) and the Einstein relation for the
diffusion coefficient (D = µkBT/e), yielding

+ (cid:174)Jp = eµp (cid:174)E p − eDp∇r p

and

τ

+ (cid:174)Jn = eµn (cid:174)En + eDn∇rn

(1.28)

∂ (cid:174)Jp
∂t

τ

∂ (cid:174)Jn
∂t

For drift-diffusion modeling, the time derivative of the current density is usually neglected. The
final drift-diffusion equations for holes and electrons are

(cid:174)Jp = epµp (cid:174)E − eDp∇r p

and

12

(cid:174)Jn = enµn (cid:174)E + eDn∇rn

(1.29)

These equations describe the contributions to electrical currents in the device. The first term
describes carrier drift due to the built-in electric field, and the second term describes carrier
diffusion due to the concentration gradient (see Figure 1.1). These equations are usually coupled
with Poisson’s equation to solve for the electric field.

Drift-diffusion modeling has been widely used for semiconductor device simulation due to
being computationally inexpensive and accurate for many situations. For solar cells, drift-diffusion
modeling allows us to understand the carrier dynamics (generation, transport, recombination, and
extraction) as well as simulate practical efficiency by accounting for various recombination losses
in multilayer devices. Recently, there has been work on developing models that are a more accurate
approximation of the BTE by taking higher-order moments (e.g. hydrodynamics and energy
transport equations), including quantum effects, or even directly solving the full BTE [18, 20, 21].
Some devices, such as nanoscale transistors where the carriers can behave ballistically and quantum
effects become important, need these more sophisticated and computationally expensive methods.
However, in most solar cells, the carriers experience enough resistance to make their motion
diffusive. Also, most solar cells have active layers that are thick enough that quantum confinement
effects are not significant. Therefore drift-diffusion modeling, while being one of the simplest
models in the hierarchy of semiconductor device models [19], remains accurate for most PV
devices. Chapters 2-4 of this dissertation are focused on drift-diffusion modeling.

1.4 Photovoltaic System Modeling

One of the main goals of PV system modeling is to predict annual energy production in order to
size a PV system to the energy demand and estimate return-on-investment. Unlike device models,
which are generally focused on device performance under standard laboratory test conditions,
models of PV systems must consider the realistic environmental conditions. For example, solar
modules in hot and sunny locations may have a significantly lower efficiency than when measured in
the lab. Solar modules in locations with less directly normal irradiance will produce less power than
if a global average solar irradiance is assumed. Since weather cannot be reliably predicted a year in

13

advance, such models use what is known as Typical Meteorological Year (TMY) weather data. The
TMY data represents the median weather conditions (based on a set of weather parameters) for a
location for a given historical time frame (30 years is the most common) and can be a combination
of measured and modeled data [22].

Solar irradiance, ambient temperature, and wind speed are the three most important parameters
from the TMY data for PV modeling. Incident irradiance is the main factor that determines energy
production. The combination of incident irradiance, ambient temperature, and wind speed allows
to estimate the module temperatures, and therefore, the associated thermal losses. The system’s
orientation with respect to the sun throughout the day is also important since device efficiency is
higher under directly normal irradiance. Even the impact of particulate matter (e.g. dust, dirt,
pollutants) accumulating on the panels must be considered, as it can lead to significant losses in
energy production. The process of particulate matter accumulating on PV modules is known as
soiling. Losses due to soiling (known as soiling losses) are significant enough in some areas of
the world that PV systems must be regularly cleaned to maintain their profitability. Such cleaning
should be optimized to balance the cost of cleaning with the cost of energy lost due to a certain level
of soiling. The importance of soiling on PV systems has only recently been realized, and the topic
is not yet well understood. Chapter 5 discusses the development of improvements to current soiling
loss estimation techniques. While not the focus of this dissertation, other losses affecting a PV
system’s performance are those due to mismatch (electrical loss due to slight differences between
connected modules), resistance in the wires, shading of solar irradiance (e.g. by nearby buildings,
trees, or hills/mountains), snow, light-induced degradation, and losses in the direct-current to
alternating-current inverter.

For standard PV installations (e.g. free-standing modules or rooftop systems), accounting for
the system’s orientation with respect to the sun is a relatively straightforward trigonometry problem,
and module temperature can be estimated through empirical equations. This is no longer the case
when modeling emerging PV system technologies such as the integration of PV modules or films
onto irregular surfaces such as vehicles. Estimation of incident irradiance and cell temperature for

14

PVs integrated into irregular surfaces is discussed in Chapter 6.

1.5 Outline

The central focus of this dissertation is multi-physics modeling and simulation of PV devices
and systems. The first part (Chapters 2-4) focuses on electrical modeling of PV devices, which can
help our understanding of the physics of new solar cells and aid their development. The second part
(Chapters 5-6) focuses on the modeling of PV systems, such as rooftop modules and utility-scale
power plants.

In Chapter 2, I present an introduction to the numerical methods for solving the drift-diffusion
I also demonstrate that the drift-diffusion results are consistent with the analytical
equations.
In
expression for current in single-carrier devices in the space-charge-limited current regime.
Chapter 3, I describe investigations into the numerical properties of a drift-diffusion model for
bilayer organic solar cells (DD-BI model) that was developed by Non Thongprong and Kanokkorn
Pimcharoen, who are Ph.D. alumni from our research group. Initially, I investigated the numerical
instabilities which were often encountered when running the DD-BI. In order to further study
these stability issues and more effectively prototype improvements, I re-implemented the Fortran
code in Matlab, and then developed and investigated several variations of the iterative scheme,
some of which resulted in improved convergence. After acquiring an understanding of the drift-
diffusion modeling techniques and their numerical properties, I developed a drift-diffusion model for
perovskite solar cells, which are one of the most promising emerging PV technologies. Chapter
4 focuses on drift-diffusion modeling of perovskite solar cells and analysis of the fundamental
physics to explain experimentally measured current-voltage curves. I describe the adaptation of
the drift-diffusion model for perovskites and validation of the model by comparison to literature
results. Finally, I converted the code into C++ for significantly improved performance and applied
the model to help explain a new experimental result where it was found that ultrathin fullerene
layers can have a drastic impact on a particular planar perovskite solar cell architecture. C++,

15

Matlab, and Python examples of the drift-diffusion codes that I developed are available1.

Chapter 5 describes the development of PV energy production forecasting models that consider
environmental conditions and soiling losses. I propose improvements over existing soiling loss
models where several semi-physical models for soiling accumulation and a model for the angular
reflection due to soiling are combined. Additionally, I demonstrate the potential of using a machine
learning approach to soiling modeling.
In a research and development collaboration with PV
startup Enlighten Energy Ltd., the semi-physics soiling models were combined with the PV system
performance models provided by the open-source Python package PVLIB to create PV SoilSayer,
a new commercial web-based software tool. PV SoilSayer predicts a power plant’s yearly energy
production, losses due to soiling, and calculates optimized cleaning schedules. The simulation
results were compared with experimental data collected at large solar power plants. Appendix A
describes PV SoilSayer in more detail.

Chapter 6 describes the development of coupled thermal-electrical PV models by integrating
temperature-dependent equivalent circuit solar cell models with a commercial heat-transfer software
(TAITherm).
I analyze the effects of temperature and irradiance on PV device performance.
Additionally, I describe the development of an improved algorithm for determining equivalent
circuit model parameters. Then I discuss several practical multi-physics PV applications that can
be simulated with the models developed in this dissertation. These include residential roof-mounted
PV systems, vehicle-integrated transparent photovoltaics, and PV-battery systems.

Chapter 7 presents the conclusions of this dissertation and an outlook on potential future

research directions.

1

https://github.com/tgolubev

16

PART I

DRIFT-DIFFUSION MODELING OF SOLAR CELLS

17

CHAPTER 2

IMPLEMENTING DRIFT-DIFFUSION MODELS

2.1 1D Drift-Diffusion Equations

Steady-state one-dimensional modeling along the thickness dimension is often sufficient to
represent a solar cell [17]. In the device, the electric potential (ψ) is due to the free electrons (n)
and free holes (p) and is described by the 1D Poisson equation

2

∂

∂x2 ψ(x) =

[n(x) − p(x)]

q



(2.1)

where q is the elementary charge (always positive) and 
 is the dielectric constant. The above
equation assumes that 
 is constant within each layer of the device. Defining x = 0 as the anode
and x = L as the cathode, the boundary conditions for the electric potential are

ψ(L) − ψ(0) + Vext = Vbi

(2.2)

where Vext is the externally applied bias and Vbi is the built-in potential, which is defined as
(Wc − Wa)/q where Wc and Wa are the cathode and anode work functions respectively.

The current density is separated for electrons and holes (Jn and Jp, respectively) and related to

the net free carrier generation rate, U(x), by the continuity equations

(x) = −qU(x)

∂Jn
∂x

and

(x) = qU(x)

∂Jp
∂x

(2.3)

The equations to find U generally involve charge recombination and dissociation rates and will vary
depending on the solar cell device that is being modeled.

The current densities are also related to the electric potential and charge carrier densities through

the drift-diffusion equations

Jn = −qnµn

∂ψ
∂x

+ qDn

∂n
∂x

Jp = qpµp

∂ψ
∂x

− qDp

∂p
∂x

(2.4)

18

where Dn(p) is the diffusion coefficient and µn(p) is the charge carrier mobility. Usually the Einstein
relation D = µVt is used, where Vt = kBT/q is called the thermal voltage.

The boundary conditions for the charge densities at the anode are

n(0) = Ncexp

and

p(0) = Nvexp

− φa
Vt

where Nc and Nv are the density of states of the conduction and valence bands respectively, Egap
is the active layer bandgap, and φa is the injection barrier at the anode. Similarly, for the cathode

(cid:19)

(cid:18)

(cid:18)

Vt

− Egap − φa
(cid:19)

− φc
Vt

(cid:18)

(cid:19)

(cid:19)

(cid:18)

− Egap − φc

Vt

(2.5)

(2.6)

n(L) = Nvexp

and

p(L) = Ncexp

where φc is the injection barrier at the cathode. The drift-diffusion and current continuity equations
are usually combined.

2.2 Discretization

In order to solve these equations numerically, we need to choose a discretization scheme.
For 1D simulations of solar cells, a square mesh can be used so finite-difference discretization
is appropriate. The Poisson equation is discretized in a straightforward way, using the central-
difference approximation for the second derivative
ψi−1 − 2ψi + ψi+1

(2.7)

= q(ni − pi)


i

(δx)2

where δx is the distance between mesh points. The variable ψ is many orders of magnitudes smaller
than n and p, which can lead to numerical issues such as rounding errors, overflow, and underflow.
This can be avoided by using dimensionless scaled variables ψ(cid:48)
= pi/N
i
where N is of the same order as the density of states of charge carriers. The rescaled Poisson
equation is

= ni/N, and p(cid:48)

= ψ/Vt, n(cid:48)

i

i


r,i(ψi−1 − 2ψi + ψi+1) =

(2.8)
where 
r = 
/
0 is the relative permittivity. For points i that are on the interface of two layers with
different relative permittivity (
r,1 and 
r,2), we change the left-hand side of the equation to

(n(cid:48)
i − p(cid:48)
i)

qN(δx)2

0Vt

i−1 − 2(
r,1 + 
r,2)
(cid:48)

2


r,1ψ

(cid:48)
i + 
r,2ψ
ψ

(cid:48)
i+1

(2.9)

19

To solve for the electric potential at all mesh points, Equation 2.8 is rewritten as a matrix equation




r,2

−2
r,1
0 0

r,1 −2
r,2 
r,3 0
...
...

...

...

· · ·
· · ·

. . .

0

0

...

...

0

...
· · ·

...

0

. . .

...
· · · 0 
r,L−2 −2
r,L−1







ψ(cid:48)
1
...

...

...

ψ(cid:48)
L−1



(n(cid:48)
1 − p(cid:48)

1) − 
r,0ψ(cid:48)
0
...

...

...
L−1) − 
r,Lψ(cid:48)

L

(n(cid:48)
L−1 − p(cid:48)



= CV

(2.10)

where CV = qN(δx)2/(
0Vt). We keep the relative permittivity in the matrix to allow treatement
of devices with differing dielectric constants in different layers. At the interfaces, the elements will
need to be changed according to Equation 2.9. Note that we omit solving for the potential at the
endpoints (i = 0 and i = L) because it is already defined there by the boundary conditions, which
are enforced by moving the terms for the potentials at the endpoints to the right-hand side of the
matrix equation. Notice that the matrix is tridiagonal and therefore can be stored as a sparse matrix
(storing only the non-zero elements) and solved by using a matrix solving technique such as the
straightforward Thomas algorithm or the QR algorithm (available as functions in Matlab and also
in the LAPACK library, which can be used with Fortran or C++). One could also use the theorem
for inverting a tridiagonal matrix derived by R.A. Usmani [23].

For the continuity equations, one needs to approximate the current in between the mesh points
in order to have a result for the derivative of the current on the grid points. Scharfetter and Gummel
found that if one estimates the current between mesh points using a simple central-difference method
with a step size greater than 2Vt, numerical instability will arise due to the exponential dependence of
the charge densities on the electric potential [24]. Since using such a small mesh is computationally
expensive, they introduced a method (now known as Scharfetter-Gummel discretization) that allows
for use of a courser mesh. It reduces numerical instabilities by assuming that Jn, Jp, and ∂ψ/∂x are
constant between mesh points. Then the drift-diffusion equations for n and p between mesh points
become differential equations of one variable (n or p respectively). For example for the electrons,

20

we have

Jn = qµnVt

∂n
∂x

The solution of this first-order differential equation is

− qµnn ∂ψ
∂x

(cid:18) ∂ψ/∂x

Vt

(cid:19)

x

(2.11)

(2.12)

n(x) = − Jn
∂ψ
qµn
∂x

+ b exp

We now use the assumption that n and ∂ψ/∂x are the same at neighboring mesh points to write the
equation for n at neighboring points

ni = −
ni+1 = −

Jn,i+1/2

qµn(∂ψ/∂x)i+1/2
qµn(∂ψ/∂x)i+1/2

Jn,i+1/2

+ b exp

+ b exp

(cid:19)

xi

(cid:18)(∂ψ/∂x)i+1/2
(cid:18)(∂ψ/∂x)i+1/2

Vt

xi+1/2

Vt

(cid:19)

(2.13)

(2.14)

(2.15)

Combining these allows to solve for b

Jn,i+1/2

ni+1 +
qµn(∂ψ/∂x)i+1/2
exp((∂ψ/∂x)i+1/2/Vt)xi+1

b =

(cid:20)

(cid:18) δψi+1

(cid:19)

Vt

(cid:18)

(cid:19)

(cid:21)

Substituting back into Equation 2.13, we solve for Jn,i+1/2, and simplifying we get the Scharfetter-
Gummel discretized expression for current

qµn,i+1

δx

B

ni+1 − B

− δψi+1
Vt

ni

Jn,i+1/2 =

(2.16)
where δψi+1 ≡ ψi+1 − ψi and B is the Bernoulli function B(x) = x/(ex − 1). Rewriting in terms of
the dimensionless scaled variables that were introduced earlier (ψ(cid:48)
= ni/N), we have
i
i+1 − B(−δψ
(2.17)

= ψ/Vt, n(cid:48)
(cid:48)
i+1)n(cid:48)
i]
The discretization proceeds in an analogous way for Jp,i+1/2, resulting in
i+1)p(cid:48)
(cid:48)
i]

Jp,i+1/2 = −qVt N µp,i+1

i+1 − B(δψ

qVt N µn,i+1

Jn,i+1/2 =

[B(δψ

(cid:48)
i+1)n(cid:48)

[B(−δψ

i+1)p(cid:48)
(cid:48)

(2.18)

δx

i

δx

Then, the first order finite-difference approximation can be used to rewrite the continuity equations
as

∂Jn,i
∂x
∂Jp,i
∂x

=

=

Jn,i+1/2 − Jn,i−1/2
Jp,i+1/2 − Jp,i−1/2

δx

δx

21

= −qUn,i

= qUp,i

(2.19)

(2.20)

Finally, the Scharfetter-Gummel discretized expressions for current (Equations 2.17 and 2.18) are
substituted into the above two equations to give

µn,i+1B(δψ

i+1)n(cid:48)
(cid:48)

i+1 − [µn,i+1B(−δψ

(cid:48)
i+1) + µn,iB(δψ

i)]n(cid:48)
(cid:48)

i + µn,iB(−δψ

i−1 = −(δx)2
i)n(cid:48)
(cid:48)

Vt N

Un,i

(2.21)

µp,i+1B(−δψ

(cid:48)
i+1)p(cid:48)

i+1 − [µp,i+1B(δψ

(cid:48)
i+1) − µp,iB(−δψ

i)]p(cid:48)
(cid:48)

i + µp,iB(δψ

i)p(cid:48)
(cid:48)

i−1 =

(δx)2
Vt N

Up,i

(2.22)

These equations are used to solve for carrier densities at all mesh points by rewriting them as matrix
equations, which are analogous to the matrix form of the Poisson equation.

2.3 Iteration Schemes

Two main types of iteration schemes are used to self-consistently solve the 1D drift-diffusion
equations [17]. One method is the Gummel iteration, where the equations are decoupled and
solved separately as three sets of matrix equations: one for the Poisson equation, and two for the
drift-diffusion/continuity equations (one for electrons and one for holes) [25]. The other is the
Newton-Raphson method (also known as simply Newton’s method), which is a general technique
used to solve non-linear systems of equations where all of the equations are solved simultaneously
using a single matrix.

2.3.1 Gummel Method

Flow charts of two general Gummel iteration schemes are shown in Figure 2.1. First, initial guesses
for the carrier densities and potential are entered and the discretized Poisson equation is solved.
There are two possible approaches: one is to solve for a correction δψ to the potential (Figure 2.1a);
the other is to solve for the new potential directly (Figure 2.1b). In this study, it was found that
the first approach does not give any advantage to stability and requires more CPU time due to the
additional loop for the Poisson equation. Therefore, the later approach was used in this work.

22

Figure 2.1: Two Gummel iteration approaches. Approach b) is used in this study.

At each iteration, Equation 2.10 is solved first to compute the new value for the electric potential
(ψ(cid:48)). Once ψ(cid:48) is found, it is used in the continuity/drift-diffusion equations (Equations 2.21 and
2.22) formulated as tridiagonal matrix equations. These equations are solved separately for n and
p, which are then substituted back into the Poisson equation for the next iteration. The iterations
are repeated until the change in n and p between two consecutive iterations (old and new) become
less than a set tolerance (e.g. 10−12). The convergence condition can be written as

(cid:18)|nnew,i − nold,i| + |pnew,i − pold,i|

(cid:19)

max

nold,i + pold,i

< tolerance

(2.23)

Depending on the specific problem, Gummel’s method as described above will not always converge.
The convergence can be improved by using a weighted average of the solutions from the last two
iterations for the initial guess of the following iteration. This technique is called mixing or under-
relaxation and has the effect of decreasing the possibility of solution estimates overshooting the
true solution, thus preventing divergence or oscillations in the solution. The simplest mixing is the

23

linear one, given by

nnew = wnnew + (1 − w)nold
pnew = wpnew + (1 − w)pold
Vnew = wVnew + (1 − w)Vold

(2.24)

(2.25)

(2.26)

where w is the mixing factor. Note that the method will also work without mixing of the voltage
values but requires more iterations to converge and lower mixing factors.

The Gummel method works well for initial guesses that are far from the true solution, and
saves computational power by decoupling the equations. Therefore, this scheme has become very
popular for drift-diffusion models. However, convergence is problematic at higher voltages. We
find that as the applied bias is increased, we must decrease w, in order to maintain convergence.
When Va approaches a solar cell’s built-in voltage, the Gummel scheme no longer converges, even
when very small w values are used (e.g. 10−5). This convergence issue at high biases is known in
the literature and is attributed to the strong coupling of the equations at high voltage [17]. If one
wants to study the devices at high bias, the suggested solution is to use the fully coupled Newton’s
method described in the next section.

2.3.2 Newton-Raphson Method

If we have a function
Newton’s method is based on first-order multivariate Taylor expansion.
f(x1, x2, ..., xn) ≡ f((cid:174)x) = 0, let (cid:174)xm be an approximation to the exact solution (cid:174)x, and define
(cid:174)h = (cid:174)x − (cid:174)xm (the error between the approximation and the exact solution), we can write [26].

f((cid:174)x) = 0 = f((cid:174)xm + (cid:174)h) = f((cid:174)xm) +

24

n


j=1

hj

∂ f
∂xj

((cid:174)xm) + ...

(2.27)

If we have a system of n coupled equations that are all dependent on the same set of n variables,
we can write an equivalent expression in matrix form





f1((cid:174)xm)

...

fn((cid:174)xm)



0 =

+

∂ f1

∂x1((cid:174)xm)· · · · · ·

∂ f1
∂xn

...

∂ fn

∂x1((cid:174)xm)· · · · · ·

∂ fn
∂xn

((cid:174)xm)

...
((cid:174)xm)







h1

...

hn

(2.28)

Fp,i ≡ µp,iB(δψ

(cid:48)
i)p(cid:48)

i−1 − [µp,iB(−δψ

(cid:48)
i) + µp,i+1B(δψ

... + µp,i+1B(−δψ

(cid:48)
i+1)p(cid:48)

where C = (δx)2/(Vt N).

25

We call the matrix of partial derivatives the Jacobian (J) and can write the equation as J(cid:174)hm =
− (cid:174)f((cid:174)xm). Since f((cid:174)xexact) = 0, the right-hand side of the matrix equation is the error between the
approximate and exact solution. An m + 1th iteration of Newton’s method proceeds as follows.
First, we compute the Jacobian matrix elements using the solution from the mth iteration, (cid:174)xm. Then,
we solve the matrix equation for the update vector (cid:174)hm. Finally, we update the solution by simply
adding the update vector to the mth solution: (cid:174)xm+1 = (cid:174)xm + (cid:174)hm. The iterations are repeated until
the maximum error (cid:174)f((cid:174)xm) is less than a desired tolerance condition.

For the drift-diffusion model, we will have a system of 3L coupled equations, where n is
the number of interior mesh points. The equations can be grouped as L Poisson equations and L
continuity equations for both electrons and holes. The only difference from the equations in Section
2.2 is that we must rearrange them so that one side equals zero as follows
i + pT(cid:48)

(cid:48)
i+1 − CV[(n(cid:48)

(cid:48)
i−1 − 2ψ

i + nT(cid:48)

i ) − (p(cid:48)

FV,i ≡ ψ

i )] = 0

(cid:48)
i + ψ

Fn,i ≡ µn,iB(−δψ

(cid:48)
i)n(cid:48)

i−1 − [µn,iB(δψ

(cid:48)
i) + µn,i+1B(−δψ

... + µn,i+1B(δψ

i+1)n(cid:48)
(cid:48)

(2.29)

(cid:48)
i+1)]n(cid:48)

i + ...
i+1 + CUn,i = 0

i+1)]p(cid:48)
(cid:48)

i + ...
i+1 − CUp,i = 0



The Newton’s method matrix equation (Equation 2.28) becomes

∂FV,1
∂ψ(cid:48)
1
...

∂FV,L
∂ψ(cid:48)
1
∂Fn,1
∂ψ(cid:48)
1
...

∂Fn,L
∂ψ(cid:48)
1
∂Fp,1
∂ψ(cid:48)
1
...

∂Fp,L
∂ψ(cid:48)
1

· · ·

. . .
· · ·

· · ·

. . .
· · ·

· · ·

. . .
· · ·

∂FV,1
∂ψ(cid:48)
L
...

∂FV,L
∂ψ(cid:48)
L
∂Fn,1
∂ψ(cid:48)
L
...

∂Fn,L
∂ψ(cid:48)L
∂Fp,1
∂ψ(cid:48)
L
...

∂Fp,L
∂ψ(cid:48)
L

∂FV,1
∂n(cid:48)
1
...

∂FV,L
∂n(cid:48)
1
∂Fn,1
∂n(cid:48)
1
...

∂Fn,L
∂n(cid:48)
1
∂Fp,1
∂n(cid:48)
1
...

∂Fp,L
∂n(cid:48)
1

· · ·

. . .
· · ·

· · ·

. . .
· · ·

· · ·

. . .
· · ·

∂FV,1
∂n(cid:48)
L
...

∂FV,L
∂n(cid:48)
L
∂Fn,1
∂n(cid:48)
L
...

∂Fn,L
∂n(cid:48)
L
∂Fp,1
∂n(cid:48)
L
...

∂Fp,L
∂n(cid:48)
L

∂FV,1
∂p(cid:48)
1
...

∂FV,L
∂p(cid:48)
1
∂Fn,1
∂p(cid:48)
1
...

∂Fn,L
∂p(cid:48)
1
∂Fp,1
∂p(cid:48)
1
...

∂Fp,L
∂p(cid:48)
1

· · ·

. . .
· · ·

· · ·

. . .
· · ·

· · ·

. . .
· · ·

∂FV,1
∂p(cid:48)
L
...

∂FV,L
∂p(cid:48)
L
∂Fn,1
∂p(cid:48)
L
...

∂Fn,L
∂p(cid:48)
L
∂Fp,1
∂p(cid:48)
L
...

∂Fp,L
∂p(cid:48)
L



∆ψ(cid:48)
1
...
∆ψ(cid:48)
L
∆n(cid:48)
1
...
∆n(cid:48)
L
∆p(cid:48)
1
...
∆p(cid:48)
L





FV,1
...

FV,L

Fn,1
...

Fn,L

Fp,1
...

Fp,L



= −

(2.30)

where Fv, Fn, and Fp label the Poisson, electron continuity, and hole continuity equations respec-
tively. Many of these derivatives equal zero because the equations for the ith mesh point depend
only on the other variables at the same mesh point and nearest neighbor mesh points (e.g. FV,2
has no dependence on n20). Therefore, the Jacobian consists of tridiagonal blocks. For example,
∂Fv/∂ψ(cid:48) results in the Laplacian operator, which can be written in matrix form as

(2.31)



0

0

...

0



−2

1

...

...

0

1

−2

...

...
· · ·

0

1

...

...
· · ·



0 · · · · · · · · · · · ·
· · · · · · · · ·

0

...

...

· · ·

1

−2

...

...

0

26

so this will be the upper-left block of the Jacobian. In order to minimize numerical error, where
possible, the derivatives should be taken analytically to yield equations that can be evaluated at

each iteration’s solution values. In cases where an analytic expression for the derivative cannot be
found, numerical derivatives using central-differences are used as follows

∂F
∂xi

=

F(x1, ..., xi + 
, ..., xn) − F(x + 1, ...., xi − 
, ..., xn)

2


(2.32)

where 
 is a small perturbation to the value of xi.

Each Newton iteration is more computationally expensive than a Gummel iteration since here
we solve a 3n × 3n matrix instead of an n × n matrix. Because Newton’s method requires initial
guesses relatively close to the real solution, one can use Gummel’s method first to establish a better
initial guess for input into Newton’s method. A flow chart of the main elements of a Newton
iteration for solar cell simulation is shown in Figure 2.2.

Figure 2.2: Flow chart of the main steps in the Newton-Raphson method.

For the first Newton iteration, at equilibrium conditions, we use the solutions from Gummel’s
method. First, we calculate the elements of the Jacobian using values of the three functions FV,
Fn, and Fp from the previous iteration. Next, we solve the matrix equation (Equation 2.30) for
the update vector, which we use to update the solutions for ψ, n, and p. Next, we update the net
generation rate. Finally, we recalculate the values of FV, Fn, and Fp and check for convergence.

27

Since these functions should equal zero at the exact solution, the convergence condition is that the
values of the functions are less than a tolerance (e.g. 10−12).

2.4 Model Verification

2.4.1 Single-Carrier Device in Space-Charge-Limited Current Regime

In order to verify the behavior of our implementations of the drift-diffusion model, the results for
a simulation of a single-carrier device in the space-charge-limited current (SCLC) regime were
compared with known analytical equations. For these simulations, the influence of carrier traps
was neglected. Additionally, since there is only one carrier and traps are ignored, there will be
no bimolecular or trap-assisted recombination. Thus, the recombination rate was set to zero in
these simulations. An analytical expression for the SCLC of a trap-free single-carrier device with
an Ohmic contact, negligible diffusion current, and no photogeneration can be derived from the
Poisson-drift-diffusion equations. We will next outline the derivation for a hole-only device (an
analogous derivation can be performed for an electron-only device) [27]. The Poisson equation is

2

∂

∂x2 ψ(x) = −q




p(x)

(2.33)

Since diffusion current is negligible in the SCLC regime (also known as the drift regime), the total
hole current is determined solely by the drift contribution

Rearranging the Poisson equation in terms of p and substituting into the current equation, we have

J = qµp ∂ψ
∂x

(2.34)

J = qµ(− ∂

2

ψ

∂x2 ) 


q

∂ψ
∂x

We can rewrite the above equation in terms of the electric field (E = −dψ/dx)

or

J = µ
E ∂E

∂x

=

µ

2

∂(E2)
∂x

∂(E2)
∂x

=

2J

 µ

28

(2.35)

(2.36)

(2.37)

The general solution is

[E(x)]2 = ax + b

(2.38)

Since there is no injection barrier due to the Ohmic contact, the electric field must be zero at x = 0.
Therefore,

E(x) =

√

ax1/2

The potential across the device is given by the integral of the electric field

Þ L

0

ψ(x) = −

E(x)dx = −2
3

√

ax3/2 + C

(2.39)

(2.40)

(2.43)

The potential at x = 0 must be 0 due to the Ohmic contact. The potential at x = L is due to the
externally applied voltage, Va. Using these boundary conditions, one can find that C = 0 and

a =

9
8

V2
a
L3

Therefore, we have arrived at the Mott-Gurney equation
V2
a
L3

9
8 
 µ

J =

(2.41)

(2.42)

Note that the same equation applies for electron current in an electron-only device. The Mott-
Gurney equation represents the maximal possible current in a semiconductor device and is often
used to estimate the carrier mobility of semiconductor materials by fitting the equation to measured
JV curves [28]. Combining Equations 2.39 and 2.41, the electric field is

(cid:115)2J x

µ


E =

Figure 2.3 shows a comparison of the current density and electric field results of the drift-diffusion
simulation of a device in the SCLC regime and the analytical Mott-Gurney equations. We see
excellent agreement between the simulation and the analytical expressions in this limit, therefore
verifying the correctness of our implementation of the drift-diffusion model. Note that the model
was simply set-up to ensure SCLC conditions, and no attempt was made to model a real material.
The modeling of real devices is described in the next two chapters.

29

Figure 2.3: Comparison of results from drift-diffusion model of an arbitrary single-carrier device
under SCLC conditions with Mott-Gurney analytical equations. a) Current density versus voltage.
b) The electric field distribution through the thickness of a device under high voltage.

Figure 2.4 shows the effect of injection barriers (φa), which cause the JV behavior to diverge
from the SCLC regime. This demonstrates that the approach of fitting JV curves to determine
material mobilities will yield inaccurate mobility values if there is a significant injection barrier.
(The challenges of determining carrier mobility using analytical expressions for JV curves are
discussed in [28]).

30

020406080100Voltage (V)102104106Current Density (A/m2)Drift-DiffusionMott-Gurneya)00.20.40.60.81Position (m)10-7051015Electric Field (V/m)108Va =100 VDrift-DiffusionMott-Gurneyb)Figure 2.4: Drift-diffusion (DD) model results for JV curves of a hole-only device with varying
sizes of injection barriers compared with the Mott-Gurney equation.

2.4.2 Two-Carrier Device

A two-carrier (also known as bipolar) device supports the transport of both electrons and holes. In
a trap-free device, the recombination rate and the locations where it occurs are determined by the
probability of electrons and holes finding each other, which is influenced by the carrier mobilities.
Figure 2.5(a-d) shows the results of a drift-diffusion simulation of a bipolar device with balanced
hole and electron mobilities (µp/µn = 1) when there is no illumination. In this device, the balanced
mobilities cause the carrier densities, current densities, electric field, and recombination rate to
be distributed symmetrically across the thickness of the device. This symmetry is broken if the
mobilities are unbalanced, as shown in Figure 2.5(e-i), where µp/µn = 100. The higher hole
mobilities result in higher carrier density and current for holes than for electrons throughout most
of the device. There is a build-up of electrons near the electron-injecting contact due to their
low mobility. This causes a higher electric field and a higher recombination rate near the electron
contact. These simulation results are qualitatively consistent with previously published results [29].

31

020406080100Voltage (V)10-5100105Current Density (mA/cm2)Mott-GurneyDD with a = 0.0 eVDD with a = 0.2 eVDD with a = 0.4 eVDD with a = 0.6 eVFigure 2.5: Results from drift-diffusion models of single-layer, two-carrier devices: one with
uniform mobilities (a-d) and another with 100x larger hole mobility than electron mobility (e-i).

32

2.5 Conclusions

In this chapter, we presented the main numerical methods for solving the 1D drift-diffusion
equations. The primary aspect of these methods is to solve the coupled Poisson, drift-diffusion,
and continuity equations iteratively until the carrier density profiles converge. We presented two
iteration schemes: Gummel’s and Newton’s method. We also described verification of our imple-
mentation of the drift-diffusion model by comparing results from single-carrier device simulations
to the Mott-Gurney law. Additionally, we presented example results from simulations of single-
layer, two-carrier devices. The two-carrier device results are found to be consistent with our
expectations based on an understanding of the device physics. Example Matlab, C++, and Python
implementations of the drift-diffusion model are available at https://github.com/tgolubev.
While the focus of this work is on solar cells, the drift-diffusion model can be applied to a variety
of semiconductor devices, including light-emitting diodes (LEDs) and transistors.
In the next
chapter, we investigate the sometimes problematic convergence behavior of drift-diffusion models
and compare the performance of Gummel’s and Newton’s iterative approaches.

33

CHAPTER 3

IMPROVING CONVERGENCE OF A DRIFT-DIFFUSION MODEL FOR BILAYER

ORGANIC SOLAR CELLS

The general drift-diffusion model needs to be modified in order to properly describe the physics in
various thin-film solar cells. Here we present an example of this modification for a bilayer organic
solar cell (OSC). A bilayer (also known as planar heterojunction) solar cell consists of a donor
and acceptor material, charge transport layers, and the contacts (Figure 3.1a). When photons are
absorbed by the donor material, this produces excitons (bound excited electron-hole pairs). These
excitons can either recombine or dissociate to create free charge carriers, which cause a current in
the device. Exciton dissociation is usually described as a two-step process [30]. First, these excitons
can diffuse to the donor-acceptor interface where there is a high field due to the energy level offsets
between the donor and acceptor materials [31]. At the interface, the field causes the electrons to
be transferred from the donor’s lowest unoccupied molecular orbital (LUMO) to the acceptor’s
LUMO, while the holes remain in the donor’s highest occupied molecular orbital (HOMO) (Figure
3.1b). These paired charges at the interface are called polaron pairs (PPs) or geminate pairs or
charge transfer states. The second step of dissociation is when the charges migrate away from each
other due to the built-in field in the device to form free charge carriers. The newly-generated free
charge carriers will flow towards the electrodes due to the built-in field. There is also a chance that
the PPs recombine since there is a significant Coulomb attraction between the electron and hole in a
PP. The competition between recombination and dissociation of PPs is described by rate equations.

34

Figure 3.1: Basic architecture (a) and energy level alignment (b) of a bilayer solar cell. Adapted
from [32].

3.1 DD-BI Model

Two former Ph.D. students in our group, Kanokkorn Pimcharoen and Non Thongprong, devel-
oped a 1D drift-diffusion bilayer (DD-BI) model [16, 27] based on the models by Anton Koster
[33] and Giebnik, Wiederrecht, Wasielewski, and Forrest (GWWF) [15]. The DD-BI model uses
the Gummel method (Figure 2.1b) to solve the Poisson-drift-diffusion system of equations with an
added step of recalculating the generation and recombination rates in each iteration.

For simplicity, the mesh size is chosen to be 1 nm, which matches the average separation
between electron and hole in PPs at the interface and defines the interface width a0. The mesh
endpoints are defined to be at 0 and L, and the interface is defined to be at mesh points l and l + 1.
Boundary conditions for the potential (ψ) and charge densities (n and p) are

p0 = NHOMO

nL = NLUMO

pL = NHOMO · exp
n0 = NLUMO · exp

(cid:18)
(cid:18)

ψL − φ0 = Vbi − Va
− ELUMO,A − EHOMO,A
− ELUMO,D − EHOMO,D

Vt

Vt

(cid:19)
(cid:19)

(3.1)

(3.2)

(3.3)

where Vbi is the built-in field (difference between the cathode and anode work functions), NHOMO
and NLUMO is the density of states of the HOMO and LUMO levels, and E is the energy level (in
eV) of the molecular orbitals, with the subscripts A and D denoting the acceptor and donor levels
respectively. The smallest value from the boundary conditions is used for the initial values of n and

35

p everywhere inside the device. A linear function from ψ0 to ψL is used for the initial potential.
The discretization and solution of the Poisson and drift-diffusion equations is done as described in
Sections 2.2 and 2.3.

We also need to consider the energy barriers for carriers crossing the donor-acceptor interface.
The energy barriers are simply equal to the energy offsets between the donor and acceptor conduction
bands (for electrons) and valence bands (for holes). This can be done by adding the energy barrier
height (in dimensionless units) to the numerical derivative of the potential at the interface (δψ(cid:48)
l+1)
inside of the Bernoulli functions in the continuity equations [34]. Note that since the energy barriers
for electrons and holes are different, the δψ(cid:48) for holes and electrons in Equations 2.17-2.18 will
be different at l + 1. At applied bias greater than the open-circuit voltage, current flows towards
the interface. Since the charge carriers cannot cross the barrier, this leads to an accumulation of
carriers of opposite charge at the opposing sides of the interface, which results in both a very high
field and a high recombination rate (due to high carrier densities).

Figure 3.2: Carrier densities at short-circuit (a) and open-circuit (b) conditions for the bilayer
device.

Figure 3.2 shows the carrier densities inside the device at short-circuit and open-circuit condi-
tions. The densities at open-circuit conditions increase near the interface due to charge accumu-
lation. There is a larger drop in hole densities than electron densities over the interface since, for
this simulation, the energy barrier for holes (difference between the HOMO levels of donor and

36

102030405060701011051010101510201025 p n(a)0102030405060701011051010101510201025 p n(b)acceptor material) was set to be greater than the barrier for electrons (difference between LUMO
levels).

The treatment of net generation rate must consider the effects of the bilayer interface. GWWF
describe the possible competing processes that can occur at the interface (Figure 3.3) and use
detailed balance to derive an expression for the total PP density at the interface (ζ). An exciton
current density Jx/a0 flows to the interface, where a0 is the separation between the electron and
hole in a PP. At the interface, the excitons become PPs where they can either dissociate into free
charge carriers with the rate kPPd ζ or recombine back into the ground state with the rate kPPr ζ.
The free charge carriers can recombine back into PPs at the rate R. Also the PPs can be regenerated
from the ground state at the rate kPPr ζeq where ζeq is the PP density in thermal equilibrium (in the
dark). Writing a detailed balance equation for these processes, we have

Solving for ζ, yields

− kPPr(ζ − ζeq) − kPPd ζ + R = 0

Jx
a0

J x/a0 + kPPr ζeq + R

kPPd + kPPr

ζ =

(3.4)

(3.5)

Once we calculate ζ, we can define the net carrier generation rate of electrons (Un) and holes (Up)
at the interface as the dissociation rate minus the free carrier recombination rate

Un = Up = kPPd ζ − R

(3.6)

Figure 3.3: Recombination and generation processes at the bilayer interface. Adapted from [15].

37

In bilayer OSCs, the exciton binding energy is strong; and therefore, we can assume that the
thermal dissociation rate of free charge carriers is very small and can be neglected. This means that
free carriers are only generated at the interface where there is a strong field that separates the PPs.
In our discretization, we define the interface as the region from mesh point l to mesh point l + 1.
Therefore, our arrays Un(xi) ≡ Un,i and Up(xi) ≡ Up,i, which we use in the continuity equations,
only have one non-zero element in each: the element corresponding to mesh point l for the holes
(donor side) and the element corresponding to mesh point l + 1 for the electrons (acceptor side).
In order to calculate Un(p), we must have expressions for the various rates in Equation 3.5. In a
bilayer OSC, the PP dissociation rate is dependent on the electric field at the interface and can be
described by the Onsager-Braun equations [35]

kPPd =

3krec
4πa3
0

√−2b)
B1(2
√−2b

e−Eb/kBT,

where

b =

q3FI
8π
 k2
BT2

(3.7)

where B1 is the first-order Bessel function of the first kind, FI = (ψl+1 − ψl)/a0 is the field at the
interface, and Eb = q2/(4π
a0) is the PP binding energy. Note that we define positive field as
the field that enhances the dissociation of PPs. krec is the Langevin bimolecular recombination
coefficient and is given by

krec =

q(µn + µp)




(cid:34)

(cid:35)

(3.8)

(3.9)

where µn and µp are the electron and hole mobilities respectively. For purposes of numerical
calculation, the Bessel function is expanded up to the fourth order and Equation 3.7 becomes

kPPd =

e−Eb/kBT

3krec
4πa3
0

1 + b +

b2
3 +

b3
18 +

b4
180

The GWWF model improved on these equations by considering the case when the field acts as
a barrier to dissociation. When the field points towards the interface, it suppresses the dissociation,
and when the field points away from the interface, it enhances the dissociation. This can be
described by adding the additional energy FIrccosθ to Eb, where rc = q2/(4π
 kBT) is the Onsager
radius and θ is the angle between the vector connecting the PP and the vector normal to the interface.
Then the dissociation rate expression is changed by adding this additional energy and integrating

38

over all possible θ values

kPPd =

Þ π/2

−π/2

1
π

3krec
4πa3
0

exp

(cid:18)−Eb + FIrccosθ

Vt

(cid:19)

dθ

(3.10)

The integration is only over a semi-circle because, at the interface, the holes will always be on the
donor side and electrons on the acceptor side. Note that this equation is only used when the field
points in a direction that acts as a barrier to the dissociation. Otherwise, the original Onsager-Braun
equation (3.7) is used.

For the recombination of PPs, we use the empirical recombination coefficient equation from

paper II of GWWF [36]

kPPr = kPPr,0exp

where kPPr,0 is the coefficient in zero field.

(cid:18)−qFI d

(cid:19)

kBT

(3.11)

Several models for free carrier recombination can be used. A widely-used model is Shockley-

Read-Hall (SRH), which in its simplest form is

RSRH =

CnCpNtrap(np − n1p1)
Cn(n + n1) + Cp(p + p1)

(3.12)

where n1 is the number of electrons in the LUMO band when the electron’s quasi-Fermi energy
equals the trap energy Et, and p1 is the number of holes in the HOMO band when the hole’s
quasi-Fermi energy equals Et. n1 and p1 are given by

n1 = NLUMOexp

− ELUMO − Et

kBT

and

p1 = NHOMOexp

− Et − EHOMO

kBT

(3.13)

(cid:18)

(cid:18)

(cid:19)

(cid:19)

The quasi-Fermi level is the effective Fermi level for a charge carrier when a semiconductor is not
in thermal equilibrium, but the electron population is in equilibrium within the conduction band
and the hole population is in equilibrium in the valence band. The "regular" Fermi level is defined
as the HOMO level at the temperature of 0 K.

The net carrier generation rates are updated at each iteration as follows. The new voltage (from
the solution of the Poisson equation in the case of Gummel’s method or solution of the entire system
in the case of Newton’s method) is used to find the new field at the interface FI. The field is then

39

used to recalculate the PP recombination and dissociation rates, kPPr and kPPd. The free carrier
recombination rate, R, is also calculated. Finally ζ and Un = Up are calculated using Equations
3.5 and 3.6. The Un and Up are used in the continuity equations.

3.2 Convergence Improvements

In the Fortran implementation of the DD-BI model by Pimcharoen and Thongprong, the solu-
tions for charge densities oscillate from iteration to iteration. The calculation is unable to converge
unless mixing (Equation 2.24) is used. With higher applied voltages, progressively smaller mixing
factors are required in order to achieve convergence. This is implemented by dividing the mixing
factor by two every time the convergence of the Gummel iterations stagnates. Figure 3.4a shows
the behavior of a converging solution for carrier density and Figure 3.4b shows the behavior of a
solution where the mixing factor is insufficient to achieve convergence. At voltages starting from
slightly above the built-in voltage, the algorithm cannot converge at all even with mixing factors
as small as 10−5. Another issue related to the instabilities is that at certain voltages, due to very
strong oscillations, negative carrier densities are computed.

Figure 3.4: The hole density at a point close to the interface inside the device (which exact point
does not matter) is plotted verses the number of iterations of the Gummel loop. a) A converging
solution, which occurs when the mixing factor is sufficiently small for the applied voltage. b) A
solution that cannot converge.

In order to further study these convergence issues, in this work, a Matlab version of the

40

010203040Iterations1.161.181.21.221.241.261.28p(x=43 nm) m-3 10-5Convergent solutiona)0100200300400500600Iterations3.13.153.23.25p(x=43 nm) m-310-5Non-convergent solutionb)DD-BI model was implemented. This allowed to take advantage of Matlab’s intrinsic matrix-
solving functions and confirm that the convergence issues are not due to a mistake in the Fortran
implementation. A few improvements to the Gummel iterative method were made and also a
version using the Newton-Raphson iterative method was implemented. An improvement to the
treatment of negative charge densities is to set each negative element of nnew and pnew to zero.
Note that when mixing is applied, the densities after mixing will no longer be zero. This allows for
the computation to continue without reducing the mixing factor and repeating the calculations each
time there are negatives carrier densities. However, at higher voltages, the convergence stagnates,
and the mixing factor still needs to be reduced.

3.2.1 Convergence Acceleration through Pulay Mixing

Convergence can be accelerated by applying the Pulay mixing method (also known as direct
inversion in the iterative subspace), which is widely used for quantum chemistry calculations [37].
Pulay mixing uses a weighted average of several of the previous iterations as the trial solution for
the next iteration. Let (cid:174)pi be the solution at the ith iteration. We can approximate the solution at
iteration m + 1 as a linear combination of the previous trial solutions

(cid:174)pm+1 =

ci (cid:174)pi

(3.14)

Now we expand each trial solution as the exact solution (cid:174)pexact plus the error between the trial and
exact solution (cid:174)ei, so (cid:174)pi = (cid:174)pexact + (cid:174)ei. Substituting into Equation 3.14, we have

(cid:174)pm+1 = (cid:174)pexact

ci +

ci(cid:174)ei

m


Therefore, in order for (cid:174)pm+1 to be close to the exact solution, we need

ci = 1

In other words, we would like to minimize
m
L = | m


i=1

and

ci(cid:174)ei = 0

i=1 ci(cid:174)ei under the constraint
m
ci(cid:174)ei|2 − λ(cid:0) m


ci − 1(cid:1)

done using the Lagrange multiplier method with the Lagrangian

i=1

(3.15)

(3.16)

(3.17)

i=1 ci = 1. This can be

m


i=1

m


i=1

m

m


i=1

i

i

41

where λ is the multiplier. Taking the partial derivatives with respect to the coefficients ci and λ and
setting them to zero results in m + 1 equations, which can be written as a matrix equation





c1

...

cm

λ



=





0

...

0

−1



B11 B12

...

...

Bm1 Bm2
−1
−1

...

· · · · · · B1m −1
...
. . .
· · · · · · Bmm −1
· · · · · · −1
0

(3.18)

where Bi j =< (cid:174)ei|(cid:174)ej > is the scalar product of the error vectors. Now since we do not know the
actual error from exact solution, we approximate it as the change in the trial solution between
successive iterations (cid:174)ei = (cid:174)pi − (cid:174)pi−1. Solving the above matrix equation for the ci coefficients, we
i ci (cid:174)pi. In practice, only a few previous

then define the solution at the next iteration as pm+1 =
m

iterations are used in the sum.

With Pulay mixing, we found that the Gummel scheme requires about 20% fewer iterations to
converge than with linear mixing. However, the mixing factors still need to be reduced at higher
voltage, and the voltage at which the program no longer converges is the same as when linear
mixing is used.

3.2.2 Newton vs. Gummel Iterative Methods

Literature claims that the reason for the observed convergence difficulties in drift-diffusion models
at higher voltages is due to use of the Gummel method, which decouples the equations [25]. At
high voltages, the equations are strongly coupled. For example, a small change in the voltage
significantly changes the net generation rates and currents. In the literature, it is recommended to
use Newton’s method for cases of high applied bias [17]. Here, Newton’s method for drift-diffusion
was implemented as described in Section 2.3.2 and the convergence properties of the method were
investigated. Analytic expressions were used for all of the derivatives in the Jacobian (Equation
2.30), except for the derivatives of the net generation rates, which were found numerically by
Equation 2.32.

42

For Newton’s method, a damping (also known as relaxation) factor needs to be used to achieve
convergence. The damping factor is simply a number less than one that scales the correction
vector in a Newton iteration to prevent oscillating solutions, which can happen when the correction
overshoots the value of the true solution. This is analogous to the mixing factors that we used for
Gummel’s method. For the P3HT:PCBM bilayer device we studied, the Voc is around 0.69 V, so
voltages above this are in a region where the device no longer produces any power. Above 0.69 V,
the current now flows into the interface, causing a large build-up of charge. This causes a highly
negative recombination rate at the interface, which rapidly becomes more negative with higher
voltage. At this point, the device is approaching the breakdown limit, with the current densities at
1.0 V becoming 1000 A/m2. Under these conditions, a physical device would most likely either
overheat or the built-up electrons and holes at the interface would have enough energy to overcome
the energetic step and spill over into the other half of the device.

Figure 3.5 shows an analysis of the convergence behaviors of the Gummel and Newton methods
when applied for the simulation of the bilayer device. We see that both Gummel’s and Newton’s
method can converge with a mixing factor of 60% for low voltages, but Gummel’s method requires
a reduction of the mixing factor at Va = 0.62 V. Comparing the number of iterations required for
convergence to a tolerance of 5 × 10−13, we see that the number of iterations is similar for lower
voltages, but at higher voltages, Gummel’s method requires many more iterations. This is related
to the lower mixing factor needed. The numerous spikes occur near the points where the solutions
are no longer converging, which are followed by a decrease in the mixing to allow for convergence.
Mixing was decreased once there were more than 1000 iterations where the error increased.

43

Figure 3.5: Comparison of convergence behaviors between Gummel and Newton methods. a)
Mixing factor needed to achieve convergence. b) Number of iterations required to converge.

These models are not computationally expensive, and calculations can be easily done on a
standard personal computer (Windows 10 laptop with 8GB RAM and Intel i7-6820HQ CPU at 2.70
GHz base speed and overclocked up to 3.3 GHz). CPU times for the Matlab implementation of these
models are reported below. (Implementations in C, C++ or Fortran are expected to be significantly
faster). Each iteration of the Gummel method takes only 0.02 − 0.035 seconds. Therefore, even
with the large number of iterations required for convergence, an entire calculation of a JV curve
sweeping over applied voltages from 0 V to 0.8 V with 0.01 V steps takes only about 4 minutes.
For the Gummel method, most of the CPU time per iteration is spent on the recalculation of the net
generation rate because this requires evaluation of several non-trivial equations, including integrals
that are computed numerically.

Newton’s method requires significantly more CPU time per iteration. For the same bilayer
model, one Newton iteration takes about 0.25 − 0.65 seconds (lower time for lower voltages). This
ten-fold increase over the Gummel method is due to having to calculate all of the partial derivatives
for the Jacobian elements and also having to solve a 3n × 3n matrix instead of three n × n matrices.
The number of iterations required to converge depends on the damping factor that is used. The
same simulation from 0 V to 0.8 V with 0.01 V steps took 794 seconds with Newton’s method and
240 seconds with Gummel’s method. Thus, for the generation of JV curves within the operating
range of a solar cell, Gummel’s method is more efficient.

44

3.2.3 Slotboom Variables Approach

During the process of designing ways to improve the algorithm, an alternative approach was investi-
gated which did not yield improved results. Nevertheless, it is overviewed here for completeness of
the analysis of possible variations in computational techniques.The Slotboom approach [38] was in-
troduced as an alternative to Scharfetter-Gummel discretization. In this approach, the drift-diffusion
equations are rewritten as

Jn = qNVt µn

eψ/Vt

dFn
dx

and

Jp = −qNVt µp

e−ψ/Vt

dFp
dx

where

Fn = e−ϕn/Vt

and

Fp = eϕp/Vt

and ϕn and ϕp are the electron and hole quasi-Fermi potentials given by

(3.19)

(3.20)

ϕn = ψ/Vt − ln(n/N)

and

ϕp = ψ/Vt + ln(p/N)

(3.21)

Note that Equations 3.20 and 3.21 can be combined as

Fn =

e−ψ/Vt

n
N

and

Fp =

eψ/Vt

p
N

When discretizing the continuity equations, we get

µn,i+1(Fn,i+1 − Fn,i)eψi+1/Vt − µn,i(Fn,i − Fn,i−1)eψi/Vt = −(δx)2
µp,i+1(Fp,i+1 − Fp,i)e−ψi+1/Vt − µp,i(Fp,i − Fp,i−1)e−ψi/Vt = −(δx)2

Vt N

Vt N

Un,i

Up,i

(3.22)

(3.23)

(3.24)

Therefore, the matrix equations will now be solved for Fn and Fp instead of n and p. A benefit
of this approach over Scharfetter-Gummel discretization is the removal of the approximation using
Bernoulli functions, yielding more elegant expressions for the equations. While this is useful
for analytical calculations, when attempting to solve these numerically, the eψ terms often cause
overflow or underflow errors. Note that in the Scharfetter-Gummel method this is avoided because
the exponentials are eδψ (using the difference in potential between neighboring mesh points),

45

which, provided that the mesh is small enough, will result in a reasonable magnitude exponential.
An interesting observation is that if we substitute the expressions for Fn and Fp into Equations 3.23
and 3.24, we can get a matrix equation for n and p that, while appears similar to the ones from
Scharfetter-Gummel’s discretization, does not have any Bernoulli functions. The terms generating
the matrices will be
(ψ(cid:48)
i−ψ(cid:48)
i−1)
n(cid:48)
µn,ie
i−1−ψ(cid:48)
(ψ(cid:48)
i)
p(cid:48)

i−1 −(cid:0)µn,i+1e
i−1 −(cid:0)µp,i+1e

i+1 = −(δx)2
i+1 = −(δx)2

(ψ(cid:48)
i+1−ψ(cid:48)
(ψ(cid:48)
i−ψ(cid:48)

i) + µn,i
i+1) + µp,i

i + µn,i+1n(cid:48)
i + µp,i+1p(cid:48)

(cid:1)n(cid:48)
(cid:1) p(cid:48)

(3.25)

(3.26)

Un,i

Up,i

Vt N

Vt N

µp,ie

Using these equations for the bilayer device simulation, gave almost identical results (to several
decimal places) to those from the Scharfetter-Gummel discretization. There was no significant dif-
ference in the convergence properties when compared to using Scharfetter-Gummel discretization.
It appears that the Slotboom variables approach is a valid and usable alternative to the discretization
as long as one substitutes back the definitions of Fn and Fp as done above.

3.3 Conclusions

In this chapter, we investigated the impact of several numerical techniques on the convergence
behavior of a 1D drift-diffusion model for bilayer organic solar cells. We found that Pulay mixing
accelerates the convergence of Gummel iterations by about 20%, but does not resolve the divergences
experienced at higher applied voltages. When the coupled Newton’s method was used, the model
could use higher mixing (relaxation) values and converge at higher applied voltages. This supports
the idea that the convergence difficulties are due to a strong coupling of the bilayer drift-diffusion
equations at higher applied voltages due to the build-up of space charge at the bilayer interface.
In this situation, the decoupled Gummel iteration approach leads to oscillations and instabilities,
while Newton’s method suppresses such oscillations by solving the entire system simultaneously.
The Slotboom variables discretization approach, which avoids the need to use Bernoulli functions,
was found to lead to overflow and underflow errors unless the equations are reformulated to solve
for the carrier densities. Using the Slotboom variables in this way, gave almost identical simulation

46

results to the more-traditional Scharfetter-Gummel discretization approach, but did not improve
convergence. The results of these investigations can help one select an appropriate discretization
and iteration scheme to use for each drift-diffusion problem. Pulay mixing can be used to improve
performance, Newton’s method is desirable for situations where the equations become highly
coupled, and Slotboom variables may be chosen as a way to simplify the implementation of the
matrix equations.

47

CHAPTER 4

DRIFT DIFFUSION MODELING OF PEROVSKITE SOLAR CELLS

Perovskite solar cells are thin-film cells based on hybrid halide perovskites, the most common being
CH3NH3PbX3 where X is Cl, Br, I, or a mixture of these. They have attracted great interest in recent
years [39–43] due to their lower material and production costs, as well as rapid improvements in
efficiencies to above 25% [44]. These materials have favorable properties such as a direct bandgap,
high absorption coefficients, and high charge mobilities [45, 46]. The devices can be made in both
planar heterojunction and bulk heterojunction (also known as mesoporous) architectures. This
chapter discusses the application of the drift-diffusion model to planar perovskite solar cells.

A typical planar perovskite solar cell has a perovskite absorber layer sandwiched between an
electron transport layer (ETL) and a hole transport layer (HTL). The perovskite layer absorbs light,
which results in the photogeneration of free electrons and holes. The charge carriers will diffuse
from higher to lower concentration and drift due to the device’s built-in electric field. Electrons
will, on average, move towards the ETL and holes towards the HTL. The ETL and HTL’s primary
role is to transport the respective carriers to the electrodes while blocking the opposite carrier type
with energy barriers. Holes are unlikely to pass into the ETL and electrons are unlikely to pass into
the HTL. This results in very low hole densities in the ETL and very low electron densities in the
HTL, which greatly reduces free carrier recombination. Therefore, in the model, we can assume
that there is no recombination in the transport layers.

4.1 Numerical Model

To model perovskite solar cell devices, we use a one-dimensional drift-diffusion (1D-DD)
approach based on the model reported in Sherkar et al. [47] to self-consistently solve the Poisson,
continuity, and drift-diffusion equations using Gummel iterations together with Scharfetter-Gummel
discretization, as described in Chapters 2 and 3. The device is discretized into a one-dimensional
mesh with a grid size of 0.25 nm, with the anode and cathode located at x = 0 and x = L,

48

respectively. When discretizing the Poisson equation, we use Equation 2.9 for mesh points located
at the boundaries between two layers. The dielectric constant of perovskite is generally much
higher than that of the transport layer materials, causing a smaller electric field in the perovskite
than in the transport layers. The implementation of the energy barriers between layers is done in
the same way as in the DD-BI model, namely, by modifying the derivative of the potential δψ(cid:48) at
the interfaces (see Section 3.1).

A significant difference between perovskites and organic bilayer devices (discussed in Chapter
3) is that perovskites have no donor-acceptor interface where polaron pair formation and separation
occurs. In perovskites, the exciton binding energy is on the order of the thermal energy of the
carriers [48]; therefore, excitons separate into free charge carriers throughout the device. This
means that in simulation, the free carrier generation rate can be treated as directly related to the
absorption density of photons without calculating any exciton densities.

Both radiative and non-radiative recombination of carriers decrease device performance. Ra-
diative recombination is the annihilation of an electron from the conduction band with a hole in
the valence band, which results in the emission of a photon. Non-radiative recombination (also
known as trap-assisted recombination) occurs through a defect or trap, which is an energy level in
the bandgap. The trap can capture either an electron or hole (depending on the trap’s energy level),
which can recombine with a hole in the valence band or an electron in the conduction band, respec-
tively. We include both radiative and non-radiative (trap-assisted) carrier recombination. Radiative
recombination is modeled by Langevin bimolecular recombination with the recombination rate
given by

Rb = kb(np − n2
i )

(4.1)

where kb is the bimolecular recombination constant and ni is the intrinsic carrier density. Bimolec-
ular recombination is assumed to occur only in the perovskite layer. It is not considered in the
transport layers because of the very low density of minority carriers in those layers which makes
the recombination rate many orders of magnitude less than in the perovskite. Non-radiative trap
assisted recombination is assumed to occur only in 5 nm thick regions inside the perovskite layer

49

near the HTL and ETL interfaces and is described by the Shockley-Read-Hall equation (3.12).
The trap energies are assumed to be located in the middle of the bandgap since this energy gives
the most effective recombination [49]. As in Sherkar et al., the traps are assumed to have neutral
charge.

4.2 Model Validation

The 1D-DD model for planar perovskites was validated by comparing to the literature results
from Sherkar et al. The devices studied use gold as the cathode and poly(3,4-ethylenedioxy-
thiophene):poly(styrenesulfonic acid) (PEDOT:PSS) and indium tin oxide (ITO) as the anode. The
HTL is negatively doped poly[N,N-bis(4-butylphenyl)-N,N-bis(phenyl)benzidine (polyTPD) and
the ETL is positively doped [6,6]-phenyl C61-butyric acid methylester (PCBM). Figure 4.1 shows
the device architecture.

Figure 4.1: Device structure of planar heterojunction perovskite solar cell modeled by Sherkar et
al. [47].

The effects of various physical parameters on device performance can be studied with the help of
this drift-diffusion device model. From simple physical arguments, we expect an efficient perovskite
solar cell to have high carrier mobility in the perovskite layer, low trap densities, good blocking of
the minority carrier at the transport layers, a high built-in field, and low injection barriers. High

50

carrier mobility leads to faster extraction of carriers from the perovskite, which reduces the carrier
density and thus reduces recombination. We would also like to have materials with minimum trap
densities where trap-assisted recombination could occur. Good blocking of minority carriers by the
transport layers prevents recombination near the electrodes. A high built-in field creates a stronger
drift current for the separation of electrons and holes in the perovskite layer, which leads to faster
charge extraction. We can use our model to test these arguments with numerical experiments.

Figure 4.2 shows the effect of the perovskite mobility on the JV curve and the power conversion
efficiency η. The most considerable effect of lower mobility is the reduction of fill factor, which
reduces the maximum power output and thus efficiency. Jsc is also reduced with lower mobility
since fewer free carriers reach the electrodes. Notice that above a mobility of 0.5 cm2/Vs, further
increasing the mobility does not have much of an effect on the JV curve. This indicates the existence
of some mobility threshold that ensures that carriers are able to reach the electrodes.

Figure 4.2: Simulated JV curves for different carrier mobilities in perovskite layer from this work
(a), and Sherkar et al. (b).

Figure 4.3 shows the effect of trap densities at the HTL/perovskite and perovskite/ETL interfaces.
The interfaces are assumed to be 5 nm in width. Note that the surface trap density values are
converted to volume trap densities (by dividing by the 5 nm interface width) before use in the
Shockley-Read-Hall equation [50]. Higher trap densities reduce fill factor, Voc, and Jsc. Notice

51

that the trap density at the perovskite/ETL interface (Ntrap) has a much smaller effect than the
traps at the HTL/perovskite interface. This is because the light is incident from the HTL side, and
therefore, there is a much higher photogenerated free carrier density near the HTL, so the traps
can significantly reduce the free carrier density. Therefore, when designing a perovskite solar cell,
choosing materials and manufacturing techniques that reduce traps at the HTL/perovskite interface
is important.

Figure 4.3: Simulated JV curves for different trap densities at HTL/perovskite interface (a, b) and
ETL/perovskite interface (c, d). Results from this work’s model (a, c) are compared with results
from Sherkar et al. (b, d).

The built-in voltage (Vbi) and electric field are determined by the work function difference
between the cathode and anode. Therefore, lowering the injection barriers at the electrodes will
increase Vbi. Figure 4.4 shows the effect of injection barriers on the JV curve. As expected, zero
injection barriers are predicted to give the best performance.

52

Figure 4.4: Simulated JV curves for different injection barriers at the anode and cathode (φa and
φc respectively) on the JV curve from this work’s model (a), and from Sherkar et al. (b).

In this section, we have confirmed that our 1D-DD model can reproduce results from a validated
model in the literature. We have also shown an example of how such a model can be used to
understand the effects of carrier mobility, trap density, and injection barriers on device performance.
In the next section, we describe the application of this validated 1D-DD model to explain the
physics behind experimentally-observed drastic performance differences in perovskite devices with
and without fullerene layers.

4.3 Understanding the Impact of C60 at the Interface of Perovskite Solar

Cells

There is increasing interest in the application of fullerene compounds to perovskite photovoltaics
[51, 52]. Recently, Liu et al. [53] observed that an ultrathin fullerene (C60) layer could have a
drastic impact on the performance of planar perovskite solar cells. Through a combination of
fluorescence microscopy and impedance spectroscopy, they showed that the main role of the C60 is
to efficiently extract electrons from the perovskite film. They demonstrated that even an ultrathin
vapor-deposited 1 nm C60 layer works efficiently in the devices, while devices without C60 exhibit
poor performance.

The device architecture is as follows. The cells use PEDOT:PSS for the hole transport layer and

53

bathocuproine (BCP) for the electron transport layer. The devices have a 1 nm, 20 nm, or no C60
layer between the perovskite and BCP layers. All devices have an 80 nm Ag cathode and 100 nm
indium tin oxide (ITO) anode. Figure 4.5 shows the interior layer thicknesses and electronic energy
levels of the devices, with the rectangles representing the bandgap of the materials. The energy
alignment allows electrons to flow from the perovskite to the cathode and holes to flow from the
perovskite to the anode. It has been reported that electrons travel through BCP via gap-states which
are thought to be caused by diffusion of the Ag into the BCP layer [54–56]. The energy levels of
the gap-states, as predicted by our numerical model, are shown within the BCP rectangle.

Figure 4.5: Energy levels and thicknesses of the device layers with the rectangles representing the
bandgap of the material. The levels were found from a combination of literature values and fitting
of the numerical model (more details in Table 4.1). The levels within the BCP bandgap are
gap-states due to diffusion of Ag into BCP.

In this work, we provide an explanation for the observed effects of ultrathin C60 on device
performance by use of the drift-diffusion model. The differences in electron extraction with
C60 layer thickness are modeled by a reduced effective carrier mobility in the C60 layer or at
the perovskite/BCP interface (for the devices without C60). Our numerical results support the
conclusions of Liu et al. regarding the role of C60 as essential for efficient electron extraction. Based
on fitting of the model to the experimental results, the reason for the differences in performance

54

ŶĞƌŐLJ;ĞsͿWͲϱ͘ϵͲϰ͘ϮͲϯ͘ϱͲϱ͘ϰͲϯ͘ϵͲϱ͘ϭWKd͗W^^/dKϲϬͲϰ͘ϴͲϰ͘ϯϮWĞƌŽǀƐŬŝƚĞͲϳ͘ϬŐͲϰ͘ϯϮͲϯ͘ϱͲϯ͘ϵϯϳ͘ϱŶŵϯϮϬŶŵϭ͘ϱŶŵdžŶŵbetween the devices with and without C60 is mostly due to a much lower electron mobility at a
perovskite/BCP interface than at a perovskite/C60 interface. These results help clarify the role of
the C60 layer as part of the ETL in a perovskite solar cell.

Table 4.1 provides the complete set of parameters used for the model with references for
parameters taken from literature listed. Less efficient electron transport at the perovskite/BCP
interface is modeled by a decreased electron mobility value in a 0.5 nm interface region. For the 1
nm C60, less efficient electron transport due to incomplete coverage of C60 is modeled by a reduced
effective mobility for C60.

55

Table 4.1: Complete set of parameters used for the model. References for parameters obtained
from literature are listed.

Source
Ref. [57]
Ref. [58]
Ref. [58]
Ref. [47, 57, 58]
Ref. [47, 57, 58]
Fit
Fit
Fit/Ref. [54–56]
Ref. [59]
Ref. [59]/Fit
Ref. [47]
Ref. [47]
Ref. [47]
Ref. [60]
Fit
Ref. [61]/Fit

Value
Parameter
-4.8 eV
Anode (ITO) work function
-5.1 eV
HTL (PEDOT) HOMO Level
-3.5 eV
HTL (PEDOT) LUMO Level
-3.9 eV
Perovskite conduction band minimum
-5.4 eV
Perovskite valence band maximum
-5.9 eV
ETL (C60) HOMO level
-4.2 eV
ETL (C60) LUMO level
-3.93 eV, -4.32 eV
BCP gap-state levels
-7.0 eV
BCP HOMO
-4.32 eV
Cathode (Ag) work function
8.1 × 1024 m−3
Density of states (DOS)
24.1
Perovskite relative permitivity
3.0
PEDOT:PSS relative permitivity
4.25
C60 relative permitivity
4.25
BCP relative permitivity
10−4 m2V−1s−1
Carrier mobility in perovskite
4.5 × 10−6 m2V−1s−1 Ref[62]
Carrier mobility in PEDOT:PSS
1.6 × 10−4 m2V−1s−1 Ref. [58]
Carrier mobility in 20 nm C60
1.6 × 10−4 m2V−1s−1 Ref. [58]
Hole mobility in 1 nm C60
1.6 × 10−5 m2V−1s−1 Fit
Electron mobility in 1 nm C60
2 × 10−9 m2V−1s−1
Fit
Carrier mobility in BCP bulk
Electron mobility at perovskite/BCP interface 5 × 10−13 m2V−1s−1
Fit
1.97 × 1028 m−3s−1
Fit
Maximum charge generation rate
10−13 m3s−1
Electron and hole capture coefficients
Ref. [47]
6 × 10−17 m3s−1
Ref. [47]
Bimolecular recombination constant
5 × 1021 m−3
Ref. [47]
HTL/perovskite interface trap density
6 × 1020 m−3
Perovskite/ETL interface trap density
Ref. [47]/Fit
Ref. [53]
1.5 nm
PEDOT:PSS thickness
Ref. [53]
320 nm
Perovskite thickness
BCP thickness
7.5 nm
Ref. [53]
0.25 nm
Grid spacing

56

Charge generation profiles were determined using an open-source transfer-matrix optical model

[63]. A sample profile for the device without C60 is shown in Figure 4.6.

Figure 4.6: Carrier photogeneration rate profile for the device without C60 as determined from the
transfer-matrix optical model using the sum over wavelengths 350-750 nm. Position is displayed
relative to the beginning of the HTL (PEDOT:PSS) layer.

We find a good agreement between the experimentally measured and numerically predicted
current-voltage (JV) curves (Figure 4.7). We see that the C60 layer is essential for good device
performance and that a 1 nm C60 layer is sufficient for good performance, with only a small
improvement achieved when the layer thickness is increased to 20 nm. The fit to experiment
suggests that the electron mobility at a perovskite/C60 interface is significantly higher than at a
perovskite/BCP interface (case of no C60). This higher mobility models a more efficient extraction
of electrons from the perovskite to the ETL.

57

050100150200250300Position (nm)00.511.52Generation rate /(s cm3)1022CH3NH3PbI3PEDOTBCPFigure 4.7: Comparison of the JV curve results from the experiment and numerical model.

To determine the reason for the experimentally observed performance loss in the 0 nm C60
devices, the influence of electron mobility and traps at the perovskite/BCP interface was compu-
tationally studied (Figure 4.8). For the mobility trials, we set all parameters as in Table 4.1 and
only vary the perovskite interface mobility. For testing the effects of the interface trap density,
we first raise the perovskite interface mobility to the same value as in the device with 20 nm C60
(2 × 10−9 m2/Vs) to remove the effect of lower mobility, and then vary the trap density. We
find that mobility mostly influences the short-circuit current, while trap density mostly influences
the open-circuit voltage and fill factor. Figure 4.8a shows how a decrease in perovskite interface
mobility significantly affects the short-circuit current and fill factor of the device. For the devices
without a C60 layer, fitting the model to the experimental results suggests an electron mobility of
5 × 10−13 m2/Vs. An increased trap density at the interface (Figure 4.8b) does not result in the
drastic short-circuit current loss that is experimentally observed; therefore, the dominant factor is
a low interface mobility.

These results support the fluorescence microscopy experiments of Liu et al. which showed

58

00.20.40.60.81Voltage (V)-20-15-10-505Current Density (mA/cm2)C60 - 0 nm experimentC60 - 0 nm modelC60 - 1 nm experimentC60 - 1 nm modelC60 - 20 nm experimentC60 - 20 nm modelthat while adding thin C60 layers to perovskite cause a significant decrease in photoluminescence
(PL), placing BCP alone onto perovskite does not significantly change the PL efficiency (neat
perovskite films are as bright or brighter with BCP) [53]. A quenched PL intensity indicates a
smaller accumulation of electrons. If the improvement in performance due to C60 was due to small
molecules passivating defect states, we would expect the PL quenching with BCP to be similar
to that of C60. Since this is not the case, it was concluded that C60 acts to greatly improve the
efficiency of electron extraction from perovskite. Similar results were reported in the literature for
perovskite devices using TiO2 as the ETL, where the addition of fullerene compounds to the ETL
enhanced electron extraction at the perovskite/ETL interface [64, 65].

Figure 4.8: JV curves showing the effects of perovskite/BCP interface carrier mobility (a) with
ETL trap density set to 6 × 1020 m−3 and trap density (b) on the 0 nm device with perovskite/BCP
interface mobility set to 2 × 10−9 m2/Vs (the same value as for the 20 nm C60 device). All other
parameters were kept the same as in Table 4.1.

We gain more insight into the space-charge effects in these devices by considering the charge
density profiles for devices with different C60 thicknesses at zero applied bias (Figure 4.9). In the
0 nm C60 device, due to the low mobility at the perovskite/BCP interface, the electron extraction
is poor and results in a lower extraction of electrons than holes from the device, as seen from
the low electron density in the BCP layer at the right-hand side of Figure 4.9. This can also be
understood as an imbalance in mobilities of the majority charge carrier through their corresponding

59

00.20.40.60.81Voltage (V)-25-20-15-10-505Current Density (mA/cm2)2.010-105.010-115.010-123.010-121.010-125.010-13ExperimentInterfaceMobility (m2/Vs)(a)00.20.40.60.81Voltage (V)-25-20-15-10-505Current Density (mA/cm2)6.010206.010216.010221.01024ExperimentInterface Trap Density (m-3)(b)transport layer, or equivalently as the addition of an extraction barrier for electrons. The poor
electron extraction causes a charge accumulation at the perovskite/BCP interface.
In contrast,
the devices with C60 have no electron accumulation within the perovskite layer. This supports
the fluorescence microscopy measurements done on these devices, which show a much stronger
photoluminescence, indicating a larger electron accumulation for the devices without C60 than with
C60 [53]. In the devices with C60, due to the efficient electron extraction from the perovskite, there
is a high electron density in the C60 together with a very low hole density and an injection barrier
that prevents electrons from moving back into the perovskite. The high electron transfer efficiency
also reduces the loss of free carriers due to carrier recombination within the perovskite layer. This
means that more electrons can be extracted to the BCP than in the 0 nm C60 device, resulting in a
higher current.

60

Figure 4.9: Charge carrier densities for the simulated devices with 0 nm, 1 nm, and 20 nm C60
under illumination at zero applied bias. The perovskite/C60 and C60/BCP interfaces are marked
by dotted black lines. The parameters used are listed in Table 4.1.

A possible reason for the reduction of mobility at the perovskite/BCP interface could be the
formation of a back-to-back Schottky barrier [66]. The charge transport through this barrier

61

00.511.522.53Position (m)10-7101510201025Carrier Density (m-3)0nm C60HolesElectrons00.511.522.53Position (m)10-7101510201025Carrier Density (m-3)1nm C60HolesElectrons00.511.522.53Position (m)10-7101510201025Carrier Density (m-3)20nm C60HolesElectronscould be a tunneling process that is exponentially related to the height of the barrier. However,
using an energetic barrier for electron extraction instead of a low effective mobility region at the
perovskite/BCP interface yields current-voltage curves that are very different from the experimental
results (Figure 4.10). Therefore, the energy barrier approach cannot be used to model the poor
performance of the 0 nm C60 device. A more likely reason for the less efficient transport could be
related to a difference between the two interfaces at the molecular level, causing the perovskite/C60
interface to have a better electron transport pathway.

Figure 4.10: The effect of using an energetic barrier for electron extraction instead of a low
effective mobility region at the perovskite/BCP interface of devices without C60. The fit to
experiment that was achieved when using a low mobility region is shown for comparison.

Next, we offer an explanation for the origin of the S-shaped JV curve observed in the 0 nm C60
device (Figure 4.7). Several groups have proposed explanations for the appearance of S-shaped JV
curves in solar cells. They have been attributed to various factors including interface dipoles [67],
charge accumulation [59, 68], injection or extraction barriers [69], imbalanced mobilities [70], and
poor extraction of electrons near the cathode due to a mobility drop-off [71]. In the device studied
here, the origin is likely a combination of several of the above, namely an injection barrier, mobility

62

00.20.40.60.81Voltage (V)-25-20-15-10-505Current Density (mA/cm2)0nm C60Fit to experimentNo barrier0.05 eV barrier0.1 eV barrierdrop-off (equivalently imbalanced mobilities), charge accumulation, and interfacial dipole.

Figure 4.11 shows the 0 nm C60 device under 0.7 V and 0.9 V applied bias, which correspond
respectively to points before and after the onset of forward current in the device. At 0.7 V, we
see that there is no longer a dip in electron density within the BCP, which indicates that electrons
are being injected from the cathode into BCP. However, the electrons are not being injected into
the perovskite layer, as seen by the steep difference in electron density on opposite sides of the
perovskite/BCP interface. The electron accumulation in BCP is caused by a combination of the 0.42
eV injection barrier for electrons (Figure 4.5) and the low electron mobility at the perovskite/BCP
interface. The injection barrier and low interface electron mobility prevent an injection current from
flowing. As we further increase the applied voltage from 0.7 V, there is an even larger accumulation
of electrons within the BCP near the perovskite/BCP interface and now also an accumulation of
holes on the other side of the interface, forming a dipole. The dipole effectively decreases the
injection barrier for electrons; however, no injection current will flow as long as an injection barrier
is present. Therefore, the net current in the device is near zero until the applied voltage is sufficient
to overcome the injection barrier (i.e. about 0.9 V). Notice that the width of the flatter region in the
JV curve increases with decreasing interface mobility (Figure 4.8a) since a lower interface mobility
requires a stronger net field (and thus a higher applied voltage) to establish an injection current
across the interface.

63

Figure 4.11: Charge carrier densities for device with 0 nm C60 under illumination at 0.7 V (a) and
0.9 V (b) applied bias. The parameters used are listed in Table 4.1.

To ensure the reliability of the simulations, we checked the effects of variation of the HO-
MO/LUMO levels of the HTL/ETL on the JV curve. In these devices, the most important energy
parameters are the relative energy offsets between adjacent layers that are used to compute the
gradient of the electric potential at the interfaces. Therefore, we checked the effects of changes
of ± 0.1 eV in the HOMO/LUMO of HTL/ETL, which corresponds to up to a 33% change in the
relative offsets between adjacent HOMO/LUMO levels. Even with these changes, which are large
relative to the offsets, the simulation is still comparable to the experimental results (Figure 4.12).
This demonstrates that even with significant variation of the energy level offsets, our arguments
regarding the role of C60 and the reason for the device’s poor performance without C60 still hold.

64

00.511.522.53Position (m)10-7101510201025Carrier Density (m-3)Va = 0.7 VHolesElectrons(a)00.511.522.53Position (m)10-7101510201025Carrier Density (m-3)Va = 0.9 VHolesElectrons(b)Figure 4.12: The effect of variation of the HOMO/LUMO of the HTL/ETL on the JV curve of the
device with 0 nm C60. The fit to experiment is included for comparison.

4.4 Conclusions

A 1D drift-diffusion model for planar perovskite solar cells has been developed and validated by
comparison to reported literature results from an experimentally validated model. The model was
then used to describe recent experimental results for inverted planar structure perovskite devices
that use fullerene layers for electron transport. By systematically studying the effects of interface
mobility and trap density on the JV curves, we find that poor electron extraction at the perovskite
interface, modeled by a decreased carrier mobility, is the main cause of the severe decrease in
device performance when there is no C60 layer. We demonstrate that the decrease in performance
is not due to an increased trap density at the interface. We have also analyzed and provided an
explanation for the S-shaped JV curve of the 0 nm C60 device. The main origin of the S-shape is
likely due to the non-equal mobilities in the HTL and ETL, which results in charge accumulation
and the formation of an interface dipole. The modeling results support the conclusion that very
little C60 is needed to enhance carrier extraction. Ultimately, this work helps clarify the role of
fullerenes in perovskite solar cells.

65

00.20.40.60.81Voltage (V)-10-8-6-4-202Current Density (mA/cm2)0 nm C60Fit to experimentChange HTL LUMO & HOMO by -0.1 eVChange ETL LUMO & HOMO by -0.1 eVChange HTL HOMO by +0.1 eVChange ETL HOMO by +0.1 eVPART II

MODELING OF PHOTOVOLTAIC SYSTEMS

66

Drift-diffusion modeling of solar cells can improve our understanding of fundamental device
physics and limitations of device performance, and help engineer better solar cell architectures.
However, most applications of photovoltaics involve multiple PV modules, where each module
is composed of many solar cells. Unlike individual solar cells, which are generally studied in
controlled laboratory conditions, PV systems are utilized outdoors where environmental conditions
such as irradiance, ambient temperature, and particulate matter affect their performance. Thus, to
accurately model PV systems, it is necessary to develop system-level models with consideration of
the appropriate physics and environmental boundary conditions.

Additionally, improving PV efficiencies at the individual solar cell scale is becoming increas-
ingly difficult and costly. Therefore, there is an increasing focus on optimizing energy production
at the PV module and system levels. PV systems are most effective in areas with abundant sunlight.
However, such areas are often in semi-arid and desert climates, causing the systems’ energy pro-
duction to be significantly reduced by two major environmental factors: high ambient temperature,
which causes thermal losses, and high concentration of atmospheric dust, which deposits on the
panels causing soiling losses [72]. The losses due to these environmental factors can cause even
the most efficient solar cells to perform poorly when deployed in the field. For example, energy
production losses due to soiling have been reported to be between 20-70% in some desert regions
[73, 74], and losses due to temperature are commonly 0.1-0.5% for each 1◦C increase in panel
temperature [75]. In order to design new systems, improve return-on-investment assessment, and
optimize system maintenance, PV system models need to accurately predict energy output, which
requires accounting for losses due to temperature and soiling. The next two chapters analyze
existing models and develop improved models for considering these losses.

67

CHAPTER 5

IMPROVING PV SYSTEM SOILING LOSS PREDICTIONS

Soiling of PV panels is the process in which airborne particles deposit and accumulate on the panel
surface, reducing solar irradiance received by the panel by absorbing and reflecting a portion of the
incident light [74, 76]. In addition to reversible optical losses, soiling can cause permanent damage
to PV modules. In cases of omitted cleaning, cemented dust layers can be very difficult to remove,
whereas harsh cleaning with brushes can lead to scratching or abrasion of anti-reflective coatings
or panel glass. Non-uniform soiling causes partial shading, which can lead to the formation of
hot spots (areas of high temperature that result in localized decrease in efficiency and accelerated
material degradation). Even with regular cleaning, mechanical loads during cleaning or thermal
shocks when hot panels are cleaned with cold water can break solar cells or cause crack formation.
Soiling has been estimated to cause an average loss of 3-4% of the global annual energy yield
of photovoltaics [77]. Soiling also introduces additional maintenance costs and increases the
uncertainty in the estimation of PV performance, leading to higher financial risks and interest
rates charged to plant developers [78]. Improving soiling loss predictions will allow to both more-
accurately predict system energy production, which will lower financial risk and interest rates, and
form more optimal cleaning maintenance schedules to mitigate some of these losses.

5.1 Background

Soiling has been discussed in the literature for more than 70 years, and yet the fundamental
properties of aerosols and their effect on energy transfer are still not fully understood, nor is there
a clear solution to the problem [73]. Soiling has been commonly underestimated and overlooked
in the PV industry until it came to the forefront recently because of increased deployment of PV
systems in parts of the world where soiling is a major issue [79]. For example, in sunny, arid, and
dusty regions such as the Middle East, North Africa, and Asia, power losses due to soiling have
been reported to be between 20-70%, while in locations with frequent precipitation or low ambient

68

particle concentration, the energy loss is typically less than 5% [73, 74]. The vast majority of
previous research into soiling has consisted of observations of the reduction in performance due
to dust accumulation as a function of exposure time at a particular location. Many researchers use
these observations to derive estimates for annual soiling loss (e.g. 5% per year loss) or constant daily
soiling rates (e.g. 0.1%/day). These rough estimates are then used in PV performance modeling
and for cleaning schedule guidance [79]. A large proportion of research studies use artificial dust
particles, which are not representative of natural dust composition [80]. The conclusions from
these studies are often contradictory, and much of the information available in the literature applies
only to the specific location where the experimental work was conducted due to numerous very
localized factors.

Annual PV soiling losses are generally computed by considering the soiling rate (typically the
increase in loss per day) for a site combined with rainfall patterns and manual cleaning events.
For example, in 2006, Kimber et al. [81] presented a new model for predicting soiling losses in
California as a function of rainfall data and the number of manual cleanings. Their empirical model
uses typical meteorological year (TMY) data and hourly soiling rates to predict energy production.
They suggested a linear model to represent daily system efficiency reduction due to soiling between
rainfalls. When daily rainfall exceeds a threshold value, the soiling loss is assumed to drop to a
minimum value. After a study of data from many PV plants in California, they found that the soiling
rates are between 0.1-0.3% per day during dry periods, which corresponds to annual energy losses
between 1.5 and 6.2%, depending on system location. They found that their model is more accurate
than using a constant annual soiling loss factor, which is a common approach in the industry.
While more advanced approaches have been developed by academic researchers, the Kimber et
al. approach appears to be one of the most advanced soiling loss models used commercially.
Some of the proposed approaches require too many experimental measurements (e.g. particle
accumulation masses, particle diameters, etc.), making them impractical for commercial use at
this time. The goal of the work described in this chapter was to analyze some of the more
sophisticated approaches proposed by researchers, evaluate the practicality of their implementation

69

for commercial applications, and encourage some of them to be utilized by the PV industry by
implementing more advanced soiling models in a commercial software tool, Enlighten Energy’s
PV SoilSayer.

In this work, we use Kimber et al.’s approach as a starting point and attempt to improve it by

addressing some of its limitations, which are listed below:

• The soiling rates were determined by comparing energy production data to predicted ideal
production from solar resource estimates. Any errors in the ideal energy production will
contribute to an error in the soiling rate. A better approach to determine soiling rates would
be to compare actual energy production data from a soiled module to one that is cleaned
regularly.

• They did not consider the effect of angle of incidence (AOI) on soiling losses, even though
multiple previous studies have shown a dependency. This dependency means that two PV
modules in the same location, but with different tilt angles or a fixed-tilt versus single-axis
tracker design, are expected to experience different soiling losses [82, 83].

• Their study monitored 250 PV systems, but all data with an R2 of less than 0.7 when fitting
a line to the dry periods was excluded from the analysis. As pointed out by another study
[84], this could have biased the data towards sites with higher soiling where more linear
relationships are likely. Sites with smaller losses tend to have smaller R2 because random
errors in solar resource estimates dominate. Another study in California analyzed 186 PV
sites and found that soiling losses averaged 0.051%/day, which is significantly less than the
0.2%/day found by Kimber et al. [85]. However, this study also had the flaw where measured
energy data was compared with predicted ideal data, so the discrepancy could be caused
by different prediction approaches of ideal energy production or different sources for solar
resource data. Such discrepancies between studies performed in the same region demonstrate
the lack of reliability in existing approaches to model and measure soiling losses.

70

• Their model requires empirical measurements of soiling at the locations where soiling loss
predictions are desired. Often, especially when planning for new solar plant construction,
making such measurements over a long duration is not practical, yet an estimate of the soiling
losses is still needed. Such estimates could potentially be achieved by studying the impact of
environmental conditions such as particulate matter concentration, wind speed, and relative
humidity, then developing models that use historical or TMY weather data to predict soiling
losses based on these observed relationships.

When considering the approaches to address such soiling model limitations, some studies have
proposed that two independent models are required: one to correlate the amount of dust accumulated
with locally available weather or environmental conditions, and the second to correlate the amount
of dust deposited with energy production loss [80]. Most literature models further separate the
prediction of soiling accumulation based on weather into the soiling rate during dry periods and
the effects of rain. We take this approach of multiple submodels, with Section 5.2 focused on
the soiling rates during dry periods, Section 5.3 considering the effect of rain, and Section 5.4
correlating the dust deposition prediction with energy production loss.

5.2 Semi-Physical Modeling of Soiling Accumulation

Most of the reported soiling research consists of taking field measurements of the energy losses
due to soiling on particular PV modules in a particular location and deriving soiling rates (i.e.
percent loss per day) based on this data. These regression models typically only apply to the
specific location where the experimental work was conducted. It is often not feasible to perform
such measurements of soiling losses at each PV site due to lack of necessary equipment, labor
costs, or when an energy production prediction is desired for a PV plant that has not yet been built.
Therefore, it is desirable in some situations to move beyond site-specific soiling loss studies and have
a predictive model for soiling losses in any location in the world that does not require taking new
measurements at that location [79]. However, we must note that a generalized model cannot capture
all localized effects and therefore can only provide estimates of soiling and is not a replacement

71

for local experimental measurements. Nevertheless, an estimate based on local environmental
parameters is better than assuming a constant annual soiling loss or constant daily soiling rate.
This section discusses some of the existing research regarding the physical mechanisms of soiling
accumulation, explores existing semi-physical models, and proposes a general model that could be
used for such estimates, provided that the empirical parameters are tuned on a region-by-region
basis based on soiling observations.

5.2.1 Background

The physics of dust deposition and accumulation are complex due to many influencing factors,
ranging from weather, site, and system specifications to surface nano-characteristics as well as their
variation in time (Figure 5.1) [80]. Particle size, shape, and constituents, as well as deposition be-
havior and accumulation rates, can vary dramatically in different localities based on the geography,
climate, and urbanization of a region. Airborne dust concentration and rainfall are often considered
to be the major determinants of soiling. If sufficiently abundant, rain is effective at cleaning soiled
surfaces. However, rain can also increase soiling loss by depositing particles that have been washed
out of the atmosphere.

72

Figure 5.1: Soiling is a complex problem with many influencing factors over a large range of size
and time scales. Environmental parameters are shown in blue, microscopic soiling processes are
shown in green, and controllable influencing factors related to plant design and maintenance
strategies are in orange. Adapted from [86].

Wind speed also may have an important effect because it influences the particle deposition
mechanics. Wind speed determines the balance between deposition and resuspension since it
can increase both deposition of dust particles and their detachment from the surface. The bal-
ance between them is sensitive to many factors, such as the geometry of the PV system, airflow
characteristics, dust characteristics (size, shape, composition), and humidity. Multiple previous
reports demonstrated both net cleaning (wind-induced resuspension) and net soiling (additional
contribution from inertial deposition) effects of wind [87]. Some studies report that for conditions
where wind speed is less than 3 m/s, the particle deposition rate can be adequately accounted for
by sedimentation alone [76]. It has also been reported that wind is not very effective at removing
soiling for particles smaller than 50 microns because smaller particles have a larger adhesive force
and resist removal even at wind velocities greater than 50 m/s. Thus, higher wind velocity is
generally observed to deposit more dust on a PV module in a dusty environment [72]. Some have

73

unsuccessfully attempted to enhance dust resuspension during windy periods by increasing the
panels’ tilt angle to achieve a higher detachment force exerted by the wind [88]. Furthermore,
relative humidity, dew, and light rain can strongly enhance dust adhesion to PV surfaces through
capillary forces, particle caking, and cementation. These moisture-related adhesion mechanisms
can be important even in deserts where while the humidity is low, radiative cooling of glass surfaces
at night allows them to cool below ambient air temperature and reach the dew point [77].

One of the most comprehensive studies on the influence of environmental parameters on soiling
was performed by Micheli et al. at the U.S. National Renewable Energy Laboratory (NREL), who
searched for correlations between soiling and 102 environmental parameters at 20 PV sites [73].
Out of all parameters investigated, they found that the annual average of daily mean particulate
matter (PM) and precipitation pattern (quantified by the average length of the dry period) recorded
by stations near the PV systems are the best soiling predictors. PM is defined as a mixture of solid
particles and liquid droplets suspended in air. There are two types of PM generally reported: PM10
and PM2.5, representing the concentrations of airborne PM less than 10 microns and less than 2.5
microns in diameter in a cubic meter of air, respectively. Note that PM2.5 is a subset of PM10, with
these smaller particles typically remaining airborne and traveling longer distances than the larger
particles included in PM10 because of their lower mass. PM2.5 is the main cause of haze, which is
periodically reduced visibility in urban areas, causing a reduction in solar intensity and alteration
of the solar spectrum reaching the ground [89]. The NREL study found no clear conclusion about
whether PM10 or PM2.5 was a better predictor of soiling. A separate study performed in the
Atacama Desert supports NREL’s conclusions, finding that soiling at various sites was strongly
correlated with aerosol optical depth, which is known to be correlated to PM [90]. While these
works have shown that PM concentration seems to be a good explanation for variation in soiling
rates over large distances (geographic location), it has been reported to be inadequate to explain day-
to-day variations in soiling rates at a particular location because of other weather conditions [76].
To predict soiling over shorter periods such as weeks, days, or even hours, additional parameters
such as wind, RH, and ambient temperatures are thought to be important [91, 92].

74

The general approach of semi-physical dust accumulation models during the dry period is to
consider net dry deposition velocity. Dry deposition is defined as the exchange process of pollutants
from the earth’s atmosphere to surfaces in the absence of precipitation [74], and dry deposition
velocity (vd) is a way to express the rate at which particulate matter from the atmosphere encounter
a surface. Models predicting dry deposition are complex and require information that is difficult
to obtain; therefore, deposition fluxes are often estimated using PM and assumed dry deposition
velocities [93]. The simplest approach uses an empirically-derived net deposition velocity, assumed
to be a constant for each location. Then, mass flux accumulation rate (with dimensions mass per
area per time) is simply [94]

m = vdPM

(5.1)

Some studies use a weighted average of PM2.5 and PM10 concentrations [95], while others use
only PM10, motivated by the fact that coarse particles are dominant in soiling mass accumulation
[94]. Accounting for different particle sizes can improve accuracy, but measuring size distribution
requires special equipment, and equipment for measuring particles greater than 10 microns is scarce
[86]. More sophisticated approaches describe the dust accumulation as the net sum of competing
processes: dry deposition, rebound, and resuspension [76, 96]. Dry deposition, which is defined
in this context as the transport of particles to a surface regardless of whether they adhere, is often
further broken up into sedimentation (gravitational settling), and deposition due to turbulence and
boundary layer (quasi-laminar) effects, and expressed in terms of resistance analogs [74, 76, 86, 95–
99]. The total deposition velocity can be expressed as a sedimentation velocity in parallel with two
resistors that model the resistance to deposition of the turbulence (Ra) and resistance to deposition
in the boundary/surface layer (Rb)

vd = v

turb,bndr y
d

+ vsed

d

=

1

Ra + Rb

+ vs

(5.2)

A diagram of the approach is shown in Figure 5.2. Next, we individually discuss the main
mechanisms that contribute to soiling accumulation.

75

Figure 5.2: Schematic of particle dry deposition model resistance analogy. Adapted from [97].

Sedimentation

In aerosol physics, sedimentation is described by the Stokes terminal settling velocity which is
valid when particle Reynolds number is less than one (true for small particles). Only a very small
fraction of soiling particles have Reynolds number greater than one, so Stokes velocity alone can
be used to estimate sedimentation. The Stokes terminal settling velocity is the velocity of a particle
when the gravitational force Fg, drag force FD, and buoyancy FB force experienced by the particle
are in equilibrium. To derive the expression for Stokes velocity, a spherical particles with diameter
d is assumed. The balance of forces for terminal velocity dictates [100]

FB + FD = Fg

(5.3)

The buoyancy force is the product of air density ρair, particle volume V, and gravitational acceler-
ation g

(cid:19)3

(cid:18) d

2

FB = ρairV g =

4
3 π

ρair g =

π

6 ρair gd3

The aerodynamic drag on the particle is given by Stokes law

FD = 3π µvgd

76

(5.4)

(5.5)

where vg is the Stokes velocity that we are solving for and µ is the kinematic viscosity of air. The
force of gravity on the particle is

(cid:19)3

(cid:18) d

2

Fg =

4
3 π

ρpg =

π

6 ρpgd3

(5.6)

(5.7)

where ρp is the particle’s density. Substituting Equations 5.4-5.6 into Equation 5.3, we find

(ρp − ρair)gd3

18µ

≈ (ρp)gd3
18µ

vg =

where we have used the fact that ρair << ρp. To correct for the shape of real non-spherical
the Cunningham correction factor C (≈ 1 for
particles, two correction factors are often added:
particles larger than one micron), and the dynamic shape factor k (reported to be between 1.0 and
1.2 for soiling particles). The general equation for sedimentation velocity in the Stokes regime
[86, 101] is

ρd2
g
18µ

C
k

vg =

(5.8)
Many models for soiling set the correction for real particles shape C/k to one, which means they
assume that the spherical assumption is adequate, since often the proper particle shape correction
factor for a particular location’s soiling is not known. When considering the sedimentation of
soiling on PV modules, only those particles that deposit on the optical surface of the module should
be considered. Therefore, the tilt angle (θt) of the PV panel influences the particle sedimentation
velocity as follows [72, 79, 86, 97, 102].

vs = vgcos(θt) =

ρd2
g
18µ

C
k

cos(θt)

(5.9)

Turbulence and Boundary Layer effects

To account for wind turbulence and boundary layer effects, the atmosphere can be separated into
two layers: an upper layer where particle transport is described by an atmospheric turbulence, and
a quasi-laminar layer (also known as the boundary or surface layer) where particle transport is
governed by Brownian diffusion and inertial impaction. The contribution of these effects to the net

77

deposition velocity can be modeled by two resistors in series

turb,bndr y
d

v

=

1

Ra + Rb

(5.10)

where Ra = 1/(CDuwind) where CD is the surface drag coefficient and uwind is the mean wind
velocity. The product CDuwind represents the capability of wind to transfer kinetic energy to
particles in the air, with CD accounting for the turbulence when wind blows over rough surfaces
such as land and has a global yearly average value of 1.2 × 10−2 (dimensionless). Numerous
expressions exist in the literature for the boundary/surface layer resistance Rb [95, 97, 98]. The
expressions generally have the form

Rb =

1

u∗Pattach(Eb + Eim)

(5.11)

where Eb is the collection efficiency from Brownian deposition, Eim is the collection efficiency
from impaction, Pattach is the attachment probability, and u∗ is the friction velocity defined as

u∗ = uwind

κ

ln(zR/z0)

(5.12)

The logarithmic term accounts for the difference in wind speed at different heights, depending on the
roughness of the surface. zR is the reference height above ground where the wind speed is measured
(commonly 10 meters [97]), z0 is the surface roughness length (a length scale representation of the
roughness of a surface), and κ is the Von Karman constant that is widely agreed to have a unitless
value of 0.4. When wind hits a solar module, the number of particles that pass by the panel per
unit time increases proportionally to the wind speed. Therefore, resistance to wind-induced surface
deposition (Rb) is inversely proportional to wind speed.

Brownian deposition (quantified by Eb) describes the deposition of microscopic particles (less
than one micron) that diffuse in space due to random impacts with surrounding air molecules
and other aerosol particles and remain entrained in fluid streams [86]. Since such small particles
do not significantly contribute to the soiling mass accumulation, Brownian deposition can be
neglected. Very large particles only exhibit sedimentation, rapidly depositing from air regardless
of flow behavior. Medium-sized particles (about 10 micron in diameter, which is the upper limit

78

for PM10 measurements) experience viscous forces comparable to their own inertia and thus can
exhibit inertial deposition in windy conditions. In calm conditions, medium-sized particles deposit
through sedimentation [86]. The dominant size of particles accumulating on outdoor surfaces
has been observed to be 10-30 micron. Overall, PV soiling’s dominant deposition mechanism is
sedimentation, with inertial deposition contributing during windy conditions. The concept that
sedimentation is the dominant mechanism is supported by studies that found that the amount of
soiling on tilted collectors simply follows the cosine of the tilt angle, which means it simply
corresponds to the horizontal component of the collector’s surface area as expected by Equation
5.9 [72, 86].

Soiling particles can be carried by an airstream (wind) where impacts from the air molecules
keep them traveling with the airstream. If the flow hits an obstacle, like a PV panel, the streamlines
deviate from their original path (Figure 5.3). The airflow direction change is transferred to the
particles in the airstream by the air molecules impacting them. If the transferred momentum is
sufficient to overcome the particles’ kinetic inertia, the particles will follow the direction of the
airstream. If not, the aerosol particles cannot follow the air and will eventually hit the PV panel’s
surface, which is referred to as impaction or inertial deposition. The impaction collection efficiency
Eim is a function between 0 and 1, describing the proportion of particles carried past the surface
by wind that deposit on the surface due to impaction.

Figure 5.3: Schematic of deposition due to particle impaction. Adapted from [98].

79

Eim increases with increasing particle diameter since more massive particles have larger inertia,
which makes them more likely to leave the airstream at a bend in the flow. There are multiple
empirical-based step-like functions used for Eim such as

Eim =

St2

(1 + St) ∗ 2

(5.13)

was found for vegetative canopies [103], or some (e.g. [98, 102]) use a sigmoidal distribution

Eim =

1

1 + exp(− fim(St − 1))

(5.14)

where Fim is an empirical parameter that determines how steeply the function transitions from 0 to
1. Kim and Larson empirically found [104]

Eim = 10−2.8/St

(5.15)

All of these Eim equations are expressed in terms of the Stokes number, which characterizes the
behavior of particles suspended in a fluid flow and is defined as

St =

vsu2∗
g µ

(5.16)

[98] point out that only the component of the wind vector that points
Wolferststetter et al.
perpendicular towards the panel should be considered, thus impaction velocity is multiplied by
the cosine of the difference in angle between the wind direction and the panel azimuth orientation
(∆θaz,w).

The attachment probability is the probability that a particle that hits a surface will attach to
the surface instead of rebounding. Attachment probability decreases with increasing wind speed
and particle size since particles will rebound if their kinetic energy is greater than the adhesion
force with the panel surface [86]. For example, some field studies reported that impacting particles
detached when wind speed reached just 4 m/s [87]. There are no widely adopted models that
quantitatively describe rebound in arbitrary conditions. Wolferststetter et al. propose a logistic
equation for attachment probability, expressed in terms of a threshold particle diameter for rebound,

drb

Pattach = 1 −

1

1 + exp[−Crb(dp − drb)]

(5.17)

80

where Crb describes the steepness of the transition between attachment and rebound and

drb =

ζrb
uwind

+ wRH(RH)2

(5.18)

where RH is the relative humidity, and ζrb and wRH are constants determined during parameteri-
zation of the model. This model describes the physical effect of higher humidity causing particles
to adhere better to a surface, thus increasing the particle diameter threshold for rebound to larger
particles. The rebound threshold diameter decreases with increasing wind speed because particles
in higher wind speed have more kinetic energy, therefore reducing the mass required to give them
enough kinetic energy to overcome the surface adhesive forces.

Resuspension

Resuspension describes the process in which soiling that has already been present on the surface
of a PV panel for some time detaches from the panel due to wind overcoming the adhesion force.
Resuspension is very difficult to model because it varies greatly in natural conditions and is sensitive
to adhesion forces, particle and surface chemistry, and particle shapes [87]. One study found that
the wind velocity necessary for resuspension of particles from flat surfaces varied from 0.82 m/s
to 2219.8 m/s for different particles [105]. Fundamental chemistry studies of soiling adhesion
using atomic force microscopy found that particles with higher organic surface content have higher
adhesive force than particles with no organic surface content. They also found that cementation
occurs when high-organic particles are exposed to dew, high humidity, or light rain. They found
that soiling particles close to highways are contaminated with hydrocarbons from diesel fuel,
which tend to bind particles together and form cement-like films adhering to glass [106]. Such
localized effects make adhesion forces nearly impossible to predict in a generalized way, and thus
a generic resuspension equation that is empirically fitted to the observed resuspension may be the
best approach. A desert study in Qatar observed that resuspension strongly increases for wind
speeds above 4 m/s. They also observed a decrease in resuspension with increasing RH, with a
threshold of about 60%, above which almost no resuspension occurs [76].

81

5.2.2 Limitations of Existing Models

When Enlighten Energy Ltd., as part of our research collaboration, came up with the idea of
implementing a model that considers soiling based on environmental variables, the natural approach
was to start with a model from the literature. Guo et al.’s model [107] was chosen as one of the only
clearly explained and validated semi-physical soiling models in the literature. Here, we describe
the limitations which were found when trying to predict soiling using Guo et al.’s model.

For completeness, we will first outline the derivation of their model, which predicts soiling
losses due to particulate matter concentration (PM10), wind speed, and relative humidity.
It
considers the non-linear effects of dust deposition, resuspension of deposited dust, and the effect of
relative humidity on resuspension. Guo et al. define a cleanness index, which is simply a measure
of the relative performance in terms of the DC power output of a soiled PV array relative to a clean
array operating under the same solar irradiance. This is similar to what other researchers call the
soiling ratio. The semi-physical model assumes that the daily change in a PV panel’s cleanness
index is proportional to the difference between the daily average deposition and resuspension flux.
Dust deposition flux is proportional to the ambient dust concentration (PM10), while resuspension
flux is a function of wind speed and relative humidity. The dry deposition model by Kim and Larson
[104] has been adopted. Dry deposition velocity is expressed as a combination of gravitational
settling Vg and turbulent deposition Vt

Vd = Vt + Vg = uηdI +

ρpgdp
18µ

(5.19)

where µ is the dynamic viscosity of air, ρp is the density of an aerosol particle, and dp is particle
diameter. The friction velocity, u, is assumed to be directly proportional to the scalar wind speed,
usc, with the proportionality coefficient α1 determined by empirical fitting

The term ηdI is defined by

ηdI = exp

u = α1usc

(cid:19)

(5.20)

(5.21)

(cid:18)−α0gv

Vgu2

82

where α0 is an empirical parameter which controls the effect of turbulence on dust deposition
velocity, g is gravitational acceleration (9.81 m/s2), and v is kinematic viscosity of air (1.5 × 10−5
m2/s).

Therefore, the full expression for deposition velocity in terms of scalar wind speed is

Vd = α1uscexp

+ Vg

(5.22)

(cid:19)

(cid:18) −α0gv

Vgα

2
1u2
sc

Next, the dust deposition flux may be expressed as the product of deposition velocity and the
particulate matter concentration (PM10, which we label as P10 in the equations)

(cid:20)

(cid:19)

(cid:18) −α0gv

Vgα

2
1u2
sc

(cid:21)

Fd = P10

α1uscexp

+ Vg

(5.23)

Guo et al. use the following equation to model dust resuspension flux, taking into consideration
relative humidity (H), which has been reported to affect the threshold friction velocity for dust
resuspension.

β2(cid:2)α1usc − (uth,0 + β1H)(cid:3)α
1 + exp(cid:2) − 100(cid:2)α1usc − (uth,0 + β1H)(cid:3)(cid:3)

2
1u2
sc

Fr =

(5.24)

where the parameters β1 and uth,0 control the effect of relative humidity on dust resuspension. Note
that this is a logistic function (smoothed step-function). Due to the exponential in the denominator,
Fr approaches 0 when the friction velocity is less than the resuspension threshold, or mathematically
α1usc < (uth,0 + β1H). When the friction velocity is significantly larger than the threshold, the
denominator goes to 1. When the friction velocity is near the threshold, the exponential ensures a
continuous, smooth transition.

The net deposition flux is the difference between dust deposition and dust resuspension flux.

Therefore, the change in cleanness index for a 24 hour period is

∆CI = γ(Fr − Fd)

(5.25)

Table 5.1 lists the empirical parameters of the model.

83

Table 5.1: Empirical parameters of the Guo et al. model.

Parameter Value

γ
α0
α1
β1
beta2
uth,0
Vg

2.64384 × 106 m2s/kg
1.71 × 103 (unitless)
0.375 (unitless)
1.95 × 10−2 m/s/%RH
3.00 × 10−9 kgs2/m5
0 m/s
0.0199 m/s

An attempt to run the as-is Guo et al. model using weather and PM data for Riverside, California,
was unsuccessful. The model predicted very low levels of soiling deposition and high resuspension
flux such that almost no net soiling accumulation was predicted to occur (Figure 5.4a). The cause
of this seems to be the highly empirical nature of the model, which was developed for a site in
Doha, Qatar, with a high PM of 120 µg/m3. Guo et al.
tuned the resuspension flux equation’s
coefficients to be appropriate for this high PM, but the resuspension equation is independent of
PM. Therefore, when the model is used for Riverside, which has a much lower PM of 20 µg/m3
(typical of California), the resuspension flux overwhelms the deposition flux. In order to avoid this
issue, we tried simply setting the resuspension term to zero. However, without the resuspension
term to account for the fact that much of the wind-blown dust hitting a solar panel does not adhere,
the model overestimates wind-induced deposition, leading to unphysical spikes of several orders of
magnitude in the hourly soiling rate (Figure 5.4b).

84

Figure 5.4: One year of hourly soiling rates predicted by the Guo et al. model with (a) and without
(b) the resuspension term.

A third limitation is that the model does not consider the effect of the tilt angle. As we described
in the literature review above, numerous studies have found that soiling accumulation decreases
with increasing tilt angle and follows approximately a cosine law. The development of a model that
addresses these limitations is described in the next section.

5.2.3 Developing a General Soiling Accumulation Model

This section describes the development of a general semi-physical model for soiling estimation that
yields correct trends for locations worldwide. Note that this model is only intended to be used for
rough estimates of the possible soiling that a location may experience when soiling measurements
are not available (e.g. for pre-construction site planning). We propose data-driven models that
would use soiling measurements to provide more accurate soiling predictions in Section 5.8. Our
soiling model is based on many of the same principles as in Guo et al. and other literature models
discussed in Section 5.2.1. Ideas from several models were combined in an attempt to develop
a more general model that does not include too many site-specific features or require too much
detailed information about the soiling (e.g. particle sizes and chemistry) that is not practical to
obtain. The model describes soiling accumulation as the net of three contributing mechanisms:
sedimentation, wind-induced deposition, and resuspension. The equations for the three mechanisms
are presented below.

85

Sedimentation

Here we take the standard approach as described in Section 5.2.1 with the Cunningham and
dynamic shape correction factors set equal to one since it is impractical to determine these in many
situations. In contrast with the Guo et al. model, we include the influence of panel tilt angle (θt)
on sedimentation velocity. The velocity of sedimentation onto the tilted panel surface is

vs = vgcos(θt) =

ρd2
g
18µ

cos(θt)

(5.26)

where vg is the sedimentation velocity on a horizontal surface. In the initial implementation, the
particle density was assumed to be that of sand (1631 kg/m3) [108], and the particle diameter was
assumed to be 13 µm. The kinematic viscosity of air (µ) was taken at 25◦C, which is a reasonable
approximation of the average air temperature for a large portion of solar power stations and has a
value of 15.52 × 106 m2/s [109]. Kinematic viscosity varies by only about 0.6% per degree, so
using the constant temperature assumption is not expected to cause much error.

Wind-Induced Deposition

For considering additional deposition due to wind, we describe particle transport in terms of the
sum of an aerodynamic and surface resistance. Brownian diffusion is ignored since it is only
significant for very small particles. We use the expression for impaction collection efficiency from
Kim et al. [104], combined with the probability of attachment and azimuth orientation effect from
Wolfertstetter [98].

vd,wind =

cos(∆θaz,w)
Ra + Rb

=

cos(∆θaz,w)
1

1

CDuwind

+

u∗EimPattach

(5.27)

where ∆θaz,w is the difference between panel azimuth orientation and wind direction, and impaction
efficiency is given by

with

Eim = 10−2.8/St

St =

vsu2∗
g µ

86

(5.28)

(5.29)

For the surface drag coefficient, CD, the global yearly average value of 0.012 (dimensionless) is
used [97].

The attachment probability is described by

Pattach = 1 −

1

1 + exp[−Crb(dp − drb)]

(5.30)

where the parameterization constant Crb = 8×104 describes the steepness of the transition between
attachment and rebound and

drb =

ζrb
uwind

+ wRH(RH)2

(5.31)

where ζrb = 1 × 10−5 and wRH = 4 × 10−4 are parameterization constants.

Figure 5.5 shows the attachment probability as a function of particle diameter and wind speed.
As expected, we see that the attachment probability is nearly one for small particles and zero for
large particles, which have higher kinetic energy that overcomes the surface adhesion energy.

Figure 5.5: Attachment probability as a function of a) particle diameter (with wind speed of 1 m/s
and relative humidity of 70%) and b) wind speed (with particle diameter of 15 µm and relative
humidity of 50%).

Overall, the aerodynamic contribution is linear with wind speed, and the surface contribution
rises to a step-like behavior, where a minimum wind speed and diameter are needed for particles to
have enough inertia to exit the fluid flow and deposit by impaction (Figure 5.6a). The relationship
between the wind-induced deposition and relative humidity is almost linear for typical soiling

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particle diameters of 10-20 µm. For larger particles (e.g. 50 µm), the effect of rebound (described
by Pattach) begins to have a non-linear effect (Figure 5.6b), with wind-induced deposition staying
close to zero for low RH and rising more rapidly for higher RH.

Figure 5.6: Wind-induced velocity as a function of a) wind speed (for 10 µm particles in 40%
relative humidity) and b) relative humidity (for 50 µm particles at 2 m/s wind speed).

Resuspension

Resuspension is modeled based on Guo et al.’s model and assumed to occur only when the friction
velocity is larger than a threshold friction velocity. The resuspension flux (kg/m2/s) of soiling is
given by

Fresusp = β2u2∗(u∗ − u∗,th)

(5.32)

with the threshold friction velocity for resuspension linearly increasing with RH. The slope of
increase is based on the report by Neuman et al. [110], who found that the threshold wind velocity
(uth) needed for resuspension is 0.24 m/s at 0% RH and 0.31 m/s at 90% RH. The threshold
friction velocity is given by

(5.33)
where β1 = 1.95 and β2 = 3.0 × 10−10 kgs2/m2. Note that RH is defined as a decimal value
(i.e. 60% is entered as 0.6 in the equation). Resuspension flux is zero for wind speeds less

ln(z/z0)

κ(uth,RH=0 + β1RH)

u∗,th =

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than the threshold velocity and increases rapidly for wind speeds above the threshold (Figure
5.7). It decreases linearly with increasing relative humidity. Unlike wind-induced deposition, the
resuspension equation here does not consider wind direction, mainly because this effect is not yet
understood [105]. Carefully designed experiments will be needed to improve our understanding of
resuspension.

Figure 5.7: Resuspension flux versus wind speed for RH 40%.

Net Accumulation

The net soiling accumulation rate is found by multiplying the total deposition velocity by PM10
and subtracting the resuspension flux

(cid:219)maccum = P10(vs + wvd,wind) − Fresusp

(5.34)

where w = 0.1 is a dimensionless fitting parameter controlling the relative effect of wind-induced
deposition and sedimentation.

Figure 5.8 shows the dependence of net soiling accumulation flux on wind speed. At low wind
speed below the threshold for deposition due to impaction, the flux is only due to sedimentation
and, therefore, independent of wind speed. At moderate wind speeds, the flux increases linearly
with wind as more deposition due to impaction occurs. At high wind speeds, resuspension flux can

89

be (depending on the RH) higher than deposition flux, resulting in a net cleaning effect. The curve
shifts to the right with increasing RH

Figure 5.8: Net soiling mass accumulation flux at 40% RH, 50 µg/m3 PM10 concentration, and
20◦ tilt angle.

Comparison with Guo et al. Model

Figure 5.9 shows a comparison of the hourly soiling rates over one year for a fictional PV plant
in Riverside, California. We see that our model gives a more physically realistic result, showing
significant positive soiling rates with occasional negative soiling rates due to resuspension during
high-wind events. The basic shape of the curve is determined by the average PM values, with
additional wind-induced soiling appearing as the smaller noisy spikes on top.

Figure 5.9: Comparison of hourly soiling mass accumulation rates for Riverside, CA as predicted
by the Guo et al. model (a) and the improved model proposed in this thesis (b).

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This model was developed with very limited soiling data from only several sites and therefore is
mostly intended as a proof-of-concept and to provide rough estimates and trends. All parameters of
the model are expected to be needing region-specific tuning once reliable soiling data for a diverse
range of locations becomes available. The initial implementation of the model was tuned to be
consistent with soiling observed in California. With the development and industry adaption of
fully autonomous, low-maintenance soiling measurement stations (e.g. [111–114]), the availability
of reliable soiling data is expected to increase significantly in the near future, which would allow
region-specific tuning of such a semi-physical model.

5.3 Effect of Rainfall

The net effect of soiling losses is strongly dependent on rainfall. There is a consensus in the
literature that rain often effectively cleans solar panels; however, a minimum amount of precipitation
is needed for effective cleaning. Reports vary regarding the rain threshold. Also, rain can hurt the
soiling problem if the duration is brief and the intensity is light by depositing high concentrations
of residue onto a PV (e.g. water spots induced by soluble salt contained in the rainwater) or
providing water for a cementation process where soiling strongly adheres to the PV surface [72, 79].
Cementation is reported to occur in regions with high dust and humidity (possibly leading to
heavy morning dews). At high humidity, water-soluble particles form microscopic droplets of salt
solutions, which also retain some insoluble particles. When dried by evaporation, the precipitated
salt acts as a cement to anchor insoluble particles to the surface [90]. Soiling that has cemented
to the surface often cannot be removed without mechanical detergent scrubbing, thus increasing
the cost of manual cleaning [72]. Some examples of empirical observations of rainfall effects at
different locations are listed below (this list is not intended to be comprehensive)

• Rain events were found to clean soiling on panels (with the exception of bird droppings)
panels to within 1% of the manually cleaned ones. Losses due to bird droppings were
recovered to only 3% loss after heavy rain (9 mm) (Phoenix, Arizona) [82].

• Rain effectively cleaned modules as long as daily rainfall surpassed about 4-5 mm (Southern

91

Spain) [83].

• Cleaning action of rain generally only occurs when rainfall is more than 4-5 mm/day (Arizona)

[102].

• Relatively modest rainfall events of as low as 2.8 mm were sufficient to clean panel to a level
that restored power output to within 1% of the manually cleaned panel. (Portland, Oregon)
[115].

• Even light rain below 1 mm is enough to clean cover glass, reducing losses to below 5%.
Note that this study defines the soiling loss of a "clean" panel at a higher value (5%) than
most other studies [116].

• When comparing locations, some sets of locations had the same soiling rates but different
soiling losses, mainly due to different rainfall amounts. Precipitation of 0.5 mm was enough
to clean systems that had less than 10% soiling loss. For systems with higher soiling losses,
higher precipitation amounts were needed. (Atacama desert) [90].

• Precipitation amounts of 5 mm or less have the effect of promoting soiling, but amounts of

> 15 mm have a cleaning effect (Zimbabwe) [117].

• The rainfall threshold needed for effective cleaning varies from 0.3 mm to 20 mm. Variation
in values can be attributed to the complexity of surface cleaning by falling droplets, where
surface wettability, dust type, impact velocity of the droplet, surface incline angle, dust
adhesion state (cemented, caked, aged) all affect the cleaning effectiveness. Dew could also
clean the surface but has been observed to do so rarely (Review paper describing multiple
locations) [76].

• Soiling loss was reduced by 0.5-1%/mm of rainfall when daily rainfall was > 2 mm. Some

increases in soiling loss were observed when daily rainfall was <2 mm (Arizona) [118].

Given the wide variability in the observed effectiveness of panel cleaning by rainfall, our model
requires a user input (named Rainfall Clean Amount) that specifies the amount of rainfall that

92

results in a perfect or near-perfect clean. The quality of the cleaning effect of rainfall is specified
in terms of the soiling loss remaining after a rainfall event with precipitation amount reaching the
Rainfall Clean Amount. We assume that any rainfall less than the Rainfall Clean Amount results
in a partial cleaning that is proportional to the rainfall divided by the Rainfall Clean Amount. For
example, if 6 mm of rainfall is needed for a clean which leaves 5% soiling loss (Rainfall Clean
Quality = 5%), then 2 mm will result in a 0.33 · 0.95 = 31.35% reduction in the soiling amount.
Users can also specify a minimum hourly rainfall rate threshold needed to result in a cleaning effect.
Further details about the rainfall cleaning model parameters are described in Appendix A. Figure
5.10 shows an example of the reduction in soiling losses due to rain. Note that in contrast to the
soiling amount, the soiling losses, which are defined as an energy loss, are zero when there is no
sunlight.

Figure 5.10: Reduction in soiling losses due to rainfall. This plot is from Enlighten Energy’s PV
SoilSayer software, which contains an implementation of the soiling model described here.

The wetting of the ground due to rainfall has been observed in some studies to result in a slowing
or even temporary stopping of soiling accumulation for some time after the rainfall event. This can
be highly location-dependent and also dependent on the dominant source of soiling. For example,
if the source of soiling is dust from the ground, then it is likely that a wet ground will not result

93

in any significant soiling. However, if the source of soiling is pollution from a nearby factory or
pollen from nearby trees, this concept may not apply. Some models assume a Grace Period, which
is the period of time after a rainfall event where there is no soiling accumulation [81, 85, 90]. The
length of the Grace Period varies depending on the model, with typical values ranging from 5 to 21
days. Due to this effect’s location-dependence, if the Grace Period due to precipitation was used in
our model, it would have to be provided by the user based on local measurements. Since the Grace
Period phenomenon was only observed in a few studies (and even these studies observed it on only
some of their sites) and none of our measured soiling data supports the concept of a Grace Period,
we have not incorporated this into the model.

5.4 Incidence Angle Modifier due to Soiling

The losses due to soiling are also influenced by the angle of incidence (AOI) of the incoming
solar radiation. Multiple studies [82, 83, 90, 102, 116, 119] have found that the losses due to
soiling are significantly increased at larger AOIs. The AOI of direct solar radiation depends on the
sun’s position in the sky and the solar panel tilt angle, which is variable for panels that track the
sun (single-axis trackers). It may not be cost-effective to clean modules that track the sun due to
the lower impact of soiling in these modules due to the lower AOIs that they experience [82]. On
sunny days, losses have been found to be symmetric about solar noon, where they reach a minimum
[90, 116]. Changes in angular response are likely due to changes in the optical path length (OPL) of
the direct radiation through absorbers and scatterers in the dust layers over the PV module; a shorter
OPL (e.g. at noon) leads to less attenuation of direct irradiance that reaches the module surface,
while longer OPL leads to more attenuation, enhancing soiling effects [90]. On cloudy days, where
global irradiance is mostly composed of diffuse radiation, losses appear to be mostly independent
of AOI, probably due to the isotropic nature of diffuse radiation [116]. Since the proportion of
diffuse to direct solar irradiance varies from day-to-day, models that attempt to describe the IAM
due to soiling need to consider separately diffuse and direct irradiance.

Martin and Ruiz developed an experimentally-validated analytic model (we will call this the

94

MR model) that describes soiling-dependent losses due to incident light that is not perpendicular
to the solar panel surface [120–122]. These losses include angle of incidence effects and effects
due to the relative proportions of direct, diffuse, and ground-reflected solar irradiation.
In this
section, we will describe this model, and in the next section, we will combine the MR model with
the soiling accumulation model to achieve a more comprehensive model for soiling losses.

The MR model was obtained from the optical analysis of commercial PV modules of different
In order to accurately describe the angular losses, the MR model considers the
technologies.
following four solar radiation contributions: direct normal, diffuse horizontal-circumsolar, diffuse
horizontal-isotropic, and ground-reflected. Direct normal irradiance (DNI) is measured at the
Earth’s surface with a surface element perpendicular to the Sun and is equal to the extraterrestrial
irradiance above the atmosphere minus the atmospheric losses due to absorption and scattering.
Diffuse horizontal irradiance (DHI) is the radiation at the Earth’s surface from light scattered by
the atmosphere. Under the Hay and Davies diffuse model, DHI is divided into circumsolar and
isotropic components with circumsolar referring to the light that has the apparent origin in the
region of the sky around the sun, and isotropic referring to the remaining portion of diffuse that
is approximately isotropic across the sky. Ground-reflected irradiation is the irradiation received
from above a horizontal surface after it is reflected from the ground. It depends on the reflectivity
of the ground surface, known as albedo. Each of these irradiances can be further transposed to
calculate the plane-of-array (POA) irradiance, which is the portion of the radiation that hits a tilted
surface (in our case, the solar panel).

The short circuit current of a solar cell can be expressed in terms of the plane-of-array irradiance

from the four solar radiation contributions described above as

(cid:21)

(cid:20)

Isc =

E POA
dir
, E POA

dir

di f ,circ

di f ,iso(1 − FD) + E POA

alb (1 − FA)

(1 − FB) + E POA
, E POA
di f ,iso

di f ,circ(1 − FB) + E POA
, and E POA
alb

˜Isc
˜G
denotes POA direct, POA diffuse-circumsolar, POA
where E POA
˜Isc is the short-circuit current
diffuse-isotropic, and POA ground-reflected irradiance, respectively.
under a direct irradiance of ˜G, and the FB, FD, and FA are empirical angular loss factors that account
for soiling losses due to non-perpendicular incident light. To calculate the POA diffuse irradiance,

(5.35)

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the MR model uses the Hay-Davies transposition model [123]

E POA
di f

= E POA

di f ,circ

+ E POA
di f ,iso

= Edi f

(cid:20) Edir

Eetr

(cid:18)

cos(α)
cos(θz) +

1 − Edir
Eetr

(cid:19) 1 + cos(θT)

2

(cid:21)

(5.36)

where θz is the solar zenith, θT is the tilt angle of the solar panel, Eetr is extraterrestrial irradiance,
and α is the angle of incidence. Transposition of direct normal irradiance to POA irradiance is
given simply by

Ground-reflected irradiance can be expressed as

E POA
direct

= Edircos(α)

E POA
alb

= aG

1 − cos(θT)

2

(5.37)

(5.38)

where G is ground horizontal irradiance, which is the total amount of radiation received from above
by a surface horizontal on the ground, and a is the albedo coefficient.

Substituting Equations 5.36-5.38 into 5.35, we get the main equation of the MR model

(cid:20)

(cid:18)

˜Isc
Isc =
˜G
1 − Edir
Eetr

(cid:19) 1 + cos(θT)

Edircos(α)[1 − FB] + Edi f
[1 − FD] + aG

(cid:21)
cos(α)
Edir
cos(θz)[1 − FB]+
Eetr
1 − cos(θT)
[1 − FA]
2

2

+Edi f

The angular loss factors are defined by

FB(α) =

exp(−cos(α)/ar) − exp(−1/ar))

1 − exp(−1/ar)

(5.39)

(5.40)

(5.41)

(5.42)

(5.43)

(cid:19)2(cid:19)(cid:21)
(cid:19)2(cid:19)(cid:21)

(cid:20)

FA(θT) ≈ exp

(cid:20)

FD(θT) ≈ exp

(cid:18)

c1

(cid:18)
(cid:18)

− 1
ar

(cid:18)

(cid:19)
(cid:19)

(cid:18)
(cid:18)

+ c2

+ c2

sinθT +

θT − sinθT
1 − cosθT

sinθT +

θT − sinθT
1 − cosθT

− 1
ar

c1

sinθT +

π − θT − sinθT
1 + cosθT

sinθT +

π − θT − sinθT
1 + cosθT

where FA and FD are approximations to the solutions of integrals that consider the contribution of
each solid angle from which light is incident on the PV module.

96

The parameter ar is the angular losses empirical coefficient. The MR model was developed
such that a perfectly clean panel has ar = 0.16 − 0.17. A panel with "moderate dust quantity"
which is described as a 2% loss in short-circuit current, has ar = 0.20, and a panel with a "high"
amount of dust (8% loss in short-circuit current) has ar = 0.27. On clear days, angular losses are
dominated by the direct irradiance loss factor FB. Perfectly clean panels still have optical losses
at higher AOI due to the reflection of light by the panel surface. The losses significantly increase
with soiling. The diffuse and albedo angular loss factors only depend on ar and the panel tilt angle.
Since these describe losses due to scattered/reflected light, they are independent of the AOI. Figure
5.11 show these loss factors as a function of tilt angle for a clean and dirty panel. The albedo
angular loss function goes to one at zero tilt angles since there is no contribution to short-circuit
current from ground-reflected light when a panel is perfectly horizontal.

Figure 5.11: Empirical angular loss factors for direct (a), diffuse (b), and albedo (c) irradiance as
functions of angle of incidence for two angular loss coefficients (ar).

5.5 Combining Incidence Angle Modifier with Soiling Accumulation Models

The soiling accumulation models and MR soiling loss model can be combined to allow for
a predictive model of both soiling accumulation and its resulting energy production losses. We
realize that Martin and Ruiz’s soiling coefficient is approximately proportional to the amount of
soiling on the panel. Thus, while Martin and Ruiz use the soiling coefficient only to describe
angular reflection losses, we can also use it to describe the transmission losses due to surface area
coverage of soiling. We model the total hourly soiling losses (Lsoiling
) in terms of a transmission

tot

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loss (Lsoiling
trans
are multiplicative, with the overall loss being

) and additional angular reflection losses due to soiling (Lsoiling

). Percentage losses

ang

Lsoiling
tot

= 1 − (1 − Lsoiling
trans

)(1 − Lsoiling

ang

)

(5.44)

For the angular losses, we want to isolate the additional losses due to soiling from the total
angular losses to properly calculate the impact of soiling on PV power production. Therefore,
angular losses that exist on perfectly clean panels are excluded from the calculation of the soiling
loss. The overall contribution to soiling losses due to increased angular losses is expressed as the
product of the relative percent increase in angular losses due to soiling and the percent angular
losses of a clean panel.

(cid:18) Iclean

sc

Lsoiling
ang

=

(α)/ ˜Isc − Idirt y
sc
(α)/ ˜Isc
Iclean
sc

(α)/ ˜Isc

(cid:19)

(5.45)

where Isc(α)/ ˜Isc is found using the MR Equation 5.39. Note that by expressing the loss in terms of
the ratios of short-circuit current at an AOI to the nominal short-circuit current ( ˜Isc), the loss can
be calculated in terms of the irradiance components and angular loss factors, without knowledge
of the nominal short-circuit current. A value of 0.16 is used for the angular loss coefficient for the
clean panel’s Isc ratio. The dirty panel’s Isc ratio is calculated using an angular loss coefficient
estimated from the soiling mass accumulation model as described below.

We adapt our soiling accumulation model to calculate the daily accumulated soiling in terms of
an MR angular loss coefficient (we refer to this as the soiling coefficient) by scaling the accumulated
soiling mass to produce soiling coefficients that match with experimental angular soiling loss data.
The scaling coefficient is similar in concept to the γ parameter which is used to convert soiling
mass to clearness index change in Guo et al.’s model (see Section 5.2.2). A multiplicative scaling
of the soiling mass (in kg) by 28.5 was chosen. The scaling will likely need to be adjusted for some
geographic regions, due to different compositions of soiling. This adaptation can also be done with
other soiling mass accumulation models.

The hourly transmission loss is empirically related to the difference between dirty and clean
panel soiling coefficients. According to Martin and Ruiz, an angular loss coefficient of 0.27

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corresponds to a panel with 8% soiling loss at zero AOI, and a coefficient of 0.16 corresponds to
a perfectly clean panel. Therefore, an approximation can be made that for each 1% soiling loss
at zero AOI (this isolates the transmission portion of the soiling loss), the angular loss coefficient
must increase by c = (0.27− 0.16)/8 = 0.01375. The transmission loss due to soiling is defined as

trans = c(adirt y
Lsoiling

r

− aclean

r

)/100

(5.46)

For soiling accumulation models that express soiling in terms of a percentage energy loss, the
percentage is approximately converted to a soiling coefficient, based on the soiling loss at solar
noon (equivalent to the transmission loss) using Equation 5.46. In this way, we can use hourly
or daily weather data (rainfall, wind speed, PM10, and relative humidity) to calculate the hourly
soiling coefficients for use in the MR equations that compute angular soiling losses.

Figure 5.12 shows the soiling loss profile on a clear day as predicted by our combined model.
We can see the characteristic shape of the curve as reported in experimental studies where losses
are highest close to the beginning and end of each day due to the higher angle of incidences at these
times [124].

Figure 5.12: Hourly soiling loss profile over a single day for two different panels: a "no clean"
panel that was never cleaned and a "clean" panel that is periodically cleaned according to a
cleaning cycle. This plot is from Enlighten Energy’s PV SoilSayer software, which contains an
implementation of the soiling model described here.

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5.6 Model Validation

The semi-physical soiling accumulation model was validated by comparing its predictions with
a soiling study published by NREL [125].
In this study, NREL monitored the soiling rate and
precipitation for 20 sites. For verification, we selected Site 3 (Calipatria, CA) and 11 (Lamont,
CA) from the study, which had less noisy data and more significant observed soiling (many of the
sites had yearly average soiling ratios that were greater than 99%). Site 3 is a single-axis tracker,
while Site 11 is a fixed tilt (20◦) system, so this site selection allowed testing the soiling models for
both types of systems. Inputs to the soiling model were daily precipitation data provided in NREL’s
publication, hourly weather data for the year when the soiling measurements were taken downloaded
from NSRDB [126], and monthly average particulate matter data taken from the US Environmental
Protection Agency’s (EPA) database [127]. The daily precipitation data was converted to an hourly
format by dividing by 24. We compared the yearly average soiling rates, which are computed by
taking the average of the soiling rates during the dry periods. For both sites, the model’s soiling
rates matched with NREL’s measured values to within 0.01%/day (Site 3 had 0.1%/day and Site 11
had 0.13%/day).

Additionally, we compared the soiling accumulation time-series. Figure 5.13a shows the
comparison for NREL Site 3 when using unmodified input data, as described above. The predicted
soiling rate during the dry period matches the measurements, but there are discrepancies in the
effect of rainfall. One possible reason for these discrepancies are that the daily rainfall data lacks
information about rain intensity (mm/hr), while the same amount of daily rain can be more or less
effective in cleaning depending on its intensity. Secondly, NREL’s publication provides rainfall
data from an online database (PRISM [128]), instead of measuring it directly at the sites. This could
mean that the off-site PRISM data measured a different amount of rain than occurred at the site.
For a proof of concept, we adjusted the rainfall amounts of a few rain events where the measured
effect of rain was different than predicted (Table 5.2). Note that we did not change the main rainfall
pattern (dates when significant rain of more than 1 mm occurred).

100

Table 5.2: Daily rainfall adjustments needed to account for measured soiling ratio time-series for
NREL Site 3.

Date

Original
Rainfall (mm) Rainfall (mm)

Adjusted

Apr 24 1.04
Jun 8
0.354
3.64
Jul 18
0.4
Sep 7
Oct 15
3.1

0.13
1
4.64
0
7

Figure 5.13 shows the forecast with these adjustments to the rainfall input data. Changes to
decrease rainfall can be justified as follows. The rainfall could have been very light and distributed
throughout the day, leading to almost no cleaning effect. For example, often rainfall of less than 0.5
mm/hr does not result in a cleaning effect [118]. However, the dataset does not contain information
about rainfall intensity. Changes to increase rainfall can be justified by the chance that the rainfall
could have been higher at the site than the PRISM database value or more effective at cleaning than
predicted.

Figure 5.13: Comparison of model predictions with NREL Site 3 (Calipatria, California) when
using precipitation data provided by NREL (a) and when using modified rainfall data (b).

We repeated the same validation approach for NREL Site 11 (Lamont, CA) with the results
without and with adjustment to the rainfall shown in Figure 5.14. With this site, we had to add a

101

few partial rainfall clean events to account for the measured increases in the soiling ratio that did
not correspond to any recorded rainfall. The changes to rainfall are listed in Table 5.3.

Table 5.3: Daily rainfall adjustments needed to account for measured soiling ratio time-series for
NREL Site 11.

Date

Adjusted

Original
Rainfall (mm) Rainfall (mm)
Apr 29
0
May 24 0
0
Aug 15
Oct 8
0.65

0.25
0.75
0.2
2.3

Figure 5.14: Comparison of model predictions with NREL Site 11 (Lamont, California) measured
data when using precipitation data provided by NREL (a) and when using modified rainfall data
(b).

In conclusion, model validation showed an excellent agreement between measured and predicted
dry period soiling rates but a difficulty in predicting the effectiveness of rainfall in cleaning the
panels. Modifying the rainfall to change its effectiveness in cleaning allowed to account for most
of the discrepancies between measured and predicted data. This demonstrates the need for reliable
hourly rainfall data measured on-site and further research into understanding rainfall’s effectiveness
in cleaning panels as a function of the rain intensity and amount.

102

5.7 Integration into Commercial Software

The ultimate goal of PV soiling modeling is to evaluate the impact of soiling on a PV system’s
energy production and optimize panel cleaning schedules. Here we briefly describe the integration
of the soiling loss prediction models into a brand new commercial software, Enlighten Energy’s
PV SoilSayer. PV SoilSayer is a web application that provides users with soiling loss estimates
and optimal cleaning cycle forecasts in order to improve the accuracy of PV energy production
models, minimize operational costs, and maximize PV system performance [129]. It consists of
models from the open-source PV modeling Python package PVLIB (version 0.7.2) [130], soiling
models developed in this dissertation, and Enlighten Energy’s proprietary optimal cleaning cycle
algorithms, which find an optimal cleaning schedule for the power plant that will minimize financial
losses due to soiling, taking into account the cost of cleaning. PVLIB only has two soiling models:
the industry-standard Kimber model, which assumes a constant daily soiling rate and rainfall
cleaning threshold, and the newer HSU (Humboldt State University) model, which uses particulate
matter data and user-specified deposition velocity (or an assumed default value) to estimate the
effect of deposited mass on the soiling ratio through an empirical equation. Both of these models
have limitations. The Kimber model requires knowledge of the soiling rate, is not able to capture
seasonal changes in the soiling rate, and uses over-simplified assumptions about rainfall cleaning.
The HSU model does allow capturing local changes in soiling rates through the particulate matter
data, but requires knowledge of the deposition velocity (which can vary by location), does not
consider any effects of wind speed or humidity, and makes the over-simplified assumption that
panels are perfectly cleaned with each rainfall greater than 1 mm [95]. In an attempt to overcome
some of these limitations and gain better accuracy in predicting soiling losses and optimal cleaning
cycles on the daily resolution, the soiling loss models described in this chapter were integrated into
PV SoilSayer.

Figure 5.15 shows a flow-chart of the approach used to incorporate the soiling loss models that
were developed in this work into PV SoilSayer. The work-flow of using PV SoilSayer is as follows.
First, users select weather data (typical meteorological year, historical, or next year forecast) from

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one of several sources by entering the PV system’s location. Users then specify system design
parameters such as power ratings, tilt angle, module type, cell type, array type, and ground-reflection
coefficient (albedo). Users also specify the DC-to-AC inverter efficiency and other fixed losses
such as those due to wiring, age, light-induced degradation, shading, and snow. Next, users select
one of several soiling loss models with options including using the semi-physical model described
in this chapter, specifying constant daily soiling rates for each month, and uploading measured
weather and soiling data. Users also have control over the effects of rain, such as the amount of
rain necessary to significantly clean a panel, the minimum amount of rain needed to achieve a
partial reduction of soiling, and the amount of soiling left after non-perfect rain cleanings. These
inputs are expected to be determined from empirical observation. Finally, users can select one
of several cleaning cycle optimization algorithms. The soiling loss and cleaning cycle algorithms
predict a one year PV system performance forecast and an optimal cleaning schedule. The results
include a comparison (both through yearly values and dynamic yearly, monthly, and daily plots)
of the predicted losses when no panel cleaning is performed ("No Clean" case) and losses when
panel cleaning is performed according to the calculated cleaning schedule ("Clean" case). The
recommended calendar dates for cleaning are reported. Appendix A describes PV SoilSayer in
more detail.

Figure 5.15: Flow-chart of the approach used to incorporate the soiling loss models that were
developed in this work into PV SoilSayer.

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Figures 5.16-5.18 show an example results report from PV SoilSayer for a fictional 300 megawatt
solar power plant in Calexico, CA. In this sample forecast, the cleaning optimization predicts that
two manual clean events, on April 25 and June 13, results in the highest net revenue. We see from
the Total Value of Energy Gain that the first clean event is expected to save approximately $0.6
million and the second saves $1.1 million in soiling losses, before considering the cleaning cost of
$250,000 per clean. The Payoff Date shows the estimated date after which the gain in energy due to
cleaning will become greater than the cost of cleaning. The average daily and hourly soiling rates
during dry periods for each month are also shown. The daily soiling rates range from 0.16% to
0.33%, which is consistent with Kimber’s findings of daily dry period soiling rates of 0.1%− 0.3%
for California [81]. The average monthly soiling ratios for the plant if it is cleaned according to the
optimal cleaning schedule versus not cleaned are also shown.

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Figure 5.16: Beginning of PV SoilSayer results report, showing the forecast inputs and
comparison of the predicted annual energy production and soiling losses for a system that is never
cleaned ("No Clean" case) versus a system that is cleaned according to an optimized cleaning
schedule ("Clean" case).

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Figure 5.17: Optimal manual cleaning dates and soiling rates results page from a sample PV
SoilSayer forecast.

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The last portion of the report contains interactive plots for visually comparing the predicted
soiling losses, energy production, and energy losses for the plant with and without manual cleanings.
Figure 5.18 shows two of these plots.

Figure 5.18: Visualization of soiling losses and energy production from a sample PV SoilSayer
forecast. The sharp reductions in soiling loss correspond to manual and rainfall cleanings of the
panels.

One can also view plots that show the energy production and soiling losses for any month or
day in the year. Figure 5.19 visualizes the clean versus no clean system comparison for June. We
see the drop in soiling loss of the "Clean" panel starting on June 13, which is the recommended
cleaning date. The soiling loss drops over the course of ten days, since that is the user-specified

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time required to clean the entire power plant. Note that the clean panel soiling loss never drops
to zero because portions of the plant start accumulating soiling, while other portions of the plant
are still undergoing cleaning. This demonstrates the importance of considering the time required
to clean. After the cleaning event, we see a larger difference between the no-clean versus clean
panel’s energy production.

Figure 5.19: Visualization of soiling and energy production for a single month from a sample PV
SoilSayer forecast. A manual cleaning event occurs from June 13 to June 22, reducing the soiling
loss on the "Clean" panel and increasing the difference between "Clean" and "No Clean" panel
energy production.

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Effect of Inverter Clipping

Our model considers the reduction of absorbed irradiance due to soiling, instead of applying an
assumed soiling loss directly to the AC power output. Therefore, the model captures the impact
of inverter clipping (the situation where the DC power production is greater than the AC inverter
rating) on the AC power loss due to soiling. During times when inverter clipping occurs, meaning
that part of the DC power is not converted to AC power, any DC soiling loss that is lower than
the power loss due to clipping will not translate to an observable AC soiling loss. Thus on high
irradiance days when inverter clipping occurs, the daily total soiling loss will be lower than on
lower irradiance days. With alternating higher and lower irradiance days, the AC soiling losses will
oscillate, as seen in Figure 5.20.

Figure 5.20: AC soiling loss with oscillations caused by inverter clipping.

To confirm that these oscillations are, in fact, due to inverter clipping, we can rerun the same
forecast with the DC and AC system sizes set to be equal, thus removing the possibility of clipping.
We now see an approximately linear increase in soiling since now the AC soiling loss is equal to
the DC soiling loss (Figure 5.21).

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Figure 5.21: AC soiling loss for same forecast as Figure 5.20, but without inverter clipping.

Sensitivity to Weather Parameters

PV soiling is highly dependent on the weather; therefore, it is important to consider the variation in
soiling and optimized cleaning cycle predictions when weather predictions from different sources
are used. As an example, Table 5.4 compares the one-year soiling and optimal cleaning forecast for
a site in Lamont, California, when using four different weather resources available to users of PV
SoilSayer: Meteonorm 30 year TMY, AccuWeather last year, AccuWeather next year prediction,
and 10-year precipitation-based TMY generated from AccuWeather historical data.

Table 5.4: Variability of forecast for a site in Lamont, CA (NREL Site 11 from [125]) when using
different weather prediction resources.

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The different weather resources available for predicting next year’s weather vary significantly
(coefficient of variability 44%) in their precipitation prediction, which impacts the soiling forecasts.
For California, historical data suggests that the past ten years have on-average been drier than the
past 30 years, which explains the low precipitation value in the precipitation-based 10-year TMY
compared with the 30-year TMY. However, it seems like last year (2019) had significantly more
precipitation than the 10-year median. AccuWeather’s prediction for next year shows precipitation
that is more similar to last year than the 10-year TMY. The different weather resources also find
different lengths of the summer dry period, ranging from 4-7 months and differing in the start and
end month. As a result of these differences, the predicted soiling losses of a never cleaned panel are
5.5%-7.7%, and the expected revenue gain due to cleaning is 2.1%-4.4%. The average daily soiling
rate is nearly identical for all weather resources since it is defined during dry periods only and most
strongly dependent on particulate matter data that was taken from the same EPA data source for all
four forecasts. Figure 5.22 compares the one-year soiling loss time-series prediction when using
the four different weather resources. We see that the overall soiling pattern is consistent, but the
soiling loss amounts and start and end of the summer dry period vary.

Figure 5.22: Comparison of soiling time-series predictions for a site in Lamont, CA (NREL Site
11 from [125]) when using different weather prediction resources.

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While long-term weather prediction is outside the scope of this thesis, these results show that
soiling loss prediction accuracy is limited by the accuracy of the weather prediction (especially of
precipitation patterns). Even with a perfect soiling model, soiling forecasts may be very inaccurate
if the year’s precipitation pattern prediction is inaccurate. Therefore, future soiling models need
to be combined with advanced long-term weather forecasting. (AccuWeather claims to provide
this with their Next Year forecasts, but we could not find any peer-reviewed validation of their
predictions). Additionally, it is important to consider the potential error due to weather predictions
when interpreting soiling forecast results. Knowing the forecast’s uncertainty can be just as
important as the forecast itself since it allows investors, PV developers, and maintainers to make
decisions based on the relationships between probability, risk, or reward. For example, we cannot
be sure that a certain manual clean will save revenue (e.g. there could be a strong rainfall that cleans
the panels right after investing in the manual cleaning). However, if the probability that a cleaning
rainfall occurs during a traditionally dry period is very low and the soiling loss during this period
is high, the risk to reward comparison would suggest that the plant be cleaned. In future soiling
and cleaning optimization software, the forecasts should be rerun with 10th and 90th percentile
precipitation predictions to bound the soiling prediction and show the possible variations in the
optimal cleaning cycle.

5.8 Predicting Soiling with Machine Learning

Even with reliable soiling data, semi-physical models will likely need to be custom-tuned to
each region, and the equations that describe soiling in one region may not be fully applicable
to another region. While optimization packages for deriving equation parameters are readily
available, attempting to explain a very complicated process with a particular set of equations may
over-constrain the problem and limit the ability of the model to predict soiling at some locations
where non-typical processes may be occurring (e.g. additional soiling from a nearby factory, farm,
or highway). Additionally, it is difficult, costly, and time-consuming to conduct experiments that
isolate the coupled effects of particulate matter, wind speed, humidity, rainfall, and tilt angle, which

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would be needed for a careful empirical fit of the equations. Therefore, once soiling data becomes
widely available, data-driven machine learning (ML) approaches such as multiple linear regression
(MLR) and artificial neural networks (ANN), where the predictive models are trained on site-
specific historical data, will likely be preferable. Previous studies have employed MLR and ANN
for soiling modeling and were able to achieve good prediction accuracy [91, 107, 117, 131–134].
In this section, we explore two ideas for using ML for soiling modeling. The first approach is to
use ML for predicting a soiling mass accumulation rate from environmental variables, and then
use a semi-physical model (based on known/measured optical properties of soiling) to predict the
soiling loss based on the accumulated soiling mass. This approach could allow for the development
of ML-based soiling accumulation models on a regional level that can be used to calculate a
soiling loss based on each PV site’s configuration (considering properties such as PV panel type,
orientation, inverter efficiency). In the second approach, one would use ML to directly predict the
power production loss due to soiling based on environmental variables. This approach would likely
be site-specific since both environmental variables and site configuration influence the soiling loss.
In this section, we discuss the development of proof-of-concept MLR and ANN models using the
above two approaches.

5.8.1

Introduction to MLR and ANN

The multi-variable linear regression approach aims to build a simple linear equation that relates
a set of independent variables (i.e. predictors) to the dependent variable (i.e. response variable).
The response variable y is expressed as a linear function of environmental variables (x1, ..., xi)

y = a0 + a1x1 + a2x2 + ... + ai xi

(5.47)

The values of coefficients resulting in the smallest least-squares difference between modeled and
measured results are found. A disadvantage of the MLR approach is that it assumes linear rela-
tionships between the independent variables and the response, which is not necessarily true for
soiling.

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An artificial neural network model is a mathematical representation of an interconnected system
of neurons that interchange communication among themselves [132]. For curve fitting, ANNs are
used to capture the relationships between a set of numerical input data and a set of numerical
target data. An ANN consists of multiple layers of neurons: an input layer, one or more hidden
layers, and an output layer (Figure 5.23). The input layer receives the input data points. A hidden
layer manipulates the incoming data using an activation function, G, to define the output of each
ith hidden neuron based on the received inputs. Finally, the output layer receives the processed
information from the last hidden layer and computes the final prediction using

Nhidden


i=1

y =

(cid:174)βiG( (cid:174)wi (cid:174)xi + bi)

(5.48)

where (cid:174)wi is the input weights vector that connects inputs to each ith hidden neuron, bi is the input
bias of each hidden neuron, and (cid:174)βi is the output weights vector that connects the outcomes of each
hidden neuron to the output neuron.

Figure 5.23: Diagram of an ANN. From [135].

For evaluation of machine learning model predictions, three metrics are often used: coefficient
of determination (R2), mean squared error (MSE), and accuracy defined in terms of the mean
absolute percent error (MAPE). The R2 value describes the variability in the output based on the

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input variables as

R2 = 1 −

n

n


i=1

i=1

(ymeasured,i − ypredicted,i)2
(ymeasured,i − ¯ymeasured)2

(5.49)

where the ¯ymeasured is the average of all the measured values. A value of R2 = 100% means that
all variability of the data is explained by the input variables, while a value of R2
< 100% means
that other unaccounted factors are affecting the target data. The MSE is

n

n


i=1

1
n

i=1

MSE =

1
n

The accuracy is defined as

Accuracy = 1 − MAPE =

5.8.2 Soiling Mass Accumulation

(ymeasured,i − ypredicted,i)2

(5.50)

(cid:18)

1 −

(cid:12)(cid:12)(cid:12)(cid:12) ymeasured,i − ypredicted,i

ymeasured,i

(cid:12)(cid:12)(cid:12)(cid:12)(cid:19)

× 100%

(5.51)

Due to a lack of reliable experimental data of soiling mass accumulation, this approach was
tested on data generated from the semi-physical soiling model. Since the soiling model represents
realistic soiling patterns, if the data-driven approaches (which receive no knowledge of the physical
equations) are able to replicate the results of the soiling model, there is a reasonable chance that
they will also be able to replicate measured soiling data. To develop the machine learning approach
work-flow, we treat the soiling model’s results (from a forecast for Riverside, California) as real
measured data. The first step in developing a machine learning model is to analyze the correlations
between the dataset’s features. Here, we looked at both the Pearson Product-Moment and Spearman
Rank-Order correlation coefficients by using the open-source Python packages Pandas and Seaborn.
The Pearson correlation coefficient evaluates the linear correlation between two variables. Figure
5.24 visualizes the Pearson correlations between all of the input environmental variables used for
the soiling model and the predicted soiling rate.

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Figure 5.24: Heatmap of Pearson correlation coefficients between environmental variables and
predicted soiling rate.

This heat map can elucidate the interactions between variables in a dataset. For example, we
see a strong positive correlation between PM10 and air temperature. This could make sense since
California (where the weather data is from) has less rainfall in the summers, which results in higher
PM10 levels. Relative humidity is negatively correlated with air temperature and irradiance (GHI,
DNI, DHI), which can again be explained by the drier summers. It is important to remember that
the existence of a correlation between two variables does not necessarily mean there is a causality
relationship. For example, while air temperature could be causing lower humidity, it likely does
not directly cause a rise in PM10. Since our goal here is to develop an ML work-flow for predicting
the soiling rate, we treat the predicted soiling rate as if it was a measured quantity and observe
the correlations with other environmental variables. Looking at the correlation heat map, we see
that soiling has a significant positive correlation with PM10 and a negative correlation with wind
speed. Wind direction and relative humidity are found to have no linear correlation with the soiling
rate. However, we cannot assume that there is no relationship between soiling and environmental
parameters with near-zero Pearson correlations because the relationships may be non-linear. We

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should also evaluate the Spearman Rank-Order correlation coefficients, which measure how well
a relationship between two variables can be described by a monotonic function. Figure 5.25
visualizes the Spearman correlations in our dataset.

Figure 5.25: Heatmap of Spearman correlation coefficients between environmental variables and
predicted soiling rate.

As expected, the variables with strong Pearson correlations also have strong Spearman correla-
tions since a line is a simple form of a monotonic function. We also now see a significant negative
correlation between the soiling rate, relative humidity, and air temperature. Since the Pearson
correlation between RH and soiling is zero, but the Spearman coefficient is significant, we can
conclude that these two variables are non-linearly related. This is expected since the soiling model
that this data was generated with includes RH in non-linear terms only. Since air temperature and
humidity are also strongly correlated, we chose to use only humidity for training our models (it is
generally desirable to use independent variables for training an ML model).

The following approach was used for creating the MLR and ANN models. First, the soiling data
(generated from the semi-physical model) was normalized by dividing each variable by the mean
of that variable. ANN requires normalization to achieve good results. The same normalization

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was also done on MLR for consistency in data preprocessing, although it was found that using
normalization did not have any effect on the MLR prediction. Then, the data was split 80%:20%
into training and testing data. Four independent variables that are believed to be the dominant
predictors of soiling were chosen for the models: PM10, wind speed, RH, and the difference
in angle between panel azimuth and wind direction. The quantity to predict was the hourly
soiling accumulation rate. The MLR and ANN models were run using the LinearRegression
and MLPRegressor (Multi-Layer Perceptron) methods from the Python package Scikit-Learn. The
MLP Regressor used the default parameters except for changing to the L-BFGS solver (documented
to work better than the default for smaller datasets) and using three hidden layers of ten neurons
each. The numbers of hidden layers and neurons to use were determined through a series of
5-fold cross-validation trials performed on the training data (80% of the dataset), searching for the
values that gave the best prediction metrics (R2, MSE, and MAPE). K-fold cross-validation is the
procedure of further splitting up a training dataset into k groups, and repeating model training k
times, where each time a different group is used for testing and the rest of the groups are used for
training. By changing the parts of the dataset that are used for training and testing, cross-validation
calculates more representative model accuracy metrics than if only a single test:train split was used.
To evaluate the importance of PM10, wind speed, wind direction relative to the panel, and
relative humidity variables for the MLR and ANN predictions, the prediction accuracy was com-
pared when the models were trained with one, two, three, and all four of these variables. For the
single variable training, PM10 was chosen since it has the highest positive Pearson and Spearman
correlations to the soiling rate. Wind speed was added as the second variable since it is the only
other variable that has a strong linear correlation with the soiling rate. The Spearman correlations
show that relative humidity has a significant correlation with soiling, so it was added as the third
variable. While showing only small correlation coefficients, we added wind direction as the fourth
feature because we know that the soiling model attempts to include its effect.

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Table 5.5: Comparison of ANN and MLR performance when different numbers of features
(environmental variables) are used for model training.

Features

PM10

R2

MSE

Model
MLR 0.119 1.03 × 10−6
1.03 × 10−6
ANN 0.119
MLR 0.379 7.24 × 10−7
PM10, wind speed,
1.73 × 10−7
ANN 0.851
PM10, wind speed, MLR 0.379 7.23 × 10−7
1.63 × 10−8
RH
PM10, wind speed, MLR 0.381 7.22 × 10−7
1.25 × 10−8
RH, wind direction ANN 0.989

ANN 0.986

1-MAPE
59.5%
59.6%
69.3%
86.9%
69.1%
93.7 %
69.6%
94.6 %

When just using PM10, the ANN does not perform any better than MLR. The accuracy of
MLR and ANN predictions both increase when including both PM10 and wind speed. As could be
expected, adding variables that have non-linear correlations with soiling (relative humidity and wind
direction) result in no improvement for the MLR model, but do improve the ANN model, which is
capable of capturing non-linear relationships. The prediction accuracy had almost no significant
improvement when adding wind direction, which showed only a small Spearman correlation with
soiling. The lack of dependence of soiling on wind direction, even when the model which was used
to generate this data includes wind direction, can be understood by the fact that wind’s contribution
to resuspension is dominating over its contribution to deposition. We know this from the negative
Pearson correlation between soiling and wind speed. Since the model only includes wind direction
for wind-induced deposition, we do not see its effects. This exercise demonstrates that looking at
significant Spearman correlations allows choosing a set of features for an ANN model that results
in high prediction accuracy.

Figure 5.26 shows comparisons of predicted hourly soiling rates for Riverside, California when
all four environmental variables were used. With all four variables, the ANN model achieved a
much higher prediction accuracy, higher R2, and lower MSE than the MLR model. We see that the
MLR model struggles to predict many of the values because it attempts to find linear relationships

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between soiling and environmental variables when the real relationships are non-linear. MLR is
unable to capture the periodic spikes of large negative soiling rates due to resuspension at high
wind conditions. In attempting to capture these spikes, the MLR model found a negative linear
coefficient between wind speed and soiling, incorrectly implying that wind speed always gives rise
to a reduction in soiling. MLR also has a wide scattering of points missing the actual data when
trying to predict soiling at lower wind speeds. In contrast, ANN can reasonably capture the negative
soiling rate spikes and achieve a very good agreement with actual values at low wind speeds, as
seen by the thinner band of points that better match the actual values.

Figure 5.26: Comparison of hourly soiling rates predicted by MLR and ANN models to rates
predicted by the semi-physical model (with resuspension) that they were trained on. Only data
points from the testing set are shown.

The soiling data generated for the above ML exploration assumed that a strong resuspension
occurred at high wind speeds, resulting in a strong cleaning effect and spikes of negative hourly
soiling rates. The actual resuspension depends strongly on RH, particle chemistry and size, and PV
surface properties. It is possible that no significant resuspension occurs at many PV sites and many
soiling models neglect resuspension. Therefore, we retrained the ML models using simulated data
from a version of the semi-physical model without the resuspension term. Generating data without
resuspension allows us to test MLR and ANN on a different dataset that exhibits different trends.
The main difference is the lack of spikes, resulting in a smoother curve. In contrast with the last
test, we find that both MLR and ANN achieve similar prediction accuracies of about 98%.

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Figure 5.27: Comparison of hourly soiling rates predicted by MLR and ANN models to rates
predicted by the semi-physical model (without resuspension) that they were trained on. Only the
test data points are shown.

Table 5.6: Statistical metrics comparing MLR and ANN model performance when using a dataset
generated with the semi-physical soiling model without the resuspension term.

Model
MLR
ANN

MSE

R2
0.983 3.47 × 10−9
0.982 3.49 × 10−9

1-MAPE
97.8%
97.8%

These results confirm that both ANN and MLR are promising approaches for predicting soiling
mass accumulation as a function of environmental variables. For sites where soiling phenomena
are highly non-linear (e.g.
those experiencing non-linear resuspension due to wind), the ANN
approach is expected to achieve more accurate predictions than MLR. Additionally, we have shown
that analyzing statistical correlations of results from physical or semi-physical models can be a
method for verifying their qualitative behavior.

5.8.3 Soiling Loss

To investigate the possibility of using machine learning to directly predict future soiling loss from
historical soiling and environmental data, the approach was tested on approximately 1.5 years
(February 2019 to June 2020) of measured data from a PV power plant in California. Data from
2019 was used to predict soiling in 2020. The following environmental parameters were measured

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with an hourly resolution at the plant: PM10, relative humidity, wind speed, wind direction,
rainfall, and air temperature. Soiling loss was measured using a standard soiling measurement
station comprised of two panels, one of which is regularly manually cleaned, while the other is
allowed to accumulate soiling. The ratio of the short-circuit current of the cleaned and dirty
panels was measured every minute. Soiling loss measurement is most reliable when there is high
irradiance; therefore, only soiling measurements taken between 9 am and 3 pm (local time) were
used for the analysis. Additionally, unphysical outliers in the soiling and weather data were removed.
Following previous literature studies of soiling data [73], rainfall data was used to generate a dry
period length time-series variable, which is simply equal to the number of days that have passed
since the last rainfall.

First, the Spearman Rank-Order correlation coefficients between the variables in the dataset
were calculated. The correlations between a ten-day moving average of all variables are stronger
than between hourly or daily values. This is likely because the daily fluctuations in the variables
make it more difficult to capture relationships between them. Since we are mostly interested in
long-term soiling loss trends (e.g. the soiling loss after a few weeks of exposure), ten-day moving
averages were used for the rest of the ML model development. Figure 5.28 shows a heat map of
Spearman correlation coefficients between ten-day moving averages of the variables.

Figure 5.28: Heatmap of Spearman correlation coefficients between ten-day moving averages of
soiling and environmental variables measured at a PV power plant in California.

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Next, the MLR and ANN models (from the previous section) were trained on measured data
from 2019 using dry period length, PM10, and relative humidity as the input variables. Wind
speed and wind direction were excluded since the correlations analysis above showed almost no
correlations of these variables to soiling loss in this dataset. The air temperature was also excluded
since a large period of missing temperature data in 2020 prevents using this variable to predict
soiling. Also, the correlation between air temperature and soiling is likely a side effect of more
soiling in the summer, and this effect is already captured by the dry period length and relative
humidity variables. As a baseline comparison, we also attempted to fit the soiling data using the
industry-standard model from Kimber et al. [81] (see Section 5.1), which uses a constant soiling
rate and assumes that the soiling drops to some minimum value at each significant rainfall event.
Figure 5.29 compares the attempt to fit the training data with the three different models. The MLR
and ANN models were able to fit the training data significantly better than the Kimber model.

Figure 5.29: Comparison of Kimber, MLR, and ANN models’ ability to fit the training data
(measured soiling and environmental parameters for 2019).

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Since the ability to fit existing data does not necessarily indicate any predictive ability, we tested
the models’ performance when predicting soiling in 2020 based on the measured environmental
parameters (Figure 5.30). Note that the models were trained solely on 2019 data; therefore, they
predict soiling in 2020 based only on the learned relationships from the previous year. The models
have trouble predicting soiling when it is low (e.g. in February and March), possibly due to the low
signal-to-noise ratio in those measurements. In practice, the PV industry is interested in predicting
significant soiling losses (those which will require manual cleaning). Therefore, we computed two
sets of prediction accuracy metrics: one for the entire 2020 region for which there was measured
data (February 25 through June 16) and one for the portion of the region where there is more
significant soiling loss (April 23 through June 16). The artificial neural network was found to have
the highest prediction accuracy.

Figure 5.30: Comparison of Kimber, MLR, and ANN models’ ability to predict soiling in 2020,
when trained on data from 2019. Statistical metrics for two different prediction periods are shown:
February 25 through June 16, and April 23 through June 16.

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These results suggest that it may be possible to predict future soiling loss at a given site based
on measurements of soiling and environmental parameters. However, we must remember that
the prediction of future soiling depends on our ability to predict future environmental parameters,
which can be challenging. While we may not know the exact values of next year’s environmental
parameters, we may be able to make reasonable estimates based on typical weather patterns or
anomalies (e.g. we may know that next year is expected to be wetter/drier than normal). Reliable
historical weather data and medium/long-term weather forecasting will be important to enable such
soiling predictions.

5.9 Conclusions

In this chapter, we explored both semi-physical and machine learning approaches to modeling
the soiling of PV panels. After reviewing the main physical mechanisms of soiling and limitations
of existing semi-physical soiling accumulation models, we developed a general model that is
expected to be reasonably applicable globally, provided that the empirical coefficients are fitted on
a region-by-region basis. Additionally, we developed a novel practical approach of integrating a
soiling accumulation prediction with an existing validated model for optical losses due to soiling
(the Martin and Ruiz model). The overall photovoltaic system soiling model was validated by
comparing with published soiling loss studies performed by NREL. We helped integrate the semi-
physical soiling models with a Python package for PV energy production prediction (PVLIB) and
cleaning cycle optimization algorithms, which resulted in the release of a brand new commercial
software (described further in Appendix A). Finally, considering the inherent limitations of semi-
physical modeling of highly coupled and complex environmental processes, we explored machine
learning approaches to soiling modeling. The feasibility of these approaches was demonstrated by
showing that MLR and ANN can replicate the results of semi-physical soiling models and predict
future soiling loss based on historical measurements of soiling and environmental parameters. This
motivates the need to install soiling, weather, and PM10 sensors at PV system sites where soiling
is a significant issue. The models and software that we developed are expected to help PV system

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developers, maintainers, engineers, and researchers have a better understanding of soiling losses at
their current and potential future sites, as well as have access to easier-to-use and more-accurate
soiling analysis tools.

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COUPLED THERMAL-ELECTRICAL MODELING OF INTEGRATED PV SYSTEMS

CHAPTER 6

Less than 20% of the solar irradiation incident on a typical PV system is converted into electricity,
with the rest being transformed into heat, causing an increase in solar cell temperatures. When
measuring and predicting solar cell performance, a standard value for cell temperature is often
used (e.g. 20◦C or 25◦C). However, the temperature of solar cells in operation are often 20-40◦C
higher than the standard value [136]. A solar cell’s efficiency generally decreases as the operating
temperature increases, primarily due to increased internal carrier recombination rates caused by
increased carrier concentrations [137]. Electrical PV models, which are often developed and fitted
to experimental data under standard test conditions, need to be corrected to consider the impact of
temperature on electrical performance. Therefore, to predict a PV system’s power generation in
realistic conditions, the cell temperature must be estimated. In this chapter, we compare empirical
and physical approaches for cell temperature predictions and describe the development of a PV
thermal-electrical modeling approach that makes use of a commercial general heat transfer solver.
Finally, we demonstrate example applications of the model to simulation of residential PV and
vehicle-integrated PV, as well as, coupling of the PV model to a thermal-electrical model for
batteries.

6.1 Empirical Models

A common approach to considering temperature-induced losses when calculating PV perfor-
mance is to first predict cell temperatures with empirical models, and then use these predicted
temperatures in equivalent circuit models. In this work, we tested six empirical models for cell
temperature, three of which (Faiman, PVsyst, and SAPM) are implemented in the open-source PV
simulation package PVLIB. These models are briefly introduced below.

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Basic NOCT Model

Most empirical cell temperature models use a PV characteristic called Normal Operating Cell
Temperature (NOCT). NOCT is the temperature of a free-standing cell under open-circuit condition
under plane-of-array (POA) irradiance of 800 W/m2, ambient air temperature of 20◦C, and 1 m/s
wind speed [138]. The most basic empirical model assumes that the cell temperature is simply the
ambient air temperature plus the difference between NOCT and 20◦C (reference temperature for
NOCT measurement) multiplied by the ratio of POA irradiance (E) to the NOCT irradiance of 800
W/m2 [139].

TC = Tair + (NOCT − 20) E
800

(6.1)

Bizzarri et al.

Bizzari et al. proposed a model that takes into account any additional convection due to wind
speeds (WS) that are greater than the NOCT-specified 1 m/s [139]. The relationship between wind
speed and convection is controlled by a convection coefficient h.

TC = Tair +

(NOCT − 20)

800 + h(WS − 1)(NOCT − 20) E

(6.2)

Faiman

The Faiman model accounts for convection by assuming a simple two-coefficient (uc and uv) linear
convection expression [140].

TC = Tair +

E

uc + uv × WS

(6.3)

PVsyst

The model implemented in the software PVsyst is the Faiman model with an additional explicit
consideration of absorption α and panel efficiency η [130].
αE(1 − η)
uc + uv × WS

TC = Tair +

(6.4)

129

The implemented model uses a default value of uv = 0.

Sandia Array Performance Model

This model accounts for potential differences between module and cell temperature. The module
temperature is expressed as the sum of the air temperature and an exponential term [130].

Tm = Ee(a+b×WS) + Tair

The cell temperature is calculated from the module temperature by

TC = Tm +

E
E0

∆T

(6.5)

(6.6)

where ∆T is the empirically-estimated difference between module back surface temperature and
cell temperature. Table 6.1 shows the empirical parameters used in the SAPM model.

Table 6.1: SAPM temperature model coefficients.

Module Type
Open-Rack Glass-Glass
Open-Rack Glass-Backsheet
Rooftop Glass-Glass
Rooftop Glass-Backsheet

b

a
-3.47 -0.0594
-3.56
-0.075
-2.98 -0.0471
-2.81
-0.0455

∆T
3
3
1
0

Dufffie and Beckman

The Duffie and Beckman model attempts to account for both wind convection and module efficiency
[141].

TC = Tair +

E
800

a

(uc + uvWS)(NOCT − 20)(1 − η)

(6.7)

Model Comparison

In order to evaluate the accuracy of the empirical models, we compared their temperature predictions
with measured module temperatures from a solar power plant in California. This comparison was

130

first made using the original (default) coefficients that are specified by the empirical models, which
is a common way the models might be used by engineers who are looking for a quick estimate
of PV energy production. One exception to the use of default coefficients is that the NOCT and
efficiency values were taken from the manufacturer datasheet. Note that other model coefficients are
not generally available on manufacturer datasheets. The comparison of modeled versus measured
results for three days is shown in Figure 6.1 and Table 6.2.

Figure 6.1: Comparison of empirical models using default parameters.

Table 6.2: Errors of empirical models for results shown in Figure 6.1.

Model
Basic NOCT
Bizzarri et al.
Faiman
PVsyst
SAPM
Duffie and Beckman

R2
0.782 6.33
0.926
3.69
0.923 3.75
0.93
3.58
0.912
4.03
0.852 5.22

RMSE MAE
5.02
2.78
2.82
2.56
2.94
4.03

For a better statistical comparison, the module temperatures were predicted for an entire year
and compared with measured data in Table 6.3. (The results plot is not shown because there are
too many data points to plot the data effectively).

131

Table 6.3: Errors of empirical models for entire year using default parameters.

Model
Basic NOCT
Bizzarri et al.
Faiman
PVsyst
SAPM
Duffie and Beckman

R2
0.908 5.18
0.953
3.70
0.952 3.75
0.959
3.44
0.955
3.63
0.915 5.00

RMSE MAE
3.97
2.38
2.42
2.68
2.60
3.60

Seeing that some of the model predictions appear to be simply shifted from the measured
data, for the second comparison, multivariable optimization was used to find empirical model
coefficients (including NOCT) that best fit the data. The optimization was done using the Python
SciPy package’s optimize.least_squares() function with bounds specified on parameters to
maintain physically realistic values. The comparison of modeled versus measured results for three
days is shown in Figure 6.2 and Table 6.4.

Figure 6.2: Comparison of empirical models with parameters fitted using least-squares
optimization.

132

Table 6.4: Errors of empirical models’ results shown in Figure 6.2.

Model
Basic NOCT
Bizzarri et al.
Faiman
PVsyst
SAPM
Duffie and Beckman

R2
0.981 2.51
0.986
2.15
0.986 2.15
0.986
2.15
0.986
2.15
0.986 2.15

RMSE MAE
2.06
1.86
1.86
1.86
1.86
1.86

The error between one-year module temperatures predictions from the fitted model are compared

with measured values in Table 6.5.

Table 6.5: Errors for one year of module temperatures predicted by empirical models using fitted
parameters.

Model
Basic NOCT
Bizzarri et al.
Faiman
PVsyst
SAPM
Duffie and Beckman

R2
0.977 2.80
0.980
2.62
0.980 2.62
0.980
2.62
0.980
2.61
0.980 2.62

RMSE MAE
2.16
1.93
1.93
1.93
1.92
1.93

When using optimization to fit the empirical coefficients to the data, all models except Basic
NOCT result in cell temperature predictions that are almost identical.
In conclusion, multiple
empirical models can accurately predict free-standing solar module temperatures if their coefficients
are fitted to measured data.

6.2 Physical Thermal-Electrical Model

6.2.1

Introduction and Motivation

In the last section, we have shown that empirical models can accurately predict free-standing solar
module temperatures, provided they are fitted to measured data. In cases where measured data

133

is unavailable, the default values provided in PV modeling libraries/software are typically used.
However, using default empirical model coefficients can lead to significant errors in temperature
predictions. A validated physical model can allow to more-accurately estimate cell temperatures for
use in temperature-dependent solar cell performance models, such as those used within mainstream
PV modeling software. A physics-based approach to cell temperature and electrical performance
prediction is to couple an electrical PV model with a first-principles heat transfer solver.

Another application of a heat transfer solver could be to predict NOCT for modules, given the
module’s known thermal properties. This application is motivated by several limitations of NOCT
measurements listed below:

• Normal "operating temperature" is defined at open-circuit condition at which the device is
not producing any current. A device that produces current has an associated efficiency (e.g.
10-25%), which means that some of the solar energy is converted into electricity instead of
being absorbed as heat. Therefore, the NOCT temperature is expected to be higher than a
device which is producing power [138].

• NOCT does not apply to non-freestanding modules (e.g.

roof mounted), where there is

generally less convective cooling, resulting in higher temperatures.

• NOCT measurement typically requires waiting for very specific outdoor conditions, which
in some locations may only occur on several days a year. An NREL study found that many
manufacturers may not be reliably measuring NOCT when performing a side-by-side test of
three modules of the same type with manufacturer-reported NOCT values that differ by 10◦C.
NREL found that the true NOCT of these modules varies by less than 0.2◦C, as would be
expected for modules of the same type. Running a heat transfer simulation, they concluded
that the variation in manufacturer reported values could be due to changing sky, ground, and
ambient temperatures during the measurements [142].

• NOCT provides no information about how temperature varies with wind or irradiance.

134

Finally, a physical model based on heat transfer can be used to predict transient temperatures of
modules when they are integrated into various structures (e.g. on buildings, vehicles, or combined
with a heat exchanger as part of a PV-thermal system). Using an empirical approach would
likely require fitting the empirical model for each specific structure, with coefficients changing
depending on the design’s thermal properties. This approach requires many experiments to gather
cell temperature data. Utilizing a physical model can avoid the need for these experiments and
allow for rapid computational testing of PV system designs. Additionally, the heat transfer approach
allows one to predict not only the module temperature but also any effects of the solar panel on
the temperatures of the building or vehicle on which it is mounted. Such a model can allow
for computer-aided engineering of solar applications without the need to test every design idea
experimentally.

6.2.2 Methods

In this work, a coupled physical thermal-electrical model of a PV was developed using the com-
mercial heat transfer software TAITherm (ThermoAnalytics, Inc.) [143], which uses a numerical,
finite volume approach based on first principles physics. The PV model is based on the works of
Sommerfeldt and Schon [144, 145], who used TAITherm for modeling photovoltaic-thermal (PVT)
systems, with several significant differences. Sommerfeldt’s work was focused on reverse engineer-
ing heat transfer coefficients for the PVT system based on measured data, so the electrical portion
of the PV was not modeled, but instead, the measured power was used to set a negative imposed
heat on the PV layer. In Schon’s work, the electrical portion was modeled with a five-parameter
equivalent circuit, but the non-linear equation for current was solved in a brute force inefficient
way using two nested "while" loops. Schon reports that the simulation took 10-20 minutes per
time-step, and it is likely that the inefficient electrical solution implementation contributed to this
unsatisfactory performance. This performance limitation would make simulating a PV module
over the course of a 12 hour day with a time-step of 1 minute (typical for transient heat transfer
simulations) take 120 hours. Most users would desire a faster simulation time.

135

This work developed a proof-of-concept extension to TAITherm for PV modeling. Improve-
ments over Schon’s work include a more efficient implementation for solving the equivalent circuit
equations, accounting for the dependence of power generation on the AOI of solar radiation, and
allowing to consider the effects of soiling. In some use-cases, one is only interested in approxi-
mating the potential power generation of a generic PV system, and the electrical parameters are
unknown (e.g. evaluating the potentials of future PV technologies). For these cases, an option was
implemented to estimate the power generation using the efficiency, incident solar radiation, and
AOI dependence. The following sections detail the calculations used to compute power, angular
dependence, and the effect of soiling losses.

Computing Power Using Equivalent Circuit Model

The five-parameter equivalent circuit describes a simple current balance where the current reaching
the module contacts (usable current) is the difference between photogenerated current, IL, and the
current lost inside the cells [146]

I = IL − ID − Ish

(6.8)

where the current losses are referred to as the shunt current, ID, and the leakage current, Ish. The
current through the shunt diode (ID) describes the current lost due to radiative recombination of
minority carriers at the semiconductor junction under forward-bias voltage. The leakage current
describes current that is lost for reasons other than radiative recombination at the semiconductor
interface. This can be due to pinholes between the layers in the cell, allowing an alternate path
for the current. A series resistance, Rs, is used to describe the resistance related to finite carrier
mobilities and imperfect contacts between the active layer and electrodes. Figure 6.3 shows the
five-parameter equivalent circuit representation of a solar cell.

136

Figure 6.3: Five-parameter equivalent circuit representation of a solar cell. Adapted from [147].

The current through the shunt diode is modeled by the Shockley diode equation

ID = I0e(V +IRs)/a − 1

(6.9)

where I0 is the reverse-bias saturation current and a is the modified diode ideality factor, which is
given by

a = NsnVt

(6.10)

where n is the diode ideality factor, which takes into account the deviation of the diode from the
Shockley diffusion theory, Ns is the number of cells in series, and Vt = kBT/q is the thermal
voltage. Using Kirchoff’s loop law on the equivalent circuit in Figure 6.3, we see that

so

V + IRs − IshRsh = 0

Ish =

V + IRs

Rsh

Combining the above equations, the current-voltage relationship is

I(V) = IL − I0(e(V +IRs)/a − 1) − V + IRs
Rsh

(6.11)

(6.12)

(6.13)

where the five empirical circuit parameters are IL, I0, a, Rs, and Rsh. The temperature-dependence
of these parameters in terms of the reference values (denoted by the subscript ref ) is described by

137

the equations below

a = are f

IL =

S
Sre f

I0 = I0,re f

Rs = Rs,re f
Sre f
S

Rsh =

Rsh,re f

T
Tre f
M
Mre f

(cid:18) T

Tre f

(cid:2)IL,re f + αIsc(T − Tre f )(cid:3)
(cid:12)(cid:12)(cid:12)(cid:12)Tre f
(cid:19)3

(cid:18) Eg

− Eg
T

(cid:20) 1

exp

k

T

(cid:19)(cid:21)

(cid:12)(cid:12)(cid:12)(cid:12)T

(6.14)

(6.15)

(6.16)

(6.17)

(6.18)

where S is the plane-of-array irradiance, αIsc is the temperature coefficient of Isc (available
on manufacturer datasheets), M
is the air mass modifier (taking into account deviations from
Mre f
standard AM1.5 solar spectrum), Eg is the semiconductor bandgap, and kB is Boltzmann’s constant.
Note that the bandgap is also temperature-dependent with the relationship typically described
empirically. For example, the temperature-dependence of silicon’s bandgap can be described by

Eg
Eg,re f

= 1 − 0.0002677(T − Tre f )

(6.19)

The non-linear equation for current (6.13) can be solved efficiently by using the fsolve function
from SciPy. fsolve is a wrapper around the highly-optimized Fortran library MINPACK’s hy-
brd and hybrj non-linear equation root-finding routines. The current, voltage, and power at the
maximum power point (P(cid:48)
mp) are determined by simply finding the current and voltage in the IV
curve whose product is maximum. This power is scaled by area to account for any discrepancy
between the solar cell area that was used by the manufacturer to determine the empirical electrical
parameters (Amanu f ) and the area used in the TAITherm model (AT AI).

Pmp = P(cid:48)

mp

AT AI
Amanu f

(6.20)

Since equivalent circuit parameters are measured for a perfectly clean module under normal (per-
pendicular to the surface) incidence of solar radiation, the power calculated here is also assuming
normal incidence and no soiling losses. Corrections for AOI-related losses and soiling losses can
be applied as described in Section 6.2.2.

138

Equivalent Circuit Parameter Extraction

One potentially challenging aspect of using the five-parameter equivalent circuit approach is finding
the values of the five parameters. These parameters are not readily available from manufacturer
datasheets and are generally found with a combination of numerical solutions and fitting of IV
curves [148]. We did not have access to IV curve data for this work and therefore needed a solution
that could rely solely on the manufacturer datasheet. We used the method developed by Cubas et al.
[14], which is a mostly analytical approach, leaving just a single equation that needs to be solved
numerically.

This approach starts by evaluating the IV equation (6.13) at short-circuit current, open-circuit

voltage, and maximum power point conditions. At short-circuit current, we get

Isc = IL − I0(eIscRs/a − 1) − IscRs
Rsh

(6.21)

Cubas et al. neglects the second term (diode term) in this equation since at short-circuit condition,
it is much smaller than the other two terms. This allows to rewrite the expression as

IL =

Rsh + Rs

Rsh

Isc

Evaluating Equation 6.13 at the open-circuit condition, we get
0 = IL − I0(eVoc/a − 1) − Voc
Rsh

Combining Equations 6.22 and 6.23, we can write the saturation current as

I0 =

(Rsh + Rs)Isc − Voc

RsheVoc/aVt

Evaluating Equation 6.13 at the maximum power point gives

(6.22)

(6.23)

(6.24)

Imp = IL − I0(e(Vmp+ImpRs)/a) − Vmp + ImpRs

(6.25)
where Cubas et al. approximated e(Vmp+ImpRs)/a − 1 ≈ e(Vmp+ImpRs)/a because of the relative
magnitudes of the terms. Combining Equations 6.22, 6.24, and 6.25, an expression that does not
depend on IL or I0 can be derived
Isc − Voc − RsIsc

(cid:18)Vmp + ImpRs − Voc

− Vmp + ImpRs − IscRs

(cid:19)(cid:20)

exp

Rsh

(cid:19)(cid:21)

(cid:18)

Imp = Isc −

Rsh

Rsh

(6.26)

aVT

139

Another equation can be generated by applying that fact that the derivative of the power P = IV

is zero at the maximum power point. Differentiating by parts, we have

(cid:12)(cid:12)(cid:12)(cid:12)mp

(cid:12)(cid:12)(cid:12)(cid:12)mp

∂P
∂V

= V ∂I
∂V

+ Imp = 0

(cid:12)(cid:12)(cid:12)(cid:12)mp

∂I
∂V

(cid:19)(cid:20)

= − Imp
Vmp

(cid:18)V + IRs

(cid:19)(cid:21)

aVT

(6.27)

(6.28)

(6.29)

(6.30)

Next, we differentiate Equation 6.13 with respect to V and combine with the above equation to

get

dI
dV

= − I0
aVT

1 +

dI
dV

Rs

exp

(cid:18)

(cid:18)

(cid:19)

− 1
Rsh

1 +

dI
dV

Rs

Combining this with Equations 6.22, 6.24, and 6.26, an implicit expression for the series resistance
Rs can be derived

aVTVmp(2Imp − Isc)

(cid:18)Vmp + ImpRs − Voc

(VmpIsc + Voc(Imp − Isc))(Vmp − ImpRs) − aVT(VmpIsc − VocImp)
= exp

(cid:19)

aVT

This expression is solved numerically for Rs using an iterative solver (in this work we used Matlab’s
fsolve function). Next, Equations 6.26 and 6.30 are combined to find an expression for Rsh in
terms of known datasheet parameters and Rs

Rsh =

(Vmp − ImpRs)(Vmp − Rs(Isc − Imp) − aVT)

(Vmp − ImpRs)(Isc − Imp) − aVT Imp

(6.31)

Now that the two equivalent circuit resistances are known, the photocurrent and saturation current
are calculated using Equations 6.22 and 6.24. Figure 6.4 summarizes the procedure.

Figure 6.4: Flow chart of the Cubas et al. procedure to find the equivalent circuit parameters.

140

Two limitations of this procedure are that it relies on an estimate for the modified ideality
factor and only uses the manufacturer values at STC from the datasheet. Since the goal of the
equivalent circuit model is to model the electrical performance of the solar cells at various operating
temperatures, we should use the data for both STC and NOCT conditions (which is almost always
available) when extracting the model parameters to ensure that the model can capture temperature-
dependent effects. In order to improve the fit of the equivalent circuit parameters to the datasheet,
we ran the Cubas et al. method iteratively through a sweep of estimates for the modified ideality
factor (a), finding the value of a that results in the least sum of relative errors between the calculated
values and measured values of eight parameters that are available on datasheets (Vmp, Imp, Voc, and
Isc for both STC and NOCT conditions). Figure 6.5 shows our modified method. The algorithm
was implemented in Matlab with all non-linear equations solved using the fsolve function.

141

Figure 6.5: Flow chart of the improved equivalent circuit parameters extraction method.

Figure 6.6 shows sample results from the Matlab parameter extraction code. We see that the IV
curves at STC and NOCT change significantly during the iterative optimization process, with the
fill factor decreasing with increasing ideality factor.

142

Figure 6.6: Example of results from our modified parameter extraction tool. a) Convergence
behavior as the ideality factor guess is incremented. b, c) IV curves generated during the iterative
process, with the optimized ones plotted in red. Fill factor decreases as ideality factor increases.
d) Values of optimal parameters for this run.

Estimating Power from a Nominal Efficiency

In some cases, the user may not have measured electrical parameters or simply is not interested in
that level of detail for their model. Therefore, we also allow the user to use a simplified option of
estimating power production from a nominal power conversion efficiency (η in %) and a temperature
coefficient (βη = ∆η/◦C). Power production of a perfectly clean module under normal irradiance
is estimated as

Pmp = (η + βη(T − Tre f ))Pinc

(6.32)

As with the equivalent circuit approach, the estimated power production can be corrected to account
for angular dependence and soiling losses as described in the following sections.

143

Correction due to Angular Dependence

Angular losses are described in terms of an incidence angle modifier (IAM), which is the ratio of
transmission of light into a PV module when the light is incident at an angle to the transmission
of light at normal incidence. We account for angular losses by using the empirical model for
IAM developed by Martin and Ruiz [120–122] (introduced in Section 5.4). We chose this model
because it can also consider the angular losses on soiled PV modules. The full MR model requires
knowledge of the diffuse, direct, and albedo components of solar irradiation. TAITherm’s API
currently does not allow retrieving the albedo (ground-reflected) component of radiation since
TAITherm combines ground-reflected radiation and diffuse radiation into one quantity. Therefore,
we use a simplification, following the approach from a previous study of PV angle-dependence
[149], where we estimate the incidence angle modifier for diffuse radiation as the average of the
IAMs for direct radiation for all angles of incidence.

The MR model defines the IAM for direct irradiance as

Kdir
α

= 1 − FB(α) = 1 −

(cid:20) exp(−cos(α)/ar) − exp(−1/ar)

(cid:21)

1 − exp(−1/ar)

(6.33)

(6.34)

(6.35)

where FB is called the angular loss factor, α is the irradiance angle of incidence, and ar is the
empirical angular loss coefficient. Increasing the value of ar allows modeling larger angular losses,
like those occurring on soiled modules. For greater flexibility, we also implemented support for a
user-specified direct irradiance IAM vs. AOI curve that is spline-interpolated to find the IAM for
direct irradiance at any AOI. For both the empirical model and the user-defined direct irradiance
IAM curve, the overall IAM (Kα) is calculated by

Kα =(cid:2)E POA
=(cid:2)E POA

α

dir Kdir
dir Kdir

α

+(cid:0)E POA

+ E POA
di f
alb
+ E POA
di f ,T AIKdir

α

(cid:3)/E POA

total

(cid:1)Kdir
(cid:3)/E POA

total

α

dir

, E POA
di f

, E POA
alb

and are the plane-of-array (POA) direct, diffuse, and albedo irradiance,
where E POA
respectively. The average of Kα over all angles of incidence is denoted byKα. Note that the "diffuse
irradiance" in TAITherm (Edi f f ,T AI) is defined as the sum of diffuse-sky and albedo irradiance.

144

The IAM is always a value between zero and one and can be considered as an efficiency due
to incident angle. The power is scaled by the IAM at each time-step to account for the angular
dependence.

Pactual = PmpKα

(6.36)

where Pmp is the PV power computed through Equation 6.20 or 6.32. Equivalently, the PV can be
thought of as having an effective efficiency

ηe f f = ηKα

(6.37)

Correction due to Soiling

Soiling both blocks plane-of-array solar irradiation (transmission loss) and results in more severe
angular reflection losses. The Martin and Ruiz model (MR) accounts for the angular reflection
losses, including the influence of soiling, through an empirical angular loss coefficient.
In our
thermal-electrical model, a correction for soiling losses can be applied by providing a set of
MR angular loss coefficients or soiling losses (in percents) for each time-step in the simulation. If
percent soiling losses are supplied, the model expects that these losses already include the additional
angular losses due to soiling and applies the soiling losses directly to the power calculations, leaving
the angular loss coefficient at the clean panel value. If soiling is specified through angular loss
coefficients, the model estimates both the transmission losses due to soiling and the angular
reflection losses. The angular losses are calculated by the IAM equations described above. To
estimate a transmission loss due to soiling, the model uses an empirical correlation based on the
MR model. As discussed in Section 5.5, an approximation can be made that for each 1% soiling
loss at zero AOI (this corresponds to the transmission portion of the soiling loss), the angular loss
coefficient must increase by 0.01375. The transmission loss due to soiling is defined as

trans = c(adirt y
Lsoiling

r

− aclean

r

)/100

(6.38)

where the loss is a percentage expressed as a decimal, adirt y
are the angular loss
coefficients of a dirty and clean panel, respectively, and c is an empirical relationship relating the

and aclean

r

r

145

coefficients to transmission loss (with a default value of 0.01375−1 = 72.7 based on the empirical
relationships mentioned above).

The overall losses due to soiling and incident angle dependence can be expressed as a multi-

plicative combination of the transmission loss and angular loss.

Ltot = Lsoiling
trans

(1 − Kα)

The PV power when accounting for angular and soiling effects becomes

Pactual = PmpKα(1 − Lsoiling

trans

)

Equivalently, the PV can be thought of as having an effective efficiency

ηe f f = ηKα(1 − Lsoiling

trans

)

Implementation

(6.39)

(6.40)

(6.41)

The thermal portion of the modeling simply involves creating the PV system geometry and spec-
ifying the material properties and boundary conditions in TAITherm. The calculations for power
production, angular dependence correction, and soiling loss correction were implemented in Python
and coupled with the thermal model through a TAITherm user routine. User routines allow the
user to interact with the heat transfer solver through an Application Programming Interface (API)
to enforce custom boundary conditions or implement custom calculations. User routines can be
coupled with the solver at various points in the solution (beginning or end of the solution, each
time-step, or each iteration within each time-step of the thermal simulation). The PV calculations
user routine is called at the end of each time-step of the transient thermal simulation. This user
routine computes the power production (either through solving equivalent circuit equations or from
nominal efficiency) at the current time-step, using the newly computed module temperature while
accounting for AOI dependence and any soiling losses. Then a negative heat is imposed on the
PV module(s) to account for the portion of solar radiation that is converted to electricity instead of
heat. The results of the power production calculations are appended to an output file at each time
step. A flow chart of the modeling approach is shown in Figure 6.7.

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Figure 6.7: Flow chart of the calculations at each time step of the coupled thermal-electric PV
model.

Using the equivalent circuit approach allowed a 5572 element TAITherm (2020.1.0) model of a
vehicle with a solar roof to solve (without parallel processing) both the heat transfer and electrical
models in under one second per time-step (on a Linux machine with a 3.7 GHz Intel i7-8700K
CPU). Running the simulation under natural weather conditions over a 12 hour day with one hour
time-steps took under two minutes.

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6.2.3 Model Validation

Thermal Model

The thermal PV model was first validated by comparing the temperature prediction of a free-
standing PLM260M-60-DG series monocrystalline silicon module under NOCT conditions to the
value reported by the manufacturer (see Appendix B for the datasheet). The thermal model
was made in TAITherm, following the approach from Sommerfeldt and Schon’s previous works
[144, 145]. These works represented the PV module by a flat plate with three layers:
the PV
cells, plastic EVA encapsulate, and the rear glass. The front glass was omitted due to TAITherm’s
inability to model transparent and opaque layers together in a single part. Since the density of
glass and silicon are similar, a combined PV and glass layer was modeled by increasing the layer
thickness and adjusting the surface properties to account for the reflectivity, absorptivity, and
emissivity of the glass. McAdams linear wind convection [150] was used on the front and back
of the free-standing PV model. The layer properties are given in Table 6.6. Note that NOCT is
defined under open-circuit conditions; therefore, no electrical model was used for this validation
since no current is produced. TAITherm predicted a temperature of 43.9◦C, which is consistent
with the manufacturer-reported value of 44 ± 2◦C.

Table 6.6: Physical properties of the solar module.

Layer

PV Front Glass and Silicon
EVA
Rear Glass

Thickness Density Conductivity Specific Heat
mm
0.225+2.5
0.5
2.5

kg/m3 W/m − k
2330
960
6000

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0.7
1.8

J/kg − K
677
2090
500

Next, the qualitative relationships between cell temperature and irradiance, air temperature, and
wind speed were confirmed (Figure 6.8). TAITherm predicts that cell temperature varies linearly
with irradiance and air temperature, and inversely with wind speed, which is consistent with the
empirical models.

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Figure 6.8: TAITherm prediction of cell temperature variation with a) irradiance, b) air
temperature, and c) wind speed.

Equivalent Circuit Electrical Model

For initial verification of the electrical model, equivalent circuit parameters for the PLM260M-60-
DG series solar module were taken from Schon’s work (Table 6.7). (Note that these parameters
were derived in Schon’s work at the cell level and therefore Rs,re f and Rsh,re f are smaller than
what would be expected for the overall module value since the module is made up of 60 cells).

Table 6.7: Electrical parameters of the solar module.

Parameter Value
I0,re f
IL,re f
Rs,re f
Rsh,re f
nI
αIsc

1.0888 × 10−8
9.0507
0.004255
56.052
1.2
-0.06 %/◦C

While measured data of the temperature-dependent electrical properties were not available, the
temperature-dependent effects were verified qualitatively. Figure 6.9 shows the simulation results
for the solar module under weather conditions representing a sunny day in Phoenix, Arizona. As
expected, the photocurrent plot shows a Gaussian-like shape, with the highest current production
occurring during the highest irradiance. Transient cell temperature lags behind the peak irradiance

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(and current) since it takes some time for the cell to heat up after peak solar loading is reached.
We see the efficiency of the cell decrease by several percent when its temperature increases. The
silicon bandgap decreases with increasing temperature, while the ideality factor and diode saturation
current increase.

Figure 6.9: One day simulation results for a) cell temperature, b) current, c) efficiency, d) silicon
bandgap, e) modified ideality factor, and f) diode saturation current.

Figure 6.10 shows the impact of cell temperature on the IV curve when using the equivalent
circuit parameters from Table 6.7. As expected, we see a lower Voc and fill factor with increasing
temperature, resulting in lower power conversion efficiency. In this example, maximum power at
65◦C is 29% lower than at 15◦C.

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Figure 6.10: Example of the effect of cell temperature on a) IV curve and b) power production.

Coupled Thermal-Electrical Model

Finally, the coupled PV thermal-electrical model was validated against measured data (provided by
Enlighten Energy, Ltd.) of First Solar Series 4 CdTe single-axis tracker module temperatures and
short-circuit currents. Since measured values of the components of solar radiation, tracking axis
orientation, and the precise sun-tracking path for these modules were not disclosed, the angular
dependence of power production was not modeled. However, the angular effects of clean modules
that track the sun are small (< 2%) [151] and, therefore, would only slightly affect the results. Since
we are interested in the overall module temperatures, and temperature gradients across such a thin
material are small, the module was modeled in TAITherm as a single element plate with thermal
properties shown in Table 6.8.

Table 6.8: Physical properties of the CdTe solar module.

Thickness Density Conductivity Specific Heat Emissivity
mm
6.8

kg/m3 W/m − k
2451

6.2

J/kg − K
718

dimensionless
0.9

The density was calculated from the module area and thickness provided in the datasheet (see
Appendix B). The specific heat was estimated using a volume-weighted average of specific heat of
glass (750 J/kg-K [152]) and specific heat of CdTe (210 J/kg-K [153]). The thermal conductivity

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of CdTe (from [153]) was used. A sensitivity study showed that varying the specific heat between
500-2000 J/kg-K and conductivity between 3-150 W/m-k had a negligible impact on the results.
The specific heat affects only the relatively fast transient thermal behavior of a material, while here,
we are interested in panel temperatures on the hourly time-scale. Conductivity likely did not have a
significant effect because a free-standing panel’s thermal behavior is dominated by the convection
and radiation modes of heat transfer.

The measured dataset includes hourly measurements of POA total irradiance, GHI irradiance,
wind speed, ambient air temperature, module temperature, and short-circuit currents for four
nominally identical CdTe modules. The averages of the four modules’ POA irradiance, GHI
irradiance, wind speed, ambient air temperature were specified as boundary condition curves
for the following TAITherm user-inputs: imposed solar flux, fluid velocity for convection, fluid
temperature for convection, and environmental bounding box temperature. The equivalent circuit
parameters were derived using the parameter extraction method described in Section 6.2.2 with
the manufacturer nominal values for Vmpp, Impp, Voc, Isc, and temperature coefficient of Isc for
the FS-4122-3 module as inputs (see Appendix B for the datasheet). Table 6.9 shows the derived
equivalent circuit parameters.

Table 6.9: Derived equivalent circuit parameters of the First Solar 4 Series FS-4122-3 solar
module.

Units
Parameter Value
dimensionless
3.46
are f
1.855
A
IL,re f
1.317 × 10−11 A
I0,re f
3.984
Rs
1.390 × 103
Rsh,re f
+0.04
αIsc

%/◦C

Ω

Ω

The coupled thermal-electrical TAITherm simulation (as shown in Figure 6.7) was run using
0.1 hour time steps for a 15 hour summer day (June 27, 2019, was chosen arbitrarily). Figure 6.11
shows a comparison of the calculated and measured results for module temperature and short-circuit

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current. The error bars show one standard deviation of the measured values for the four modules.
(Note that module currents for the four cells were nearly identical for most of the day, causing
very small error bars). This validation shows that the approach of coupling a heat transfer solver
with an equivalent circuit model developed in this chapter can allow to accurately calculate module
temperatures and currents.

Figure 6.11: Comparison of one day of modeled and measured First Solar Series 4 CdTe module
temperature and short-circuit current.

6.2.4 Comparison with Empirical Models

Next, we present a comparison of cell temperature predictions from the popular empirical models
(described in Section 6.1) with those from the heat transfer physical model. For the empirical
models, we use the default coefficients that were derived by the developers of each model since
this is how PV engineers and technicians often use the models. Figure 6.12 shows the comparison
of cell temperature versus irradiance predictions under 1 m/s and 5 m/s wind. All models yield
similar temperature predictions under 1 m/s wind (NOCT conditions at which empirical models
are typically fitted), but diverge drastically under 5 m/s wind. As expected, the models that
do not include wind as an independent variable (basic NOCT and PVsyst) result in the highest
temperature predictions, showing the importance of considering variable wind convection if panels

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are installed at a location with varying wind speeds. Clearly, all of the empirical models cannot be
simultaneously accurate (at least when using default coefficients).

Figure 6.12: Comparison of cell temperature versus irradiance predictions under 1 m/s and 5 m/s
wind.

Figure 6.13 compares the cell temperature predictions as functions of air temperature and wind
speed (only the models that include wind speed as an independent variable were compared). Again,
we see a wide discrepancy between empirical models.

Figure 6.13: Comparison of cell temperature versus air temperature and wind speed predictions.

In order to check whether an empirical model can be fitted to match a heat transfer solver’s
results for a free-standing solar cell in operation, we modified coefficients in the PVsyst model and
compared predictions of cell temperature versus wind speed and irradiance to TAITherm results.
We see that excellent agreement between the heat transfer solver and the empirical model can

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be achieved, showing that it may not always be necessary to use a heat transfer solver. However,
benefits of the heat transfer approach remain for situations where measured data for empirical model
fitting may not be available and for rapid computational testing of different engineering designs
and panel integrations into structures.

Figure 6.14: Comparison of cell temperature predictions versus wind speed and irradiance made
with a fitted PVsyst model (ηm=18%, α = 0.9, uc = 23.2, uv = 7.4) and TAITherm.

6.3 Example Applications

Here, we present three proof-of-concept example applications of the thermal-electric PV model.

6.3.1 Building Rooftop PV Systems

The coupled thermal-electric model was used to predict the temperature-corrected power production
of a 64.8 m2 residential rooftop PV system in Phoenix, Arizona. The starting point for this model
was a TAITherm tutorial model of a house (Figure 6.15). PV panels were added to the geometry
by adding three PV material layers, with properties described in Table 6.6, to a portion of the roof.
For this proof-of-concept model, the PV system is assumed to be made of the same monocrystalline
silicon modules (PLM260M-60-DG series) as in Section 6.2.3.

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Figure 6.15: TAITherm house model with roof-mounted solar panel.

We first ran the model for one typical hot and sunny summer day in Phoenix (high temperature
of 41◦C). Weather data for Phoenix was taken from July 15 in the typical meteorological year
(TMY) dataset from the National Solar Radiation Database (NSRDB). To evaluate the impact of
temperature on the system’s energy production, we compared the model’s results when module
temperatures are computed versus when a constant temperature of 25◦C (standard test conditions)
is assumed (Figure 6.16).

Figure 6.16: PV rooftop system simulation results for a typical day in July in Phoenix, Arizona.
Predicted power (a) and efficiency (b) with and without considering temperature-induced losses.

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Summing the power over one day, the total energy production with and without temperature
correction is predicted to be 99 kWh versus 107 kWh, showing that if panel temperature is not
considered, the power production would be overestimated by 8%. The average household electricity
usage in the U.S. is 900 kWh per month or about 30 kWh per day [154]. Such a PV system producing
100 kWh on summer days would satisfy the household electricity needs, partially charge electric
vehicles that the residents may own (for example, a Tesla Model 3 uses about 26 kWh per 100 miles
[155]), and even have some energy remaining to sell back to the grid.

Additionally, the annual energy production was modeled using TMY weather data from the
NSRDB. TAITherm also needs cloud cover in order to estimate the sky temperature for the infrared
radiation exchange calculations. Cloud cover, which is not available in NSRDB TMY datasets,
was estimated as the ratio of DNI under realistic sky conditions to DNI under clear sky conditions
(provided in the NSRDB as part of its clear-sky model). Soiling loss coefficients for a typical year
in Phoenix for fixed tilt roof panels, with tilt angle equal to the roof’s tilt in the TAITherm house
model (40◦) were estimated by running the physical soiling model developed in Chapter 5 using
TMY data from Meteonorm [156]. (The NSRDB TMY data could not be used because it does not
include rainfall, which is essential for soiling prediction). The predicted DC energy production for
the rooftop PV system with and without a correction for soiling losses is shown in Figure 6.17.

Figure 6.17: Predicted PV energy production with and without including soiling losses. a) Daily
DC power. b) Monthly DC energy production.

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Figure 6.18 shows the soiling coefficients and the corresponding DC power production losses

due to soiling.

Figure 6.18: a) Soiling coefficients used as input. b) Transmission and angular losses due to
soiling calculated by the thermal-electrical model. Daily rainfall totals are shown with blue bars.

One could argue that similar results can be achieved with empirical-based temperature correc-
tions such as those in PVLIB. For simple energy production prediction of a typical rooftop PV
system, a heat and radiation transfer solver is not necessary. However, such an approach may be
useful for PV system engineering related to heat transfer, such as designing passive or active cooling
solutions to improve PV efficiency, designing PVT systems, or studying the rooftop system’s im-
pact on the temperature inside the house. Additionally, more complex integrations of solar panels
on irregular surfaces (e.g. vehicle-integrated solar panels, wearables, or solar-powered clothing)
cannot be modeled by standard PV modeling approaches such as those implemented in PVLIB.
To predict the energy production of such systems, a radiation transfer solver such as TAITherm is
almost essential.

6.3.2 Vehicle-Integrated Solar Panels

The inspiration for exploring this application was recent news on the development of prototypes of
electric vehicles with rooftop solar panels, which may extend the driving range or recharge the car’s

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auxiliary battery that powers secondary systems such as screens and climate control [157–160].
Currently, the amount of range extension is limited, but it still provides some benefits, and therefore,
there is active research in this area. Hyundai is also researching incorporating semi-transparent
solar panels into the panoramic sunroofs of internal combustion engine vehicles to directly charge
batteries and reduce the engine’s workload, which is claimed to be the first technology of this
type [158]. Vehicle-integrated solar panels (VISPs) could potentially provide a larger benefit on
commercial vehicles (e.g. trucks, buses, and trains) where there is more surface area available for
installing solar panels. Several variations of VISP systems were explored in this work. All of the
models were based on TAITherm’s tutorial model of a Jeep Grand Cherokee.

Conventional Rooftop PV

In the first version of the model, the entire surface area of the Jeep’s roof was covered by conventional
monocrystalline silicon solar panels (PLM260M-60-DG series) by adding the same three PV
material layers as in Section 6.2.3. Figure 6.19 shows the TAITherm model geometry.

Figure 6.19: TAITherm Jeep model with roof-mounted solar panel.

Like in the building rooftop PV modeling example, we first ran the model for one day (July 15
in the NSRDB TMY dataset for Phoenix). Figure 6.20 compares the results when different factors
affecting PV performance are included: irradiance-only, irradiance with angular dependency, and

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irradiance with both angular and temperature dependency. The AOI dependence in these results,
while having a significant impact on the efficiency near sunrise and sunset, decreases the day’s total
energy production by only 3%. This is because the sun stays high in the sky for most of the day
in the summer in Phoenix, and the PV panel is horizontal, leading to low AOIs that correspond to
almost no angular losses. The reduced efficiency due to angular dependence occurs when there is
low irradiance, so it has a low impact on overall energy production. Angular effects will be more
important in winter months, in Northern latitudes, and for PV installations where the PVs cannot
be mounted to minimize the AOI (e.g. building facades and pitched roofs that do not correspond
to an optimal panel tilt for that latitude). The effect of temperature dependence was explored by
comparing results when module temperatures are computed versus when a constant temperature of
25◦C is used. Like in the residential roof example, we again see a significant reduction in power
production due to temperature-induced losses during the peak sunlight hours. Summing the power
over an 8 hour period (9 am to 5 pm), the total energy production with and without temperature
correction is predicted to be 3.8 kWh versus 4.4 kWh, showing that if panel temperature was not
considered, the power production would have been overestimated by 17%.

Figure 6.20: Comparison of a single day of modeled vehicle-integrated solar panel power (a) and
efficiency (b) when different factors affecting PV performance are considered (irradiance
dependence only, irradiance and angle dependence, and irradiance, angle, and temperature
dependence).

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The VISP’s predicted energy production of 3.8 kWh during eight hours (e.g. a workday during
which the car is parked outside) could be used in several ways. A ventilation system could be run
while the car is parked in the sun to lower the cabin temperature and thus reduce AC use when the
occupants return to their car. Alternatively, the power could be used to charge an auxiliary battery
that powers secondary vehicle electronic systems (e.g.
infotainment, interior lights, electrical
plugs) or to power the vehicle owner’s electronics. Such a PV battery could be especially useful if
one uses the Jeep for rustic camping, allowing campers to charge electronics without running the
engine and without potentially discharging the main battery. On a potential future electric-SUV
with a similar roof surface area, the panels could even be used to slightly supplement the charge
in the main battery. Assuming an EV energy efficiency of 26 kWh/100 miles [155], such a PV
system could give an additional 15 miles of range after the car is parked for 8 hours in the sun. This
could be enough to use the EV for short-distance commuting without ever having to plug it into a
charging station.

Assuming an idealized scenario where the Jeep is always located in an unshaded area and the
PV panel is kept perfectly clean, a total annual energy production of 963 kWh could be achieved
in a typical meteorological year in Phoenix, Arizona (weather data from NSRDB). Assuming 26
kWh/100 miles EV efficiency, this would provide 3700 miles of solar-powered range a year, which
is about 1/4 of the average driver’s annual mileage in the U.S. [161]. Of course, the actual amount
will likely be lower due to shading and soiling, but this simulation provides us an upper-bound
estimate of what is possible. The simulation shows that integrating a conventional PV in an SUV’s
roof is not sufficient for a fully solar-powered car for the average driver. A PV technology that
could allow using a greater surface area of the vehicle is explored in the next section.

Transparent PV

Transparent photovoltaics (TPVs), which selectively absorb ultraviolet and near-infrared light while
allowing high transmittance of visible light, could allow harvesting solar energy from the surfaces
of buildings, automobiles, and electronic device displays without impacting function or aesthetics.

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TPVs have caught the attention of the automotive industry, with some automakers looking at
incorporating TPVs, for example, for sunroofs where the energy could be used to power fans to
cool a car parked in the sun [162]. Transparent windows with TPVs with nearly 10% PCE are
already being commercialized [163]. In the future, TPVs could be incorporated into automobile
windows or even cover the entire surface area of vehicles for maximal energy production. Here
we use our thermal-electrical PV model to predict the potential energy production of a Jeep Grand
Cherokee for a windows-only (including a large sunroof) and entire surface area TPV system in
five U.S. cities: New York NY, Lansing MI, Honolulu HI, Los Angeles CA, and Phoenix AZ.

Since we are modeling an emerging technology, electrical parameters were not available.
Therefore, we used an estimated realistically achievable efficiency of 15% [164]. Angular effects
were considered by specifying measured IAM curves from a TPV angular effects study [149].
The capability of TAITherm to model transparent materials was used for the Jeep’s windows
with measured values of TPV spectral transmittance and reflectance [165] specified as curves.
TAITherm integrates the spectral values to determine overall transmissivity and reflectivity for use
in the transparent materials solver. To model the TPVs on the Jeep’s body panels, regular solid
materials were used due to TAITherm’s inability to model a transparent material on top of a non-
transparent one. Much of the light transmitted through the TPV film will be absorbed by the body
panel underneath, so the effect of neglecting to explicitly model the optics is likely small. However,
note that this neglects the potential additional energy production from light that is reflected from the
vehicle body panel surface back through the film. Figure 6.21 shows the spectral optical properties
and angular dependence used in the model.

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Figure 6.21: a) TPV UV-vis spectrum for front illumination. Data from [165]. b) Incident angle
modifier for a conventional thin film PV and one that is optimized for improved
angle-dependence. Data from [149].

TMY weather data for the five cities was obtained from the NSRDB. The Jeep was assumed
to be outside in an unshaded area (but under realistic weather patterns, including cloud cover)
for the entire year. The TAITherm built-in ground surface background of aged asphalt (infrared
emissivity of 0.94 and solar absorptivity of 0.85) was assumed for the thermal and radiation
modeling. Given the difficulty of predicting soiling on a vehicle that is in use, losses due to
soiling were not considered. Vehicles with PVs should be washed regularly to maximize energy
production. Predicted total yearly energy production and monthly energy production for the five
cities are shown in Figure 6.22. For angle of incidence corrections, the TPV models assumed the
conventional thin-film angular dependence from Figure 6.21b and the silicon PV model used the
empirical equations from Section 6.2.2.

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Figure 6.22: a) Predicted total annual energy production in five U.S. cities for three different
vehicle-integrated PV systems: TPVs covering entire surface of the vehicle, TPVs only used in
windows, and a conventional silicon PV roof module. b) Predicted monthly energy production for
the TPV system that covers the entire surface of the vehicle.

Assuming an EV energy efficiency of 26 kWh/100 miles [155], a Jeep Grand Cherokee that
is completely covered with 15% efficiency TPVs could drive 13,400 miles a year in Phoenix on
solar energy alone. Note that this assumes an idealized scenario where the car is always in non-
shaded areas (e.g. parked far away from other vehicles that would block sunlight from reaching the
side panels) and kept perfectly clean. If TPVs are only used for the Jeep’s windows (including a
sunroof), solar energy could provide up to 4,300 miles of range a year. A conventional silicon PV
roof could provide up to 3,700 miles. In locations with less solar irradiance, such as New York or
Lansing, the energy production is expected to be about 30% lower than in Phoenix. Rerunning the
model of the Jeep with full coverage of TPVs with an improved AOI dependence (the optimized
curve from Figure 6.21), results in an approximately 10% increase in predicted annual energy
production. This demonstrates the need to include an AOI correction in PV modeling and also
provides motivation to develop PV devices with optimized angular dependence. The TAITherm
model also predicts that TPVs that cover the entire vehicle’s surface will have an additional benefit
of causing a reduction of cabin air temperature by 1-2◦C, which would reduce cabin cooling costs
during the warm seasons. Of course, the actual driving range will also depend on the influence
of temperature on battery performance, HVAC use, and driving conditions (city vs. highway, flat
vs. mountainous). In future work, the PV thermal-electrical model could be coupled with electric

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vehicle battery, motor, HVAC, and drive-cycle models for more detailed analysis.

6.3.3 PV-Battery Systems

Batteries are often used to store energy produced by photovoltaics. When developing PV-battery
systems, it is necessary to appropriately select the capacity and type of battery that can store the
needed amount of energy and satisfy the charge and discharge rate requirements for the system’s
application. Batteries that are charged with currents that are a larger proportion of their maximum
capacity degrade faster. Additionally, the temperature of a battery during operation must be
considered for several reasons. One is to ensure that it stays within the safe operating range for the
battery and within any additional temperature restrictions due to the application (e.g.
there may
be a narrower operating range for batteries installed together with other sensitive electronics). The
temperature of a battery also impacts its performance and, therefore, should be considered for an
accurate estimate of usable battery capacity and available discharge power. In some applications,
passive or active thermal management may be used to maintain the battery within its optimal
operating range. Understanding how batteries degrade as functions of time, charge/discharge
cycles, and temperature can help extend battery lifetime by optimizing charging and discharging
with "smarter" battery management systems (BMSs). This section outlines how one may combine
the PV thermal-electrical model presented in this chapter with battery performance and lifetime
models.

In this work, the performance and lifetime of a PV-system battery are modeled using a commer-
cial thermal-electrical battery lifetime prediction tool developed by ThermoAnalytics. The model
consists of two main parts: a thermal-electrical battery model and a battery lifetime model. The
lifetime model is an empirical model with physically justifiable formulas developed by NREL [166].
The NREL lifetime model separates battery degradation into calendar and cycling fade. Calendar
fade is degradation that occurs over time and is independent of cycling. Cycling fade is degradation
that occurs with the number of cycles and is independent of time. Since calendar and cycling fade
are due to internal chemical processes in the battery, their rates depend on battery temperature,

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thus requiring a thermal-electrical model, which is run in TAITherm to predict those temperatures.
The electrical model uses NREL’s equivalent circuit battery model to calculate the voltage, state
of charge (SOC), currents, and heat generation of individual cells within a battery. The calculated
heat generation is considered in the thermal model to calculate new temperatures. These temper-
atures affect the battery’s electrical behavior, which is modeled by temperature-dependence of the
resistors in the equivalent circuit model. Figure 6.23 shows a schematic of the thermal-electrical
battery model.

Figure 6.23: Schematic of the thermal-electrical battery model. Q is the heat generated by the
battery, calculated in terms of Joule heating (electrical losses) within the battery equivalent circuit
and entropic heating (heat from chemical reactions in the battery). Adapted from [167].

As batteries age, their resistance increases and capacity decreases, influencing the battery’s
temperatures and performance. Thus, the lifetime and thermal-electrical battery models are bi-
directionally coupled, forming a multi-timescale simulation. The lifetime model’s goal is to
describe a battery’s change in resistance and capacity over the course of months and years and
therefore uses a one-day time step. The thermal-electrical model’s goal is to describe a battery’s
performance at the minute and hour resolution and therefore uses a time step on the order of
minutes. ThermoAnalytics’ co-simulation software CoTherm controls the multiple time-scale
simulation. First, the thermal-electrical TAITherm model is run for a single day to compute the
battery’s temperatures and SOC over time. This information is provided to the lifetime model
(integrated into CoTherm), which is run with one-day time-steps for the period of a few months.

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After a few months of lifetime simulation, the currently predicted values of battery capacity and
resistance are used to update the equivalent circuit coefficients, and the thermal-electrical model is
rerun for a single day. This cycle repeats to simulate many years of the battery’s life. Figure 6.24
shows the battery lifetime simulation approach.

Figure 6.24: Flow diagram of the battery life prediction model. ("Eqc" stands for "equivalent
circuit".)

For proof-of-concept modeling, we specified a PV module’s photocurrent at maximum power
as the charging current of the battery. To compute the photocurrent, we used the validated PV
thermal-electrical model to model a single First Solar CdTe Series 4 module located in Phoenix,
Arizona. Irradiance and weather for one representative day in each of the four seasons was used to
generate four representative photocurrent profiles for input into the battery lifetime model. After
every three months of lifetime simulation, the TAITherm thermal-electrical model was rerun using
the representative photocurrent for that season. We modeled two PV-battery systems that both use
the same CdTe solar module but have different sized batteries for energy storage (30 Ah and 60
Ah). For simplicity, the batteries were assumed to start the day with 20% SOC, charge during the
day with the PV module, and be discharged each night back down to 20% SOC. Figure 6.25 shows
one such charge/discharge current profile and the corresponding battery SOC. For this simulation,
the batteries were assumed to be in a climate-controlled environment maintaining 22◦C (natural
weather environmental boundary conditions can be specified in the models if desired).

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Figure 6.25: a) Battery charge/discharge current over a single summer day. Positive current is
defined as charging. b) State of charge of the 30 Ah battery due to the current profile in a).

Figure 6.26 shows the results of the predictions of relative resistance and capacity of the 30 Ah
and 60 Ah batteries over their life. Both batteries are oversized for the system, meaning that they
are not fully charged/discharged over normal use. This oversizing reduces the cycling fade of the
battery, thus increasing the battery lifetime. We see that the larger 60 Ah battery degrades slower
than the smaller 30 Ah battery. The financial trade-off between a larger, more expensive battery
and longer battery lifetime, reducing the frequency of battery replacement, needs to be considered
when designing or purchasing a PV-battery system.

Figure 6.26: Relative resistance (a) and capacity (b) of the 30 Ah and 60 Ah batteries. Values are
relative to those of a brand-new battery.

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In future work, these simulation approaches could be used to design and model batteries which
are optimized for photovoltaic systems [168]. They could also be used to model battery packs
(multiple batteries connected in series or parallel to provide a higher energy storage capacity or
power) and smart battery management systems which optimize the charging/discharging of these
battery packs to extend usable capacity, power, and lifetime of the pack [169]. Figure 6.27 shows
one possible approach envisioned for PV-battery and battery management system modeling. The
equivalent circuit battery model could be replaced with an electrochemical model if a more detailed
and physics-based battery model is desired.

Figure 6.27: PV-battery and battery management system modeling approach.

6.3.4 Other Applications

In addition to reducing thermal losses, other photovoltaic applications where coupled thermal-
electrical PV models could be an effective simulation approach include:

• Solar cells in space under extreme (both high and low) temperature conditions [170, 171]

• Photovoltaic-thermal technology: a PV panel coupled with a thermal collector [172]

• Photovoltaic collectors and bifacial PV systems [173, 174]

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• Temperature-induced solar cell ageing [175]

• Battery management systems for efficient storage of photogenerated energy [176]

• IR imaging of PV-plants [177]

• Melting of snow on heated panels [178]

• Condensation on dusty PV modules and associated formation of mud [179]

6.4 Conclusions

In this chapter, we developed coupled thermal-electrical PV system models that consider the
effects of temperature on solar panel performance by implementing custom user routines in a
commercial heat transfer solver. After validating the modeling approach by comparison with
measured data, we developed proof-of-concept modeling examples for residential PV systems
and vehicle-integrated solar panels. Through these examples, we showed that consideration of
temperature-induced losses is important to avoid overestimating a PV system’s potential energy
production. Additionally, we developed the foundations for modeling the interactions between
solar panels and batteries used for energy storage by coupling the thermal-electrical PV model with
a commercial thermal-electrical battery temperature and performance model, and a commercial
battery aging model. These models can also be used together with the soiling models developed in
Chapter 5. The approaches developed in this chapter, which leverage the power and flexibility of
a commercial heat transfer solver, can be applied for computer-aided engineering of a wide range
of PV systems, such as vehicle and building-integrated PVs, PVs in space, PV-thermal technology,
and battery management systems for PV energy storage.

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CHAPTER 7

CONCLUSIONS AND OUTLOOK

7.1 Conclusions

In this dissertation, we described the development of computational multi-physics models
for photovoltaic devices and systems. We started our modeling with the smallest functional
unit in a PV system: a single solar cell. For this goal, we used the computationally efficient
yet accurate drift-diffusion approach. We introduced the numerical methods for drift-diffusion
modeling in Chapter 2. We also demonstrated that the drift-diffusion model is consistent with
the Mott-Gurney law for space-charge-limited currents. In Chapter 3, we investigated the impact
of the choice of discretization approach (Scharfetter-Gummel versus Slotboom), iterative solution
approach (Gummel versus Newton), and convergence improvement technique (linear versus Pulay
mixing) on the convergence behavior of a drift-diffusion model for bilayer organic solar cells. The
findings can help in selecting a suitable discretization and iteration approach for each device model.
Pulay mixing can be used to improve convergence speed, Newton’s method is superior to Gummel’s
method for situations where the equations become highly coupled, and Slotboom variables can be
used to simplify the implementation of the matrix equations.

In Chapter 4, the drift-diffusion model was adapted for planar perovskite solar cells and validated
by reproducing literature results. The model was then used to describe recent experimental results
for perovskite solar cells that use fullerene layers for electron transport. Through drift-diffusion
modeling, poor electron extraction at the perovskite interface was found to be the primary cause of
the severe decrease in device performance when there is no C60 layer. We have also found a possible
explanation for the S-shaped JV curve seen in the 0 nm C60 devices. The S-shape is likely due to
an imbalance of carrier mobilities in the HTL and ETL, which results in charge accumulation and
the formation of an interface dipole. The modeling results support the experimental observation
and conclusion that while the inclusion of a C60 layer is critical for achieving good performance of

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perovskite solar cells that use BCP as their ETL, an ultra-thin C60 is sufficient to improve device
performance by enhancing electron extraction.

In the second half of this dissertation, the modeling focus was shifted to the larger scale of
PV modules and systems, where each module is made up of many individual solar cells, and each
system can be composed of many modules. To improve return-on-investment assessment for solar
energy systems, PV models need to accurately predict energy output, including considering losses
due to temperature and soiling. In Chapter 5, we explored both semi-physical and machine learning
approaches to modeling the soiling of PV panels. We developed a general semi-physical model
which is expected to be reasonably applicable globally, provided that the empirical coefficients
are fitted on a region-by-region basis. We also developed the methodology for integrating the
soiling models with a Python package for PV energy production prediction (PVLIB) and cleaning
cycle optimization algorithms, which resulted in the release of a new commercial software tool
(PV SoilSayer). Preliminary model validation was performed by comparison to published soiling
measurements from an NREL study. Additionally, we demonstrated the feasibility of using multi-
linear regression and artificial neural networks for soiling predictions. These approaches were found
to be able to both replicate the results of semi-physical soiling accumulation models and predict
future soiling losses based on historical measurements of soiling and environmental parameters.
The models and software that we developed are expected to provide the PV industry with improved
methods for predicting soiling losses and their financial impacts.

Finally, in Chapter 6, we developed coupled thermal-electrical PV system models by imple-
menting custom user routines in a commercial heat transfer solver. The modeling approach was
validated by comparison with measured data from a utility-scale PV system. We also developed
proof-of-concept modeling examples for residential PV systems and vehicle-integrated solar panels.
Additionally, we developed and demonstrated an approach for modeling PV-battery systems by cou-
pling the thermal-electrical PV model with a commercial thermal-electrical battery performance
model and a commercial battery aging model. These thermal-electrical PV models are expected to
be beneficial for the engineering of a variety of PV systems.

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7.2 Outlook

There are many possibilities to extend this research. The drift-diffusion model can be adapted
to various solar cell architectures to help improve our understanding of the physics in these devices
and perform computational experiments that explore the impact of different materials and cell
architectures on device efficiency. The models can be extended to consider transient effects and
field-dependent carrier mobilities. Additionally, the drift-diffusion model can be extended to
simulate light-emitting diodes (LEDs). LEDs that are made of organic materials (OLEDs) are a
promising technology for digital displays and lighting. The physics in LEDs is very similar to solar
cells, with the main difference being that in solar cells, photogenerated carriers are separated to
generate a current, while in LEDs, an external current forces carriers through the device where
they recombine and emit light. Two and three-dimensional extensions of the drift-diffusion model
can be applied to devices with asymmetry in the carrier transport in the in-plane directions.
(Example implementations of extending the drift-diffusion models presented here to 2D and 3D
are available1). Further fundamental understanding of materials for solar cells and LEDs can be
achieved through ab-initio quantum chemistry calculations (e.g. density functional theory). Ab-
initio calculations can also guide experimental efforts in developing new devices by helping to
select promising candidate materials (e.g. based on calculated bandgaps and carrier mobilities).
The convergence improvement methods discussed in Chapter 3 could also be applicable to other
systems of equations with similar mathematical properties to the drift-diffusion equations.

The PV soiling models and software developed in Chapter 5 can be applied to help PV system
developers, maintainers, engineers, and researchers have a better understanding of soiling losses.
With the growing realization of the impact of soiling, the installation of soiling monitoring stations
is becoming more widespread, which will lead to the availability of more soiling data. With this
soiling data, the correlations between environmental parameters (such as particulate matter, wind
speed, and rainfall) and soiling should be analyzed for numerous regions worldwide and used
to develop region-specific semi-physical or machine learning models for soiling loss prediction.

1

https://github.com/tgolubev/Drift-Diffusion_models-Cpp_Matlab

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Careful experiments should be conducted to clarify the cleaning effects of rain and attempt to
compile a database of regional rainfall thresholds required for effective cleaning. Experiments
should also be conducted to further study the influence of wind on soiling accumulation, especially
focusing on resuspension, which is currently not well understood. Additionally, the predictions of
optimal cleaning cycles should be validated by conducting experiments where panels are cleaned
according to different schedules.

In the last part of Chapter 5, we have demonstrated the potential of using MLR and ANN
for predicting soiling. However, MLR is unable to capture non-linear effects, and deep learning
(i.e. artificial neural networks) is not necessarily the best solution for all datasets since it has
many disadvantages such as requiring larger amounts of data and more computational resources to
train than other techniques, and being difficult to interpret. Therefore, it is important to explore
other ML solutions. Testing multiple ML models for a given dataset is tedious and requires
significant knowledge and experience with ML, which limits the ability of photovoltaic engineers
and researchers to use ML. The new field of automated machine learning (AutoML) provides
methods that automate the process of selecting a suitable ML model and its hyperparameters (user-
defined parameters that control the learning process) for a given dataset, thus greatly improving the
efficiency of developing an ML model and making ML accessible to non-ML-experts. In future
work, open-source AutoML packages such as TPOT (Tree-based Pipeline Optimization Tool),
Auto-sklearn, and AutoKeras can be used to select the best ML algorithms and hyperparameters
for PV soiling loss models.

The coupled thermal-electrical PV modeling approaches developed in Chapter 6 can be applied
for computer-aided engineering of a wide range of PV systems, such as vehicle and building-
integrated PVs, PVs in space, PV-thermal collectors, wearable PVs, and battery management
systems for PV energy storage. Further validation and refinement of the models should be done in
collaboration with experimental PV research. The models can also be improved by coupling with
more sophisticated system-level models. For example, a PVT collector can be modeled by coupling
the PV models developed here with a computational fluid dynamics software for modeling the fluid

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flow in the heat exchanger. Another possible application of system-level modeling is predicting
the contribution of vehicle-integrated PVs to electric vehicle (EV) driving range. To develop such
a model, one could couple the thermal-electrical PV model with a model of EV batteries that
considers the impact of temperature on battery efficiency. One could further couple the PV and
battery models with a model of the vehicle’s HVAC system and a model of human thermal comfort
(such as in [180]) in order to estimate the power needed to preserve the comfort of the vehicle’s
occupants in different seasons and climate regions. Finally, ML-based reduced-order-modeling
where physical models are replaced with ML models that replicate the physical models’ behavior
should be explored as a way of reducing the computational expense of PV system-level modeling.

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APPENDICES

176

APPENDIX A

PV SOILSAYER SOFTWARE DESCRIPTION

A significant portion of the photovoltaics soiling research completed for this dissertation involved
developing practical solutions for integrating the various soiling modeling approaches (described
in Chapter 5) with an open-source Python package for PV modeling (PVLIB) [130, 181] and the
PV cleaning cycle optimization algorithms of a PV startup company (Enlighten Energy, Ltd). The
results of this integration was Enlighten Energy’s release of PV SoilSayer, a commercial software
application for soiling loss calculations and optimal clean cycle forecasts. A description of the
software is presented here to illustrate how the soiling models developed in this dissertation were
incorporated into a broader PV simulation tool. After writing, this Appendix was adapted to create
the first version of the PV SoilSayer User Manual.

A.1 Weather

Prediction of a PV system’s energy production and soiling losses is dependent on the location-
specific weather patterns. PV SoilSayer utilizes annual weather files with an hourly resolution for
its predictions. Upon entering the location of their PV system, the user can choose from one of
several weather databases: 30-year typical meteorological year (TMY) from Meteonorm [156],
actual measured weather from last calendar year (from AccuWeather), next year weather forecast
(from AccuWeather), and an alternative precipitation-based TMY that we self-generated based on
10-year historical data (from AccuWeather). Two important parameters for energy and soiling
loss prediction, total GHI and total precipitation, are compared graphically to help select a weather
resource. The weather station locations relative to the plant locations are displayed on an embedded
Google Map (Figure A.1). Note that the Meteonorm TMY resource interpolates the weather for
particular coordinates from multiple nearby plants (all shown on the map), while the Accuweather
resources select only the closest plant. There are also options to update TMY weather datasets with
year-to-date weather and 90 day weather forecasts (from AccuWeather).

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Users also have the option of uploading their own weather data. Since precipitation is often
measured at a daily, instead of an hourly resolution, there is also an option to upload hourly weather
data with daily precipitation.
In this case, the daily precipitation will be interpolated to hourly
values to use in the forecasting algorithms. Weather parameters used for forecasting are solar
irradiance components (GHI, DNI, DHI), relative humidity, wind speed, wind direction, ambient
air temperature, and precipitation. Any weather parameters that are not uploaded by the user are
taken from Meteonorm TMY.

Figure A.1: Weather Resource UI.

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Possible advantages of using the Meteonorm 30-year TMY datasets is that this approach is
the solar industry-standard for energy prediction. Additionally, their interpolation of weather from
multiple nearby stations could yield more representative weather datasets for locations that are
not near any weather stations. A potential disadvantage is that due to rapid climate change in
many global regions, a 30-year TMY is not necessarily representative of typical weather in the
current years. For example, many locations may be seeing higher temperatures and drier or wetter
conditions than 20-30 years ago. Therefore, weather resources based on Accuweather that rely on
the past ten years of historical data, last year data, or next year forecasts could be preferable. Future
versions of the software could allow rerunning the forecast for the same PV system with different
historical weather data and calculate a statistical average soiling forecast and uncertainties based
on weather variability.

A.2 PV System Design

The design parameters of a PV system can greatly impact its energy production. This section

provides an overview of the main design parameters considered in PV SoilSayer (Figure A.2).

Figure A.2: System Design UI.

DC and AC System Size

The nominal energy production capability of a PV system is defined by its direct current (DC) and
alternating current (AC) power rating (also known as system size). These are expressed in kilowatts

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peak (kWp) at standard test conditions (STC), meaning the peak kW power production under cell
temperature of 25◦C, irradiance of 1000 W/m2, and AM1.5 solar spectrum. PV systems produce
DC power that is then commonly inverted to AC for use in the electric grid. The AC system size
describes the peak power that the inverter is capable of outputting. The ratio between DC and AC
system size is called DC-to-AC ratio or Inverter Load Ratio and is often used to describe a PV
system [182]. PV systems are generally sized with DC-to-AC ratios of 1.1 − 1.3, meaning that the
inverter power output capability is lower than the maximum possible DC power production. This
leads to what is known as inverter clipping or clipping loss where during times of peak production
(e.g. sunny day at solar noon), the DC power feeding the inverter is greater than what the inverter
can handle, so the extra DC power potential is clipped and lost. The inverter clips by raising the
modules’ operating voltage to move the array off of its maximum power point and therefore reduce
the DC power. Such undersizing of inverters is done to improve project economics (less powerful
inverters are less expensive) and to match the AC power output with the grid connection point
design’s maximum power (e.g. a home’s AC panel has a power limit). The clipping losses due
to DC-to-AC ratios that are > 1 are small since full STC conditions are rare. STC conditions
describe noon on a clear summer day, but the cell temperature is assumed to be 25◦C. Actual cell
temperatures are commonly significantly higher, resulting in reduced efficiency (see Chapter 6).

Inverter Efficiency

The conversion of DC to AC by the inverter has an associated loss due to imperfect inverter
efficiency. PV SoilSayer implements the inverter model described in Version 5 of PVWatts, which
is based on an analysis of inverter performance data from the California Energy Commission (CEC)
[183]. The inverter performance curve is given in Figure A.1, which scales the efficiency curve to
the nominal rated efficiency specified by the user (ηnom). The default nominal efficiency is 0.96.
The reference inverter efficiency (ηre f = Pac0/Pdc0) from the CEC data for a typical inverter is
0.9637, and the AC nameplate rating (Pac0) is determined from the DC rating of the system and

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the DC-to-AC ratio.

where

and

η =

ηmon
ηre f

− 0.0162ζ − 0.0059

ζ

+ −.9858

ζ =

Pdc
Pdc0

Pdc0 =

Pac0
ηnom

(A.1)

(A.2)

(A.3)

When the predicted AC output exceeds the nameplate rating, the output is clipped to the
nameplate value. Figure A.3 shows the inverter efficiency curve as a function of load fraction (ratio
of power input to the inverter to the inverter’s AC rating) for different nominal inverter efficiencies.
Note the steep drop in inverter efficiency at very low load fractions, which may result in high
inverter losses during low light conditions (e.g. sunrise/sunset).

Figure A.3: Inverter partial-load efficiency curve. Adapted from [183].

Module Type

PV modules come in two main constructions called glass-glass and glass-backsheet. Glass-Glass
modules use glass on both the front and back sides of the module, while glass-backsheet modules
use a polymer sheet on the backside. Most PV modules use the glass-backsheet construction, but

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glass-glass modules can provide better reliability and durability. The module type impacts the
coefficients used in the cell temperature prediction model, which in turn impact the cell efficiency.
Temperature-induced losses are considered without any additional input from the user. Cur-
rently, PV SoilSayer uses the Sandia Array Performance Model (SAPM) [184] to calculate cell
temperature as implemented in PVLIB. The module temperature is given in terms of POA irradi-
ance (E), wind speed (WS), ambient air temperature (Ta), and two empirical coefficients (a and
b).

Tm = E × exp(a + b × WS) + Ta
The cell temperature is calculated from the module temperature by

TC = Tm +

E
E0

∆T

(A.4)

(A.5)

with ∆T being the empirically-estimated difference between the temperatures of the module’s back
surface and the cell.

Table A.1 shows the empirical parameters used in the SAPM model.

Table A.1: SAPM temperature model coefficients.

Module Type
Open-Rack Glass-Glass
Open-Rack Glass-Backsheet
Rooftop Glass-Glass
Rooftop Glass-Backsheet

b

a
-3.47 -0.0594
-3.56
-0.075
-2.98 -0.0471
-2.81
-0.0455

∆T
3
3
1
0

The computed cell temperature impacts the DC power model as described in the section below.
Future development of PV SoilSayer could include allowing several temperature models with user-
input empirical coefficients, or uploading cell temperatures that were either modeled in an external
software or measured. (See Chapter 6 for a discussion of cell temperature modeling).

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DC Power Model

PV SoilSayer currently uses NREL’s PVWatts DC power model as implemented in PVLIB [130].
The empirical model predicts a PV module’s power output based on incident plane-of-array
(Gpoa,e f f ) irradiance and cell temperature
Gpoa,e f f
1000 Pdc0(1 + γpdc(Tcell − Tre f ))

Pdc =

(A.6)

where Pdc0 is the DC power output of the modules at STC (DC System Size input). PV SoilSayer
currently assumes a temperature coefficient of γpdc = 0.004/◦C.

Cell Type

The cell type describes the active material of the PV module. The most common cell types used
commercially are monocrystalline and polycrystalline silicon, CIGS, and CdTe. The cell type
impacts the spectral mismatch model coefficients that are used by PVLIB. The spectral mismatch
modifier, M, is based on precipitable water, Pwat, and absolute (pressure corrected) airmass, Am
[130]. It describes the effect of variation in spectral irradiance on a module’s short-circuit current.
The following empirical equation is used

M = c1 + c2Am + c3Pwat + c4A5

m + c5P5

wat + c6Am/P5

wat

(A.7)

with coefficients given below.

Table A.2: Spectral mismatch modifier coefficients.

Cell Type
CdTe
Mono-Si
Poly Si
CIGS

c1
0.86273
0.85914
0.85914
0.85252

c2
-0.038948
-0.020880
-0.020880
-0.022314

c3
-0.012506
-0.0058853
-0.0058853
-0.0047216

c4
0.098871
0.12029
0.12029
0.13666

c5
0.084658
0.026814
0.026814
0.013342

c6
-0.0042948
-0.0017810
-0.0017810
-0.0008945

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Array Type and Orientation

PV arrays can be free-standing (also known as open-rack) or roof-mounted. Roof-mounted arrays
are generally fixed in place. The choice between free-standing and roof-mounted arrays impacts
the empirical cell temperature models since free-standing systems have convection on both sides,
while roof-mounted systems have limited convection on the back, increasing cell temperatures.
Free-standing arrays can be either fixed in place or rotate around one or two axes to track the sun.
The rotating arrays are known as single-axis trackers and two-axis trackers. Figure A.4 shows the
three types of free-standing arrays.

Figure A.4: Array types. Adapted from [185].

The array’s orientation determines the amount of irradiance incident on the collector surface
and is described by a tilt (also known as elevation) angle and azimuth angle, as labeled in Figure
A.4. Fixed tilt free-standing array tilt angles are chosen to maximize the total incident irradiance. A
common recommendation is to set the tilt angle equal to the absolute value of the system’s latitude.
Higher tilt angle arrays, while increasing energy production in higher latitudes and during winter
months, can have a higher associated cost and may increase the risk of wind damage. Roof-mounted
arrays typically have tilt angles equal to the roof pitch. For single-axis trackers, the tilt angle refers
to the angle from the horizontal of the tracking axis. Tilt angle does not apply to two-axis trackers
since they move to always maintain a normal incidence of the direct normal solar irradiation.

The azimuth angle is the angle clockwise from true north that describes the direction that the
array (or axis of rotation for single-axis trackers) faces. Often the azimuth angle is around 180◦

184

(south-facing) for locations in the northern hemisphere and 0◦ (north-facing) for locations in the
southern hemisphere. The actual optimal azimuth angle can be slightly different based on local
weather patterns. For the northern hemisphere, increasing the azimuth angle favors afternoon
energy production, and decreasing the azimuth angle favors morning energy production. The
opposite is true for the southern hemisphere [185].

PVLIB allows modeling the motion of single-axis trackers as well as the effect of self-shading1.
Self-shading is a reduction in the array’s output caused by shading of neighboring rows of modules
at certain times of day and year when the sun is low on the horizon. Single-axis trackers often
use backtracking, which is a self-shading mitigation technique that rotates the array toward the
horizontal during early morning and late evening hours. The user can turn on/off backtracking
by selecting either a "True Tracking" or Backtracking" array type. The ground cover ratio (GCR)
specifies the ratio of module area to land area. Figure A.5 shows the relationship between self-
shading and GCR. In the figure, β is the tilt angle, z measures height along the array, and ψ(z) is
the screening angle, which represents a two-dimensional field-of-view reduction at height z [186].

Figure A.5: Ground cover ratio. Adapted from [185].

For single-axis trackers, an additional optional input is the Tracker Stow Angle, describing the
tilt angle of the SAT at night. This is needed only for the semi-physical soiling model, which
1For fixed-tilt and two-axis trackers, self-shading is not modeled, but its effects can be specified

as a percent loss in the Shading input in the PV System Losses UI.

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considers the effect of tilt angle on soiling accumulation. Higher stow angles result in lower soiling
losses (see Chapter 5).

Ground Reflection Albedo

The albedo coefficient determines the amount of solar irradiance reflected from the ground sur-
rounding the PV installation, which can contribute to the total irradiance incident on the module.
Note that PV SoilSayer currently uses albedo only for mono-facial panels. Common albedo values
are presented in Table A.3.

Table A.3: Sample albedo values. From [130].

Surface Type Albedo
Urban
Grass
Fresh Grass
Soil
Sand
Snow
Fresh Snow
Asphalt
Concrete
Aluminum
Copper
Fresh Steel
Dirty Steel
Sea

0.18
0.20
0.26
0.17
0.40
0.65
0.75
0.12
0.30
0.85
0.74
0.35
0.08
0.06

A.2.1 PV System Losses

PV systems have multiple sources of energy losses. Losses other than those due to temperature and
soiling, which are computed separately, are accounted for by requesting user-input of annual (and
monthly for shading and snow) loss percentage values (Figure A.6).

186

Figure A.6: PV System Losses UI.

(cid:20)

1 −


(cid:18)

(cid:19)(cid:21)

Lnet = 100

1 − Li
100

i

The losses calculations follow the simple multiplicative approach of NREL’s free PVWatts
software [183] with the exception that PVWatts does not support entering monthly resolution
losses for snow and shading. For each month, the net losses percentage (except for temperature and
soiling losses) is

(A.8)

Since snow and shading losses are highly seasonal, using monthly values gives a more accurate
estimate of the seasonal variability of panel losses. System losses that are considered by PV
SoilSayer are described briefly below.

Mismatch Loss is electrical loss due to slight differences caused by manufacturing imperfec-
tions between modules in the array that cause the modules to have slightly different current-voltage
characteristics.

Wiring Loss is the resistive loss in the DC and AC wires connecting modules, inverters, and

other parts of the system.

Connections Loss describes the resistive losses in electrical connectors in the system.
Light-Induced Degradation Loss accounts for the reduction of power output of a PV module
from the manufacturer rating because of light-induced degradation of the photovoltaic cells. A

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new PV module’s power output often decreases during the first few months of its operation. The
module’s power stabilizes after this initial period.

Nameplate Rating Loss accounts for any inaccuracies in the manufacturer’s nameplate rating.
Field measurements of the electrical characteristics of photovoltaic modules in the array may show
that they differ from their nameplate rating. For example, a nameplate rating loss of 5% would
indicate that testing yielded power measurements at Standard Test Conditions (STC) that were 5%
less than the manufacturer’s nameplate rating.

Age Loss accounts for the effect of weathering of the photovoltaic modules on the array’s

performance over time.

Availability Loss describes the reduction in the system’s output caused by scheduled and

unscheduled system shutdown for maintenance, grid outages, and other operational factors.

Shading Loss describes the reduction in the incident solar radiation from shadows caused by
faraway features such as large buildings or mountains, objects near the array such as trees, or
self-shading. The PVWatts default value of 3% only considers typical observed shading losses due
to faraway features. This value should be increased to account for any shading from nearby features
and self-shading. For accurate prediction of shading losses using an external shading software or
conducting an on-site survey is recommended. Note that PV SoilSayer automatically considers
self-shading losses for single-axis trackers; therefore, the self-shading contribution to the overall
shading loss should only be included for fixed tilt and two-axis tracker arrays.

Snow Loss describes the reduction in the system’s annual output due to snow covering the
array. The default value is zero, assuming that there is never any snow accumulating on the array
or that the array is kept clear of snow.

For reference, Table A.4 shows the default values of annual losses used in PVWatts.

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Table A.4: Default annual losses in PVWatts for reference.

Default Value
Loss Mechanism
2%
Mismatch
2%
Wiring
Connections
0.5 %
Light-induced degradation 1.5%
Nameplate rating
Age
Availability
Shading
Snow

1%
0%
3%
3%
0%

A.2.2 Annual Energy Calculation

The first calculations occur when Calculate Annual Energy is clicked at the bottom of the PV
System Info tab. Here, the PV System Design and System Losses parameters are used with PVLIB
models to estimate the DC and AC energy production for a year for a panel with no soiling losses
(perfectly clean). The energy production is calculated with an hourly resolution, and a monthly
resolution chart is displayed to allow the user to verify that the production estimates are reasonable
before proceeding with the soiling calculations (Figure A.7).

Figure A.7: Monthly energy production chart example.

189

A.3 Soiling Profile

A.3.1 Soiling Loss Calculation/Upload Options

Users have several options for calculating or uploading soiling loss data to be used for soiling loss
prediction and clean cycle calculations (Figure A.8). For calculated options, users can specify the
month at which to start the year-long forecast and the starting DC soiling ratio (the DC power loss
due to soiling).

Figure A.8: Soiling loss calculation/upload options.

Soiling Loss Calculator

The soiling loss calculator utilizes the semi-physical soiling model described in Chapter 5 to
estimate soiling losses. The user has an additional option to empirically increase the monthly
soiling rate in any month due to any localized soiling that they may observe from sources such
as local farming practices, mowing, sheep grazing, and roadways. The localized soiling increases
are arbitrary with moderate, heavy, very heavy, and extreme soiling corresponding to 50%, 100%,
150%, and 200% increases in the soiling rate. This feature was added at the request of prospective
users who were managing PV plants with abnormally high local soiling that is not representative

190

of the region’s average air quality, but at the same time did not have daily soiling data available to
use the upload option.

Figure A.9: Soiling Loss Calculator UI.

DC Soiling Losses Over a Dry Period

DC soiling losses measured over a dry period can be used to calculate the hourly losses throughout
the year. Measurements over a dry period are correlated with weather resources and interpolated to
the hourly resolution to determine hourly soiling losses. The user should input the length (in days)
of the dry period and the change in soiling loss over its duration. For example, if one measures
that the first day of a dry period has 5% soiling at solar noon and the 30th day of the dry period
has 10% soiling loss at solar noon, then the Dry Measurement Duration is 30 days, and the Total
Change in DC Soiling Loss is 5%.

Figure A.10: DC Loss Over a Dry Period UI.

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Average Daily Soiling Rates

This option is most aligned with the way soiling is treated in other industry-standard software since
it treats soiling analogously to the Kimber et al. model with the exceptions that we do not include
a Grace Period and allow for different daily soiling rates in each month instead of one annual value
(see Section 5.1). Daily average soiling rates for each month should be calculated using data from
the longest dry period during each month as the change in soiling loss over the Dry Period divided
by the length of the Dry Period in days.

Figure A.11: Daily Soiling Rates UI.

Upload Measured DC Hourly Soiling Losses

In this option, the forecast uses hourly measured soiling losses throughout the year. The measured
soiling losses must exclude any manual cleaning events. The rainfall cleaning calculations (see
A.3.3) are not run when this option is used since all cleaning effects of rainfall are assumed to be
included in the measured soiling losses.

Upload Daily Soiling Losses Measured at Solar Noon

In this option, annual hourly soiling losses are estimated from daily soiling losses measured at
solar noon. The measured soiling losses must exclude any manual cleaning events. The rainfall
cleaning calculations (see A.3.3) are not run when this option is used since all cleaning effects of
rainfall are assumed to be included in the measured soiling losses. The algorithm treats the input
as a measurement at the hour closest to solar noon (where solar noon is not necessarily at 12 pm
but is the time at which the sun is at the highest position in the sky).

192

If Calculate Soiling Losses due to Sun Angle is selected (see Angular Loss below), then the

angle of incidence effect on soiling losses will be taken into account as described in Chapter 5.

Possible Machine Learning Feature

Future releases may offer an optional feature where machine learning models are trained on user-
uploaded measured soiling loss and weather data, and applied to predict soiling loss based on TMY
or next-year forecast weather data. A proof-of-concept of this approach is presented in Section 5.8.

A.3.2 Angular Loss

PV SoilSayer utilizes semi-physical models to calculate hourly soiling losses as a function of sun
angle and direct and diffuse irradiance throughout the day. Users have the option to turn on/off
angle of incidence effects.

Figure A.12: Angular Losses UI.

A.3.3 Rainfall Cleaning Calculations

PV SoilSayer derates the soiling losses according to rainfall (unless measured soiling losses are
uploaded, in which case the effects of rainfall are assumed to be already included in the measure-
ments). Default values are standard based-on available research; however, these values may vary
from site to site. Users can configure Rainfall parameters to estimate the cleaning effects of rainfall
for a given site’s design and location.

The Minimum Rainfall Clean Amount and Maximum Rainfall Clean Amount specify the
amount of rain that is required to clean the panels during the wet season and dry season, respectively.
During periods with regular rain, the panels may be easier to clean than after a dry period where

193

dust and debris have baked onto the panels. Any rainfall below the specified Rainfall Clean Amount
is proportioned as a ratio of the Rainfall Clean Amount.

Minimum Duration of the Dry Period to Set Max Rainfall Clean Amount is the amount
of time between rainfall events required before the Maximum Rainfall Clean Amount is used to
calculate a Rainfall Clean Event. The Minimum Rainfall Clean Amount is used until this minimum
duration without rainfall events occurs. Essentially, this defines what is considered as a "wet" and
"dry" period.

The Minimum Rainfall Resulting in Performance Recovery is the amount of rain within 1
hour that will result in a decreased Soiling Ratio (i.e. will result in higher performance of the
PV panels). Different values of this quantity can be specified for the dry and wet periods.
In
some cases, especially with panels at very low tilt angles under approximately 10◦, small rainfall
amounts will not clean the panels (and may even have a reverse effect, resulting in an increased
Soiling Ratio). Panels installed at higher tilt angles and during the wet season may need minimal
rainfall to see a decrease in the Soiling Ratio.

It is often the case that while rainfall mostly cleans the panels, some residue remains. The
Rainfall Clean Quality specifies how clean the panels will be after receiving rainfall equal to the
Rainfall Clean Amount. For example, a 1% Rainfall Clean Quality says that an average daily loss of
1% remains directly after each Rainfall Clean Event (i.e. after the Rainfall Clean Amount threshold
has been exceeded).

Minimum Duration of the Soiling Ratio to Set Rainfall Clean Quality is the amount of
time required to maintain a soiling ratio greater than the Rainfall Clean Quality, in order to set the
Rainfall Clean Quality. If the Starting Soiling Ratio is less than the Rainfall Clean Quality, the
Rainfall Clean Quality will not be applied until the Soiling Ratio defined by the Rainfall Clean
Quality is exceeded for the specified number of days. In many cases, the soiling will adhere to the
surface of the glass over time, which results in higher rainfall needed to detach the soiling from the
panels. This depends on several factors, including the type of glass, the type of soiling, and the
weather conditions.

194

Figure A.13: Rainfall Parameters UI.

A.3.4 Monthly Average Soiling Losses

After setting the Soiling Profile parameters, clicking Calculate Soiling Losses will run a soiling loss
calculation and display a preview of the monthly results for a panel that is never manually cleaned.
Hovering the mouse over the plot data will display the numerical values for total precipitation and
DC and AC soiling loss. Here users can optionally adjust the monthly soiling values to better match
any known local conditions.

Figure A.14: Sample Monthly Average Soiling Losses Output.

195

A.4 Clean Cycle Configuration

The Clean Cycle Configuration (Figure A.15) allows the user to specify which cleaning cycle
should be calculated and the parameters defining that cleaning cycle. One can calculate an optimal
clean cycle that minimizes cost and maximizes performance, a cleaning cycle that records a cleaning
event when specific thresholds (average daily soiling loss, max financial loss, or soiling loss at solar
noon) are reached, or manually enter the cleaning dates. This page also contains the user inputs
for the Manual Clean Quality (for modeling non-perfect manual cleans) and the number of days
needed for cleaning.

Figure A.15: Clean Cycle Configuration UI.

196

A.4.1 Optimal Clean Cycle

This option calculates the optimal time to clean based on the energy gain from cleaning versus
not cleaning the PV panels. The calculation of the optimal clean cycle considers three different
scenarios at each hour of the year:
No Soil: This scenario calculates the energy production assuming that the panels are always clean.
No Clean: This scenario calculates the energy production assuming that the panels are never
cleaned manually.
Clean: This scenario calculates the optimal days to clean the panels throughout the year in order
to maximize revenues.

The result of this calculation is an optimal clean cycle that is expected to yield the maximum
revenue gain considering the cost of cleaning) over not cleaning the

return-on-investment (i.e.
system.

Figure A.16: Optimal Clean Cycle UI.

The Minimum Value of Energy Gain is calculated as the net revenue gain after subtracting the
cost of cleaning the PV plant. The Minimum Value of Energy Gain filters the results to ensure that
only the Manual Clean Events above a specified net revenue gain threshold are used. Setting the
value too high may result in no cleans. The default value is calculated at 100% of the cleaning cost.
Users have the option to run the Optimal Clean Cycle calculation with a weekly or daily
resolution. While both resolutions calculate soiling losses on an hourly resolution, this option
selects the precision to which the Manual Clean Events are optimized. The weekly resolution tries
to find the optimal weeks to clean the panels, while the daily resolution tries to find the optimal

197

days.

Calculations at the weekly resolution (default) select the optimal cleaning day with a possible
error in the cleaning optimization (independent of soiling and weather prediction errors) of ±4
days. This calculation is typically returned within 1-3 minutes and provides users with reasonable
estimates of clean cycles and associated costs. Since it may be impractical to schedule cleaning
services on a precise day, and since large PV plants take multiple days to clean, the weekly resolution
is expected to be sufficient for most applications. Additionally, the errors in soiling and weather
prediction may affect the optimal clean events by greater than ±4 days, and therefore the weekly
optimization provides a good balance between accuracy and computational effort. Calculations at
the daily resolution will attempt to find the exact days for manual cleanings. These may result in
slightly different calculated Manual Clean Events and resulting revenue. These calculations often
require significant additional processing times (e.g. 10-60 minutes); however, automated emails
can be configured to be sent upon calculation completion.

198

A.4.2 Clean Based on Specific Thresholds

In these cleaning cycles, Manual Clean Events are triggered by simple thresholds. This option allows
users to model common techniques used in the industry for cleaning scheduling and compare the
predicted results to results from PV SoilSayer’s optimal clean cycle. Users can choose one of
several options for triggering manual cleans in order to calculate the resulting clean cycle and
associated revenue gains/losses.

Figure A.17: Clean Based on Specific Thresholds UI.

Clean Based on Average Daily Soiling Loss: A Manual Clean Event is triggered when the
average daily soiling loss reaches the specified amount. A soiling loss threshold in terms of AC or
DC energy production can be specified.

Clean Based on Max Financial Loss: A Manual Clean Event is triggered when the accumulated

loss of revenue due to soiling reaches the specified amount.

Clean Based on the AC Soiling Loss at Solar Noon: A Manual Clean Event is triggered when

the AC soiling loss at solar noon reaches the specified amount.

A.4.3 Clean Based on Specified Clean Cycle

In this option, users select the dates or cleaning frequency to clean the system and the resulting
energy loss and revenue loss due to the specified Clean Cycle are reported.

199

Figure A.18: Clean Based on Specified Cycle UI.

A.4.4 Manual Clean Quality and Days Required to Clean

Often cleaning the panels does not remove 100% of the soiling. The Manual Clean Quality allows
specifying the quality of each manual cleaning in terms of the soiling loss (measured at solar noon)
remaining after the cleaning. The Number of Days Required to Clean allows to account for
extended cleaning procedures that take several days or weeks at larger PV systems. This assumes
that the cleaning process is evenly distributed among the days required to clean and accounts for
the combination of higher soiling losses experienced by portions of the system that have not been
cleaned with the lower soiling losses of portions of the system that have been cleaned.

A.4.5 Value of Energy and Cost of Cleaning

Finally, the financial parameters need to be specified. These are the value of energy that is produced
($/kWh), either as an annual constant rate or as monthly variable rates, and the cost of cleaning,
which can be input as a total dollar amount per clean or dollar amount per kilowatt-peak system
size rating.

200

APPENDIX B

PV MODULE DATASHEETS

201

202

203

PD-5-401-04-3 | September 2018First Solar, Inc. | firstsolar.com | info@firstsolar.comFirst Solar Series 4™  PV Module ADVANCED THIN FILM SOLAR TECHNOLOGYINDUSTRY BENCHMARK SOLAR MODULESAs a global leader in PV energy, First Solar’s advanced thin film solar modules have set the industry benchmark with over 17 gigawatts (GW) installed worldwide and a proven performance advantage over conventional crystalline silicon solar modules. Generating more energy than competing modules with the same power rating, First Solar’s Series 4™ and Series 4A™ PV Modules deliver superior performance and reliability to our customers.CERTIFICATIONS & TESTS• PID-Free, Thresher Test, Long-Term Sequential Test, and ATLAS 25+1• IEC 61215/61646 1500V, IEC 61730 1500V, CE• IEC 61701 Salt Mist Corrosion, IEC 60068-2-68 Dust and Sand Resistance• ISO 9001:2015 and ISO 14001:2015• UL 1703 Listed Fire Performance PV Module Type 102• CSI Eligible, FSEC, MCS, CEC Listed (Australia), SII, InMetroADVANCED PERFORMANCE & RELIABILITY• Compatible with advanced 1500V plant architectures• Independently certified for reliable performance in high temperature, high humidity, extreme desert and coastal environments• Visit PlantPredict.com - The only Energy Prediction Software designed for Utility Scale PVPROVEN ENERGY YIELD ADVANTAGE• Generates more energy than conventional crystalline silicon solar modules with the same power due to superior temperature coefficient and superior spectral response• Anti-reflective coated glass (Series 4A™) enhances energy productionEND-OF-LIFE RECYCLING• Recycling services available through First Solar’s industry-leading recycling program or customer-selected third party.MODULE WARRANTY3• 25-Year Linear Performance Warranty4• 10-Year Limited Product Warranty86.0%90.0%94.0%98.0%100.0%0510152025YearsPower Output122.5 WATT MODULE EFFICIENCY OF 17.0%204

APPENDIX C

LIST OF PUBLICATIONS RESULTING FROM THIS WORK

Published

• Timofey Golubev, Dianyi Liu, Richard Lunt, and Phillip Duxbury. Understanding the impact
of C60 at the interface of perovskite solar cells via drift-diffusion modeling. AIP Advances,
9, 035026, 2019.

In-Progress

• Timofey Golubev, Richard Lunt, and Phillip Duxbury. Evaluating the electricity production of
electric vehicle integrated photovoltaics via a coupled modeling approach. Will be submitted
to 2021 IEEE 48th Photovoltaic Specialists Conference (PVSC).

205

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