Post on 18-Oct-2020
Using a Financial Model to Determine Technical Objectives for Organic Solar Cells
by
Colin David Lloyd Powell
A thesis submitted in conformity with the requirements for the degree of Master of Applied Science
in the Department of Chemical Engineering and Applied Chemistry
at the University of Toronto
Supervised by:
Professor Yuri Lawryshyn Professor Timothy Bender
© Colin Powell 2010
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Using a Financial Model to Determine Technical Objectives for Organic Solar Cells
Colin Powell
Master of Applied Science
Department of Chemical Engineering and Applied Chemistry Faculty of Applied Science and Engineering
University of Toronto
2010
ABSTRACT
Organic solar cells (OSCs) are of interest because the technology offers a significant opportunity to
reduce the overall costs of solar energy. OSCs can be very inexpensive to produce given that they rely
on non‐commodity materials and can use existing manufacturing techniques that are not labour‐ and
capital‐intensive. In this research, a financial model, named TEEOS (Technological and Economic
Evaluator for Organic Solar), is developed and is used to determine financial indicators, such as
simple payback period. These indicators are used to determine technical objectives for the OSCs. Two
sample cells are evaluated in Toronto, Canada using historical data. The results show that the cell
with a higher efficiency and wider absorptive wavelength range produces a payback period of
approximately nine years, while the other cell has a payback period well over 45 years. Stochastic
modeling techniques are also used to better replicate electricity price and weather fluctuations.
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Acknowledgements
I would like to express my sincere sincere gratitude to:
Professor Yuri Lawryshyn and Professor Timothy Bender, my co‐supervisors, for their
constant support, patience, and encouragement to explore that which I did not know and
their leadership and guidance when I may have faltered or become weary;
Professor Joseph Paradi and Dr. Judy Farvolden for their leadership and support at CMTE
and providing me the opportunity to excel in such a wonderful environment;
All those at CMTE who have helped me in every way for the past 19 months: Kelsey Barton,
Steve Frensch, Pulkit Gupta, Leili Javanmardi, Erin Kim, Laleh Kobari, Elizabeth Min, Susan
Mohammadzadeh, Joe Pun, Sanaz Sigaroudi, Justin Toupin, Angela Tran, Haiyan Zhu,
Muhammad Saeed;
The Bender Group for allowing me the opportunity to have my name on two doors in
Wallberg and for doing all the hard work for the development of organic solar cell materials;
Sau Yan Lee and Dan Tomchyshyn for helping me with all my computer needs;
The administrative staff at the Department of Chemical Engineering and Applied Chemistry;
without them I’d probably still be trying to apply for this degree;
My friends and family in Toronto, St. John’s, and elsewhere that have certainly kept me
grounded for the past 19 months;
Toronto, for helping me stay grounded as well.
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TABLE OF CONTENTS
ABSTRACT………………………………………………………………………………………………….……..………………………..…i
ACKNOWLEDGEMENTS………………………..……………………………………………………...……….……..……...……..…iii
TABLE OF CONTENTS………………………………………………………………………………………………………….….….…iv
LIST OF TABLES…………………………………………………………………………..…………….………………………..……… vi
LIST OF FIGURES…………………………………………………………………………..…………….…………………….…..… viii
LIST OF APPENDICES…………………………………………………………………………..…………….…………………..….. ix
1 Overview................................................................................................................................................................................ 2
1.1 Objectives .................................................................................................................................................................... 3
1.2 Thesis Outline ............................................................................................................................................................ 4
1.3 References ................................................................................................................................................................... 6
2 A model to determine financial indicators for organic solar cells ............................................................... 8
2.1 Background................................................................................................................................................................. 8
2.2 Abstract ........................................................................................................................................................................ 9
2.3 Introduction ............................................................................................................................................................... 9
2.4 Methodology ............................................................................................................................................................ 11
2.4.1 Weather Conditions and Irradiance Data ......................................................................................... 12
2.4.2 Modelled Irradiance Data ........................................................................................................................ 13
2.4.3 Cloud Modification Factor (CMF) ......................................................................................................... 15
2.4.4 Modified Solar Irradiance ........................................................................................................................ 16
2.4.5 Electricity Pricing Data ............................................................................................................................. 17
2.4.6 Technological Characteristics ................................................................................................................ 18
2.4.7 Financial Indicators.................................................................................................................................... 19
2.5 Sample Calculation ................................................................................................................................................ 21
2.5.1 Data ................................................................................................................................................................... 21
2.5.2 Results .............................................................................................................................................................. 22
2.5.3 Optimization Scenarios ............................................................................................................................ 23
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2.6 Conclusions ............................................................................................................................................................... 23
2.7 References ................................................................................................................................................................. 25
3 Using TEEOS to determine payback periods for organic solar cells ......................................................... 28
3.1 Initial Background ................................................................................................................................................. 28
3.2 Abstract ...................................................................................................................................................................... 28
3.3 Introduction ............................................................................................................................................................. 29
3.4 Model Development .............................................................................................................................................. 30
3.4.1 Weather Conditions and Irradiance Data ......................................................................................... 31
3.4.2 Cloud Modification Factor (CMF) ......................................................................................................... 32
3.4.3 Modified Solar Irradiance ........................................................................................................................ 33
3.4.4 Electricity Pricing Data ............................................................................................................................. 33
3.4.5 Technological Characteristics ................................................................................................................ 34
3.4.6 Ancillary Costs .............................................................................................................................................. 35
3.4.7 Financial Indicators.................................................................................................................................... 36
3.5 Results and Discussion ........................................................................................................................................ 36
3.5.1 Wavelength‐Cost Distribution ............................................................................................................... 36
3.5.2 Cell Comparison ........................................................................................................................................... 37
3.5.3 Comparison with Inorganic Solar cells .............................................................................................. 38
3.5.4 Standard Offer Contract ........................................................................................................................... 39
3.6 Sensitivity Analysis ............................................................................................................................................... 40
3.6.1 CMF Sensitivity ............................................................................................................................................. 40
3.6.2 Electricity Price Sensitivity ..................................................................................................................... 41
3.6.3 Initial Cost Sensitivity ............................................................................................................................... 42
3.6.4 Wavelength Range and Efficiency........................................................................................................ 43
3.7 Conclusions ............................................................................................................................................................... 49
3.8 References ................................................................................................................................................................. 51
4 Using stochastic models in TEEOS to determine financial indicators and technical objectives for organic solar cells ........................................................................................................................................................................ 54
4.1 Abstract ...................................................................................................................................................................... 54
4.2 Introduction ............................................................................................................................................................. 54
4.3 Electricity prices ..................................................................................................................................................... 55
4.3.1 Electricity price model .............................................................................................................................. 57
4.4 Markov Chain Weather Patterns ..................................................................................................................... 61
4.4.1 Modifications for TEEOS .......................................................................................................................... 62
4.4.2 Parameter Estimation ............................................................................................................................... 63
4.4.3 Verification of the Synthetic Dataset .................................................................................................. 69
4.5 Results ........................................................................................................................................................................ 70
4.6 Sensitivity Analysis ............................................................................................................................................... 74
4.6.1 CMF .................................................................................................................................................................... 74
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4.6.2 Correlation of Weather to Electricity Prices ................................................................................... 75
4.6.3 Modelling without Jumps ........................................................................................................................ 76
4.6.4 Averaged Data .............................................................................................................................................. 77
4.7 Technical Objectives ............................................................................................................................................. 78
4.7.1 Initial Costs .................................................................................................................................................... 78
4.7.2 Wavelength Range and Efficiency........................................................................................................ 79
4.7.3 Cell Lifetime ................................................................................................................................................... 80
4.8 Conclusions ............................................................................................................................................................... 80
4.9 References ................................................................................................................................................................. 82
5 Final Conclusions ............................................................................................................................................................. 85
5.1 Limitations of the model ..................................................................................................................................... 85
5.1.1 CMF .................................................................................................................................................................... 85
5.1.2 Cost Data ......................................................................................................................................................... 86
5.2 Future Work ............................................................................................................................................................. 86
5.3 Conclusions ............................................................................................................................................................... 87
5.4 References ................................................................................................................................................................. 89
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LIST OF TABLES Table 2.1 ‐ Technological characteristics of the organic solar cells attached to solar hats……………..21 Table 3.1 ‐ Average monthly CMF for Toronto, ON based on 2006 and 2007 irradiance data and weather………………………………………………………………………………………………………………………………………32 Table 3.2 ‐ Volumetric Electricity Charges other than the HOEP from Toronto Hydro [7] ........................ 34 Table 3.3 ‐ Technological Characteristics for Two Sample Organic Solar Cells ............................................... 34 Table 3.4 ‐ Summary of solar cell cost, balance of system costs, and total initial cost for the two example cells. A low and high range is given for each according to the ranges determined in the previous section. .......................................................................................................................................................................... 35 Table 3.5 ‐ Annual Savings and Payback Period for 2006 and 2007 for the two example solar cells. Brackets indicate negative values. The two values for payback period reflect the low and high range of initial costs given in the previous section for a five year project lifetime. ................................................... 38 Table 3.6 ‐ Net Present Value for 2006 and 2007 for the two example solar cells with five and fifteen year project lifetimes. Brackets indicate negative values. The two values reflect the low and high range of initial costs given in the previous section. ..................................................................................................... 38 Table 3.7 ‐ Annual Savings, Payback Period, and NPV for the two example cells using the Standard Offer Contract. The payback period is calculated using the initial cost for a five‐year project lifetime. The two values for each reflect the low and high range of in the previous section. ...................................... 39 Table 3.8‐ The range of each numbered wavelength range category shown in Figures 7 ‐ 10. ............... 44 Table 3.9‐ Wavelength range and efficiency of cells that have similar payback periods ............................ 47 Table 4.1 ‐ Electricity price summary statistics for HOEP between May 1, 2002 and June 30, 2009. The mean, maximum and minimum have units of $/MW‐h (CAD). ...................................................................... 58 Table 4.2 ‐ Jump Parameters ‐ Mean, Standard Deviation, and Poisson Parameter of the HOEP between May 1, 2002 and June 30, 2009 .......................................................................................................................... 59 Table 4.3‐ Significance test for difference of mean of Annual Savings with respect to Markov order for weather model .............................................................................................................................................................................. 64 Table 4.4 ‐ Frequency of the historical weather series in January in the underlying dataset. ‘0’ represents a cloudy hour and ‘1’ represents a clear hour. ........................................................................................ 65 Table 4.5‐ Cloudy and clear spell probabilities of various lengths for historical and synthetic weather series. Synthetic weather series completed with 1000 iterations. ....................................................................... 69 Table 4.6 ‐ Regression analysis between wet spells and probabilities of various sequence length for historical and synthetic weather series. Synthetic weather series completed with 1000 iterations. ... 69 Table 4.7 ‐ Technolgoical Characteristics of Two Example Cells ........................................................................... 71 Table 4.9 ‐ Annual Savings of DSSC and OSC using correlated weather series and non‐correlated weather series. .............................................................................................................................................................................. 76 Table 4.10 ‐ Financial Indicators for DSSC and OSC without modelling jumps in electricity prices ..... 77
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LIST OF FIGURES
Figure 2.1 ‐ Flow Chart of TEEOS Process ........................................................................................................................ 12 Figure 2.2 ‐ CMFs for January 2006 and 2007 in Toronto, Canada. The markers represent hourly data points for that month and weather condition. The straight lines have a slope equal to the robust slope, showing the trend. ......................................................................................................................................................... 16 Figure 3.1 ‐ Flow Chart of TEEOS Process [17] .............................................................................................................. 31 Figure 3.2‐ Integrated irradiance over one year in Toronto, Canada .................................................................. 33 Figure 3.3‐ Contribution of each wavelength range to annual savings in Toronto, Canada ...................... 37 Figure 3.4 ‐ Sensitivity of payback period and NPV to CMF for DSSC in 2006 using the lower initial cost value of $1287. The vertical line represents the point of the current CMF for 2006 – 0.3324. ...... 41 Figure 3.5 ‐ Payback Period and NPV sensitivity to electricity price changes for the OSC in 2006 using the lower range of the five‐year lifetime project initial cost. ................................................................................... 42 Figure 3.6‐ Payback Period and NPV sensitivity to Initial Costs. The vertical line represents the current initial cost estimate of $1575. ............................................................................................................................... 43 Figure 3.7 ‐ Payback periods of organic solar cells with different combinations of wavelength ranges at 5% efficiency. ........................................................................................................................................................................... 45 Figure 3.8 ‐ Payback periods of organic solar cells with different combinations of wavelength ranges at 10% efficiency ......................................................................................................................................................................... 45 Figure 3.9 ‐ Payback periods of organic solar cells with different combinations of wavelength ranges at 15% efficiency. ........................................................................................................................................................................ 46 Figure 3.10 ‐ Payback periods of organic solar cells with different combinations of wavelength ranges at 20% efficiency ......................................................................................................................................................................... 46 Figure 4.1 ‐ Histogram of time between jumps for historical electricity price series .................................. 59 Figure 4.2 ‐ Probability estimates for cloud occurrence model for Toronto, Canada ................................... 64 Figure 4.3 ‐ Histogram of ΔS parameter for weather sequence 0100 in January ........................................... 66 Figure 4.4 ‐ CDF and the function Φ from eqn. (4.6) of the ΔS distribution of the Sequence 0100 in January. The R‐squared value for the fit is 0.9957. ...................................................................................................... 67 Figure 4.5 ‐ Histogram of Annual Savings of DSSC using 1000 iterations ......................................................... 71 Figure 4.6 ‐ Histogram of Annual Savings of OSC using 1000 iterations ............................................................ 72 Table 4.8 ‐ Different ranges of solar cell and BOS costs ............................................................................................. 72 The different costs represent the high and low range of initial costs, including two mid‐level prices. This will give an indication of the range of payback period and NPV that results when the large range of the estimate of the initial costs is taken into account. Figure 4.7 and Figure 4.8 show the payback period and NPV for the DSSC and OSC, respectively. ................................................................................................... 72 Figure 4.7 ‐ Payback Period and NPV for DSSC .............................................................................................................. 73 Figure 4.8 ‐ Payback Period and NPV for OSC ................................................................................................................ 73 Figure 4.9 ‐ Sensitivity of NPV to CMF in the DSSC ...................................................................................................... 75
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LIST OF APPENDICES
Appendix A – Inputs for SMARTS2 Modelling……………………………………………………………………………...90 Appendix B – Historical Hourly Volatility, Mean Reversion Levels and Mean Reversion Rates……….91 Appendix C – Synthetic Hourly Volatility, Mean Reversion Levels, and Mean Reversion Rates – 1000 iterations……………………………………………………………………………………………………………………….………….94
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CHAPTER 1
OVERVIEW
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1 Overview Photovoltaics (PVs) generate electrical energy using solar cells to convert energy from the sun into
electricity. The initial purpose for the development of photovoltaics was to generate electricity in
space for the use of orbiting satellites and other spacecraft. Since then, an increasing amount of
photovoltaic modules are used to provide electricity for grid‐connected power installations to power
homes, businesses, and industry.
There are three main types of solar cells: first, second and third generation.
First generation PVs consist of the typical silicon‐based mono crystalline cells that are seen on
buildings and solar farms around the world. These cells use the oldest technology, but still have the
highest efficiency, longest lifetime, and make up more than 90% of the total solar cell market share
[1]. The average sunlight to electricity efficiency for this technology is approximately 25% [1].
Second generation PVs consist of flexible solar cells that can be used for a wide variety of
applications, are very adaptable and are inexpensive compared to the first generation photovoltaics
systems. However, these also have a much lower efficiency at approximately 16% [1].
Third generation solar cells consist of essentially all new solar technologies, such as quantum dots,
nanocrystallines, and organic solar cells (OSCs). These cells are still in development and have not
been fully commercialized. They have the potential to be very low cost, but this is hindered by low
stability and low efficiency. This generation will be discussed in this thesis, specifically the organic
solar cells.
Dr. Alan Heeger is thought of as the father of OSCs, sharing the Nobel Prize in Chemistry in 2000 for
the discovery and development of conductive polymers. It was thought at the time that this
technology could revolutionize the PV industry with low‐cost, high‐efficiency and flexible materials
that would leapfrog current solar technologies. This is not unfounded; OSCs, due to their ease of
manufacturing, inexpensive materials, flexibility and adaptability, have the potential to be two to
three orders of magnitude less expensive than the first generation PV technologies.
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However, OSCs are still not fully commercialized. There is further research needed to increase
efficiency, increase stability (cell lifetime) and develop suitable delivery systems for this technology.
Indeed, while work has been ongoing on the material side of OSCs, there has been little work
completed on the costs of these OSCs. While it is expected that this technology will be inexpensive,
very few practical demonstrations have been performed to prove or disprove this.
1.1 Objectives
This thesis was borne out of the need for a practical application that allows solar material
researchers the opportunity to better estimate the financial viability of a novel solar material before
full commercialization. One objective of this thesis is to evaluate the financial viability of OSC
technology from a consumer (retail) perspective in order to better direct future technological
research efforts. The model developed in this study, named TEEOS (Technical and Economic
Evaluator for Organic Solar), provides a methodology to determine economic feasibility for organic
solar cells, as well as provides information regarding the economics of certain research decisions.
The model is highly dependent on available data: weather, spectral irradiance, and electricity prices.
If actual broadband irradiance data is available, the analysis is simpler; but this data is not available
for most locales, necessitating the approach outlined here. For better accuracy, an hourly step size is
used for all required data. Other applications that determine financial indicators for solar cells do
exist, but use average daily or monthly data for weather and the cost of electricity and only include
current technologies. RETScreen [http://www.retscreen.net], for example, is a valuable tool for
renewable energy decision‐making and project valuation, but is not flexible enough to be able to
financially evaluate an OSC solely based on the material used in its structure and also uses averaged
data and other assumptions unsuitable to OSCs. And although TEEOS has nowhere near the varied‐
functionality of RETScreen, it serves a specific purpose and is valuable to novel material researchers
looking to determine the practicality of the material that they have developed.
Another purpose of the TEEOS model is to determine certain technical objectives that
researchers/manufacturers of OSC need to meet in order to make the devices financially viable in
4
different locations. There are three main factors that contribute to the success of a new solar
technology, such as cost of production, efficiency, and material. While OSCs have the potential to be
very inexpensive, they have much lower in efficiency than first generation PVs. TEEOS can
determine, for example, at what cost per cell or per square metre a particular OSC has to reach so
that, given a certain efficiency, it can produce reasonable payback periods and returns on investment
for a consumer.
1.2 Thesis Outline
This study is divided into five chapters. The first chapter provides an overview of the photovoltaic
industry, the research topic, and the thesis outline.
The second chapter1 outlines the methodology behind the TEEOS model as well as sources of data.
The model is highly data‐sensitive as it requires electricity price and weather data that is sometimes
difficult to obtain, depending on locale. Similarly, it is difficult to readily obtain irradiance data
outside the UV spectrum for most locations and this must be estimated, as is done here. The TEEOS
model is used to perform a quick calculation using solar hats as an example. As a demonstration, the
hats do not provide much economic benefit and at such a low efficiency (0.013%), barely provide
enough electricity to charge a battery in a radio.
The third chapter of this thesis discusses the results from the TEEOS model using domestic electricity
production with two sample cells in Toronto, Ontario as an example. The results show that the dye‐
sensitized solar cell (DSSC), with a higher efficiency and wider absorptive wavelength range,
produces a payback period of approximately nine years. The OSC, however, has a very high payback
period, well over 45 years, as well as a negative NPV. The financial indicators dramatically improve
when taking advantage of an incentive program offered by the government in the province of
Ontario: the OSC (with an efficiency of 3.4%) achieves a payback period of approximately nine years
1Chapters 2, 3, and 4 are papers prepared as part of this research. The paper for Chapter 2 has been accepted for publication; the paper for Chapter 3 has been submitted and is under review. The paper for Chapter 4 will soon be submitted.
5
with a negative NPV and the DSSC has a very high net present value (NPV). A sensitivity analysis is
conducted on a number of input variables, as well.
The fourth chapter of this thesis uses stochastic modeling techniques to better replicate electricity
price and weather fluctuations. A mean‐reverting model is used for electricity prices and a Markov
chain rainfall occurrence model is used for weather. A seven year data set is used for determining the
parameters of both. These models are incorporated into the TEEOS model in order to provide more
accurate financial indicators. Technical objectives that need to be met for competitiveness are
evaluated using the results from this section.
The final section of this thesis outlines the limitations of this model and how it can be improved in
the future. As well, future work is discussed, such as the development of a “dashboard”‐like feature
for easy use of the algorithms outlined here.
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1.3 References
[1] Tao, M., Inorganic PV solar cells: silicon and beyond, Electrochemical Society Interface (2008).
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CHAPTER 2
A MODEL TO DETERMINE FINANCIAL INDICATORS FOR ORGANIC SOLAR CELLS
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2 A model to determine financial indicators for organic solar cells
2.1 Background
This paper begins by introducing the concept of organic solar cells in the context of financial
indicators, such as payback period, net present value (NPV) and internal rate of return (IRR). The
description of the methodology and algorithms of the TEEOS (Technical and Economic Evaluator for
Organic Solar) model is presented here in detail. A simple flow chart, shown in Figure 1, illustrates
the process and the flow of data. This paper is long by journal standards, so only a short example is
used to demonstrate the model. A much more detailed analysis is shown in Chapter 3, which is
largely taken from another submitted manuscript.
The methodology description is the anchor of this thesis; indeed, it introduces financial indicators to
those in the field of organic solar cell research. There is very little literature dealing with financial
aspects of organic solar cells; the Risø National Laboratory in Denmark has published preliminary
cost estimates of organic solar cells that have been recently demonstrated, such as the solar hat
example presented in this paper. However, this paper is the first on the modeling of financial
indicators for organic solar cells.
A host of literature is available on financial models for commercially available silicon solar cells and
is discussed briefly in this chapter. However, a more accessible, flexible and accurate approach is
taken here to allow for the different parameters of an organic solar cell, most importantly the
different absorptive wavelength ranges.
The reference for this paper is:
Powell, C., T. Bender, Y. Lawryshyn, A model to determine financial indicators for organic solar cells,
Solar Energy 83 (2009) 1977‐1984.
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2.2 Abstract
Organic solar cells are an emerging photovoltaic (PV) technology that is inexpensive and easy to
manufacture, despite low efficiency and stability. A model, named TEEOS (Technical and Economic
Evaluator for Organic Solar), is presented that evaluates organic solar cells for various solar energy
applications in different geographic locations, in terms of two financial indicators, payback period
and net present value (NPV). TEEOS uses SMARTS2 software to estimate broadband (280 ‐ 4000 nm)
spectral irradiance data and with the use of a cloud modification factor, predicts hourly irradiation in
the absence of actual broadband irradiance data, which is scarce for most urban locations. By using
the avoided cost of electricity, an annual savings is calculated which produce the financial indicators.
It is hoped that these financial indicators can help guide certain technical decisions regarding the
direction of research for organic solar cells, for example, increasing efficiency or increasing the
absorptive wavelength range. A sample calculation using solar hats is shown to be uneconomical, but
a good example of large scale organic PV production.
2.3 Introduction
Organic or polymer solar cells, made entirely from plastic and other organic materials, provide
significant advantages over traditional PV technologies, such as crystalline Si and thin film cells, but
still have significant headway to make in a number of fields. Organic PV cells are “one of the future
key technologies opening up completely new applications and markets for photovoltaics” [1]. The
main advantages of organic PV fall into two categories: inexpensive synthesis and ease of
manufacture [2]. Their low cost can be attributed to simple and established methods for synthesis
and their easy manufacture using existing printing press technologies. Also, the amount of organic
compounds used and the energy needed to produce and manufacture the organic cells can be small,
implying that ecological damage can be minimal. However, the cost and environmental impact of
organic PV depends on the process used for their manufacture. Krebs [3] states that the ideal
manufacturing process would result in a polymer solar cell that has a “low environmental impact and
a high degree of recyclability” with a minimal number of production steps. An extensive review on
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printing and coating techniques for producing organic PV is available in [3]. However, while organic
PV has the potential to have low process costs compared to other PV technologies [1,2, 4‐9], a main
stumbling block to commercialization is low efficiency and short lifetime due to low stability [1,2,4,6‐
8].
The power conversion efficiency of an organic PV cell, defined as the ratio of the energy that is
available for consumption to the energy available from the sun, is currently less than 5% [2, 4‐9, 10‐
12] for small‐area devices. The upper bound on the efficiency for organic PV has been stated at
approximately 20% [13, 14], but a more reasonable value is likely around 10% [5, 11, 12]. Sargent
[15] notes that higher overall solar power conversion can be achieved with tandem cells of different
band gaps, and this is especially true of polymers with a lower band gap [5, 6, 12]. Indeed, the highest
reported energy conversion efficiency of 6.5% was reported using polymers of different band gaps in
tandem [11]. While the vast majority of these tandems only consist of semi‐conductors in the visible
range, Bungaard & Krebs [6] have completed a thorough review of low band gap polymers with
respect to organic photovoltaics, outlining different materials, performance, and design
considerations.
Krebs et al. [4] have provided the first cost structure of the production of organic solar cells.
Specifically, the authors provide a breakdown of the different steps used in the production of organic
PV as well as material and labour costs. While their experimental tests are only based on small
individual modules, they concede that there exists great potential for cost reduction by using cheaper
labour, increasing throughput, and most importantly, reducing material costs. A recent survey of
solar energy experts has shown that organic polymer PV will have one of the best cost structures
(price per peak watt) compared to a range of cell types, such as crystalline Si, thin‐film and other
excitonic and novel technologies [14].
The model developed in this study, named TEEOS (Technical and Economic Evaluator for Organic
Solar), will provide a methodology to determine economic feasibility for organic solar cells, as well as
provide information regarding the economics of certain research decisions. The model is highly
dependent on available data: weather, spectral irradiance, and electricity prices. If actual broadband
irradiance data is available, the analysis is simpler; but this data is not available for most locales,
11
necessitating the approach outlined here. This paper will discuss the methodology behind the model
and provide a short sample calculation. Future papers will discuss results and different scenarios
that arise from this model being applied to certain locations and cells and incorporate different cost
structures outlined by other researchers.
2.4 Methodology
Figure 2.1 shows a flow chart that outlines the inputs and outputs of various stages of TEEOS. The
model is divided into three sections. The first compares actual irradiance data to modeled clear‐sky
irradiance data in the UVB range and, using actual weather conditions, produces a cloud modification
factor (CMF) to model the effect of clouds on spectral irradiance. The second step combines this CMF
and the modeled broadband spectral irradiance data to produce an estimate of the hourly irradiance
for the chosen location. The third section uses hourly electricity prices, hourly irradiance values, and
a variety of cell characteristics to produce financial indicators in order to evaluate the specific solar
cell.
12
Figure 2.1 Flow Chart of TEEOS Process
The model can be simplified if actual broadband spectral irradiance data is available for the chosen
location; the first two steps, used in the absence of spectral irradiance data, can be eliminated. This
model is applicable to any location, provided there is sufficient spectral irradiance, weather, and
electricity price data.
2.4.1 Weather Conditions and Irradiance Data
Hourly weather conditions and spectral irradiance data are obtained from government
meteorological agencies, as well as universities and research centres. While it is a safe assumption
that clear‐sky spectral irradiance data will be constant throughout the lifetime of the organic cell, the
weather conditions will not be constant. It is possible to use past data to generate weather
conditions using weather generators. See Wilks & Wilby [16] for a review of stochastic weather
models.
13
There are a variety of descriptors used by meteorological agencies to identify the actual weather
conditions occurring at a given time in a given location. For simplicity, the model initially divided
these into four groups: cloudy, mostly cloudy, clear and mainly clear. All precipitation events,
including rain, snow, drizzle, and thunderstorms, as well as visibility‐reducing events, such as haze
and fog, are included in the cloudy group. Due to ambiguity, the groups are consolidated so that the
clear group also includes the mostly cloudy and mainly clear weather conditions, leaving just two
weather conditions to consider: cloudy and clear.
The spectral irradiance data consists of irradiance data over a certain wavelength range at different
times throughout each day. If data is available that covers the absorptive wavelength range of the PV
cell in question, then the model simplifies greatly.
2.4.2 Modelled Irradiance Data
To accurately measure the economic feasibility of organic PV, one needs accurate solar radiation data
across a large spectrum. There are in fact very few locations that record spectral irradiance data
outside the UV and visible ranges. This makes the case for modeling programs that estimate the daily
solar irradiation; they make up for a lack of consistent and robust spatial and temporal irradiation
measurements and can accurately predict broadband spectral irradiance, such as IR.
There are two main types of spectral irradiance models: sophisticated rigorous code, such as
LOWTRAN and MODTRAN, and simple transmittance parameterized models, such as SMARTS2, the
Simple Model of the Atmospheric Radiative Transfer of Sunshine developed by Christian Gueymard.
SMARTS2 will be used in this paper to model spectral irradiance. This model is more appropriate for
engineering applications, where low duration and complexity of execution is desired. It uses spectral
transmittance functions for the most significant scattering and absorption processes that take place
as the sun’s rays pass through the atmosphere. The model is described in more detail in other papers
by Gueymard [17‐19].
One limitation of SMARTS2 is that it produces only clear‐sky spectral irradiance data. Solar radiation
is scattered as it passes through clouds, so it is necessary to use other means to determine the effect
clouds have on the irradiation, such as developing a CMF to simulate a cloudy day. For example,
14
Burrows [20] completed a study predicting UV radiation at the ground in the presence of
environmental factors, such as cloud, in Toronto, Canada. Herman et al. [21] and Raschke et al. [22]
describe methods for determining the cloud effects on radiation budgets (the net radiation in versus
out of the atmosphere) in the infrared. Unfortunately, these methods both deal with global radiation
near the top of the atmosphere, not at ground level. There is a lack of information regarding the
precise effect of clouds on solar radiation of different wavelengths being absorbed at ground level.
This information is necessary for predicting irradiation data at locations where no monitoring
equipment exists and is of particular importance to the organic PV industry because of the need for
wide‐range spectral irradiance data. Because of this insufficient data, we have made the assumption
here that the effect of clouds on the UV spectrum is the same across the full spectrum of light and
have applied this in the development of our CMF.
SMARTS2 is used to determine the broadband (280 – 4000 nm) spectral irradiance. It uses a series of
inputs, such as latitude and atmospheric conditions to calculate irradiance across a chosen
wavelength range.
In order to obtain the total irradiance (W m‐2), it is necessary to integrate the hourly spectral
irradiance data over the chosen range of wavelengths according to the following equation:
, , , λ 2.1
where I[h,d] = irradiance at each hour, h and day, d [W m‐2], λ2 = upper range of chosen wavelength
[nm], λ1 = lower range of chosen wavelength [nm], I[h, d, λ] = spectral irradiance at each wavelength,
λ, each hour, h, and day, d [W m‐2 nm‐1], and dλ = wavelength step size [nm].
The upper and lower ranges of the wavelengths correspond to the optical absorption range of the
solar cell being evaluated. Within the SMARTS2 model, different step sizes are used for different
wavelength ranges: 0.5 nm in the UV (280 – 400 nm), 1 nm in the visible and part of the near infrared
15
(400‐1702 nm), and 5 nm beyond, up to 4000 nm. The trapezoidal rule is used to estimate the
integral.
Eqn. (2.1) produces a matrix of clear‐sky irradiance values for each hour over the course of a given
year. It is necessary to account for days when the weather conditions are not clear, because this is
only clear‐sky data; this can be done by developing a CMF.
2.4.3 Cloud Modification Factor (CMF)
A CMF is used to adjust modeled irradiance data to better reflect actual weather conditions in
absence of actual broad‐spectrum irradiance data. CMF is defined in TEEOS as: (Calbó et al. [23])
2.2
As discussed above, the irradiance under clear skies is obtained using the SMARTS2 simulation. The
irradiance for non‐clear skies is obtained from the inputs in section one of TEEOS. The actual and
modeled UV irradiance and weather condition for each hour are arranged in a spreadsheet and then
sorted by the two weather conditions, clear and cloudy. For each weather condition, monthly CMFs
are obtained by plotting the actual hourly irradiance on the y‐axis and modeled hourly irradiance on
the x‐axis; an example is shown in Figure 2.2. The slope, which is the monthly CMF, is determined
using robust regression because of the high possibility of outliers. See [24] and [25] for more
discussion on robust regression.
16
Figure 2.2 CMFs for January 2006 and 2007 in Toronto, Canada. The markers represent hourly data points for that month and weather condition. The straight lines have a slope equal to the robust slope, showing the trend.
It is expected that for the clear condition, the CMFs will be close to 1 because the modeled (clear‐sky)
data would be similar to the actual “clear” data. Similarly, it is expected that the CMFs for cloudy
conditions are lower than 1 because clouds absorb some of the UV radiation [26].
The main limitation in the CMF calculation is a lack of broadband spectral irradiance data for a given
geographic location. Due to the lack of sufficient data, an assumption is made here that the effect of
clouds on the UV spectrum is at least similar to the effect on the rest of the spectrum. This
assumption will be tested in a sensitivity analysis.
2.4.4 Modified Solar Irradiance
The “modified” hourly solar irradiance is calculated as follows:
, , 2.3
Eqn. (2.3) produces a matrix of hourly irradiance data that better represents the effect of weather on
the modeled irradiance. The CMF used is dependent on the hourly weather condition for that time.
17
2.4.5 Electricity Pricing Data
Electricity prices are used in the model to determine the avoided cost of electricity and form the basis
of the calculation of annual savings.
There are two main components to non‐commercial retail electricity rates: fixed recurring charges
and volumetric energy charges [27]. The fixed charges generally consist of charges for the delivery
of electricity from the generator to the supplier and then to the customer, as well as general charges
for administration. The volumetric charges, which are proportional to the amount of electricity used,
include the actual price for electricity, as well as other charges, including transmission and
distribution charges. These charges vary widely depending on jurisdiction, electricity generating
source and cost structure of the generating company. In Ontario, Canada, for example, the hourly
Ontario energy price (HOEP) is the wholesale market price for electricity and is directly related to
the price for domestic customers [28]. In general, there are three rate structures for electricity
prices: rates that are set by a local utility and do not change regularly; rates that change depending
on time and quantity of electricity used; and wholesale prices [29]. In this model, wholesale rates are
used because it is expected that this is the direction that utilities will move in order to better reflect
the cost of energy.
It is evident that an organic PV installation will not produce more electricity than the demand of the
customer, but supplement the power from the grid. Because the customer is still connected to the
grid, the fixed charges on the electricity bill will still have to be paid. Therefore, the fixed charges will
not be included in the avoided cost of the electricity. However, other variable charges, mentioned
above, must be added to the base electricity price in this model.
Forward electricity prices typically follow models such as the one‐factor mean reversion jump
diffusion model. This model takes into account the mean reverting nature of electricity prices, but
can also predict the jumps that often occur. Cartea & Figueroa [30] outlined a method for predicting
spot electricity prices, as well as the forward price curve. While using this model to predict
electricity prices over the lifetime of the solar cell would be useful, it only predicts daily prices, not
hourly. Hourly forward electricity prices would provide more accuracy for the model, even over the
long lifetime of the cell. Therefore, since the calibration for such models is taken from historical data,
18
hourly electricity prices from one year in this model are assumed to be constant annually over the
lifetime of the cell.
2.4.6 Technological Characteristics
There are four main cell characteristics that are necessary for the model presented here: absorptive
wavelength range, power conversion efficiency, cell lifetime, and the cost of producing the cell.
2.4.6.1 Wavelength Range
The wavelength range of an organic solar cell is characteristic of the polymer material used to absorb
photons from the sun. The absorptive wavelength range gives a lower and upper wavelength value
in which the material can absorb photons and convert them to electrons. Bundgaard & Krebs [6]
have created a useful table that calculates the integrated photon flux and current density from 280
nm up to certain wavelengths from 500 – 1500nm. An incident photon to current efficiency (IPCE) or
quantum efficiency versus wavelength graph can provide an accurate estimate of the wavelength
range of certain materials in order to be used in this model.
2.4.6.2 Power Conversion Efficiency
The power conversion efficiency of an organic solar cell is the percentage of power converted from
absorbed light to electrical energy when the solar cell is connected to an electrical circuit. While
power conversion efficiencies of inorganic solar cells have easily surpassed 20%, organic solar cells
are still less than 7%. There has been exceptional progress made in the past number of years, but the
maximum reported power conversion efficiency is 6.5% [11]. Most laboratory cells have reported
efficiencies below 5%. For large scale applications, however, much smaller efficiencies are seen,
almost 15‐20 times poorer than laboratory models under the same conditions [4]. Hoppe and
Sariciftci [31] have compiled a list of some of the best power conversion efficiencies of polymer solar
cells.
2.4.6.3 Cell Lifetime
The lifetime of an organic solar cell depends on the extent of chemical, physical and mechanical
degradation of the materials and hardware used in the cell. The lifetime of a cell is defined as the
time that a cell has a relatively consistent current or power conversion efficiency. The best lifetime
19
for an organic/polymer cell is estimated at 20, 000 h [6], but more realistic lifetimes of cells with
small cells have reached 10, 000h, or just over one year, in standard conditions [8].
2.4.6.4 Cost of Production
The processes used to make dyes and other technologies used in the copier/printer industry are
similar to those used to make organic solar cells. The cost of producing an organic solar cell has not
been extensively studied, except by Krebs et al. [4]. As mentioned earlier, a very exhaustive costing
of the various processes and materials used in the sequence of producing the organic solar cells is
detailed. The authors also provide some much needed insight into the large‐scale production of
organic solar cells, which has also been widely ignored in literature. It is concluded that materials
cost are the dominant cost factor and should be first reduced to make organic PV competitive with
other solar technologies.
2.4.6.5 Ancillary Costs
Ancillary costs are made up of costs including installation, maintenance, and other equipment
necessary for use in a solar module to generate electricity, such as an inverter, power monitor and
electrical meter. These are all dependent on the size and complexity of the solar installation.
2.4.7 Financial Indicators
The initial cost of a traditional solar electricity system is generally higher than the typical electrical
system for domestic use. However, a solar electricity system provides significant savings on
electricity bills compared to a typical grid‐connected system because the only costs involved are
initial set‐up costs. This savings can be used to calculate a simple payback period for a solar array.
The annual savings is calculated as follows:
, , 2.4
20
where Aarray is the area of the array [m2], η is the efficiency of the solar cell, I[h,d]modified is the
modified irradiance from eqn. (2.3) [kWh m‐2], and C[h,d] is the cost of electricity at hour, h and day,
d [$/kWh].
Using this methodology, the annual savings represents the avoided cost of electricity that one would
not have to buy from the grid because of the solar electricity system.
The simple payback period is defined as the following:
2.5
The initial investment in this case represents the initial cost of the solar array. The annual savings
(AS) are calculated from the avoided cost of purchasing electricity from a utility. Payback period,
however, does not take into account the time value of money. A more accurate measure of value used
for economic analysis is net present value (NPV), defined as:
1
2.6
where Fi is the net cash flow at time i, the time period (typically years; year zero is present time), r
is the discount rate for each cash flow, and n is the total number of time periods.
This indicator takes into account the lifetime of the cell and money that is saved even after the
payback period is reached. The detailed NPV equation used is:
1
2.7
The cash flow at time zero is the initial cost (IC) of purchasing and installing the system and is
negative. The subsequent cash flows represent the annual savings (AS) that accrue by making the
21
switch from the grid to organic solar cells to produce electricity; these are positive. A positive NPV
shows that the initial investment will be paid back, with a profit, at the end of the lifetime of the
project.
An additional indicator that is used in the analysis is internal rate of return (IRR). It is often used by
firms to make decisions on whether or not to go ahead with an investment. The IRR is the rate of
interest that makes the NPV of a project equal to zero, as such:
1
0 2.8
Where the symbols are defined as such in equation (2.6).
2.5 Sample Calculation
A sample calculation is provided in this section in order to provide the reader an example from the
model. A more detailed analysis using the model will be shown in a companion paper. The first
major demonstration of large area polymer solar cells, big enough to power a small rechargeable
battery, was performed by Krebs et al. [4]. Using circular polymer solar modules, their team made
over 1000 “Solar Hats” intended to power an FM radio and other battery powered devices at a music
festival in Denmark [4]. The data from that paper will be used to calculate the financial indicators for
the use of the solar hats or a similar sized device in Toronto, Canada.
2.5.1 Data
A polymer solar cell with active layers of P3MHOCT/ZnO and P3MHOCT/PCBM/ZnO, a PET‐ITO
(indium‐tin‐oxide) substrate, and PEDOT:PSS and silver electrodes were used to make the solar hats
using screen printing as the process technology [4]. Table 2.1 shows the technological characteristics
for the cells used in the solar hats. The wavelength range is an estimate taken from the absorption
spectra of a similar solar module in [32]. The weather is taken from Environment Canada [33].
Table 2.1 Technological characteristics of the organic solar cells attached to solar hats. Wavelength Range 350‐950 nm [32]
22
Power Conversion Efficiency 0.013%[4]
Cell Lifetime ~3 Months[4] Cost of Production 0.68 – 4.538 €/module[4] Active Area 75 cm2 [4]
2.5.2 Results
Due to the short lifetime of the cells, the financial analysis was carried out using 2006 weather data
for three months, from June 1 to August 31, the peak months of solar irradiation in Toronto, Canada,
also coinciding with the time the experiment in [4] was conducted at the Roksilde music festival in
Roksilde, Denmark. The lowest estimated price for the cost of the module in Table 2.1 was used. The
very low efficiency and small active solar area mean that in Toronto, these modules would produce
only 0.47 W‐h of electricity. By using solar energy to recharge the battery in the radio instead of
electricity purchased from the grid in Toronto, the user would save just over one‐half cent (CAD) in
that time period. The payback period or NPV are not relevant financial indicators in this case due to
the absence of full costs for the FM radios and materials used to connect the cells and the radios, not
to mention the poor savings achieved in general. The authors claim that “due the poor performance
of the modules, it is not meaningful to work out a cost of electricity” [4]. Justifiably, this experiment
was conducted not to prove the economics of organic PV, but to prove that organic PV is indeed
scalable.
An interesting exercise would be to use the maximum reported efficiency and cell lifetime of an
organic solar cell to see if current technology, if ever available in the same cell, has the potential to be
economically viable. It must be noted that this exercise is purely for calculation purposes and that
such a cell does not exist. Indeed, the cell that has the highest efficiency is not the same as the cell
with the longest lifetime and each of these cells were produced on a very small scale, unsuitable for
modern electronic devices.
Using 6.5% [11] as the efficiency and 20 000 h [6] as the lifetime, as well as using the low estimate
for the cost of a module and active solar area of 0.0075 m2 [4], one module would provide
approximately 1334 W‐h over the 20 000 h lifetime, as well as saving the user just under $0.125
(CAD) in Toronto electricity charges (using 2006 weather [33] and electricity price data [28]). Even
without the inclusion of the cost of the connection diodes and the battery itself (for whatever
23
application), the payback period is well over the lifetime of the cell at about 18 years. While the
power generated was sufficient to power an FM radio, the short lifetime and very low efficiency of
the cell make it uneconomical at this time for domestic power generation.
In order for this particular cell to be economical, the price must be reduced: mainly through material
cost reduction, the use of cheaper manpower, and, in general, more practice with up‐scaling this
relatively new technology [4].
2.5.3 Optimization Scenarios
At the current efficiency and lifetime, one house would need fewer than 13 million of these solar
modules to meet the power needs of a typical Canadian household, with replacements every three
months. This would require just under 25 000 m2 of exposure area (ignoring connection areas) and
cost well over $80 million CAD to purchase using a per module cost. In practice, the cost would be
less due to the scale‐up of production, but nonetheless this particular cell is not feasible in its current
form for domestic power generation.
To power a typical Canadian household with a roof exposure area of 10 m2, the efficiency of these
cells would need to be over 130% not to mention requiring close to 1340 modules (ignoring
connection areas).
Even at best cost estimates for optimized large scale organic PV production (€1/m2), this
optimization is futile as the number of cells that need to be purchased is enormous, making payback
periods very large. From these results, there is a dire need for more work on large‐scale organic solar
cell development.
2.6 Conclusions
The TEEOS model presented here can provide useful financial indicators for all PV technologies, but
is inherently flexible in order to incorporate the unique characteristics of organic PV. The economic
feasibility of tandem organic cells that use combinations of different materials to absorb certain
wavelength ranges should be determined using the methodology outlined here. It can allow the user
to determine the best cost structure for certain locations. Indeed, the cost structure for certain
24
organic PV could be so favorable that it may be feasible to manufacture cells for different weather
conditions and different locales around the world. Future papers will use the TEEOS model in
greater detail and show the viability of certain organic solar cells that have been developed for
various applications. It is hoped that it will be possible to use this results from the model to guide
research for organic solar cells, specifically regarding efficiency and spectral range.
25
2.7 References
[1] Brabec, C., Organic photovoltaics: technology and market, Solar Energy Materials and Solar Cells, 83 (2004) 273‐292. [2] Gnes, S., Neugebauer, H., Sariciftci, N.S., Conjugated polymer‐based organic solar cells, Chem. Rev. 107 (2007) 1324‐1338 [3] Krebs, C.F., Fabrication and processing of polymer solar cells: A review of printing and coating techniques, Solar Energy Materials and Solar Cells 93 (2009) 394‐412. [4] Krebs, C.F., Jørgensen, M., Norrman, K., Hagemann, O., Alstrup, J., Nielsen, D.T., Fyenbo, J., Larsen, K., Kristensen, J., A complete process for production of flexible large area polymer solar cells entirely using screen priting – First public demonstration, Solar Energy Materials and Solar Cells 93 (2009) 422‐441. [5] Scharber, M.C., Mühlbacher, D., Koppe, M., Denk, P., Waldauf, C., Heeger, A.J., Brabec, C.J., Design rules for donors in bulk‐heterojunction solar cells ‐ Towards 10 % energy‐conversion efficiency, Adv. Mater. 18 (2006) 789‐794 [6] Bundgaard, E., Krebs, F.C., Low band gap polymers for organic photovoltaics, Solar Energy Materials and Solar Cells 91 (2007) 954‐985. [7] Krebs, F.C., Alternative PV: large scale organic photovoltaics Refocus 6 (2005) 38‐39 [8] Jørgensen, M., Norrman, K., Krebs, C.F., Stability/degradation of polymer solar cells, Solar Energy Materials and Solar Cells 92 (2008) 686‐714 [9] Shaheen, S.E., Ginley, D.S., Jabbour, G.E., Organic‐based photovoltaics: Toward low‐cost power generation, MRS Bulletin 30 (2005) 10‐19. [10] Grätzel, M., Photovoltaic and photoelectrochemical conversion of solar energy, Phil. Trans. R. Soc. A. 365 (2007) 993‐1005. [11] Kim, J.Y., Lee, K., Coates, N.E., Moses, D., Nguyen, T.‐Q., Dante, M., Heeger, A.J., Efficienct tandem polymer solar cells fabricated by all‐solution processing, Science 317 (2007) 222‐225. [12] Koster, L.J.A., Mihailetchi, V.D., Blom, P.W.M., Ultimate efficiency of polymer/fullerence bulk heterojunction solar cells, Appl. Phys. Lett. 88 (2006) 093511 [13] Forrest, S., The limits to organic photovoltaic cell efficiency, MRS Bulletin 30 (2005) 28‐32. [14] Curtright, A.E., M.G. Morgan, D.W. Keith, Expert assessments of future photovoltaic technologies, Environ. Sci. Technol. 4:24(2008) 9031‐9038. [15] Sargent, E.A., Solar cells, photodetectors, and optical sources from infrared colloidal quantum dots, Advanced Materials 20 (2008) 3958‐3964. [16] Wilks, D.S., R. L. Wilby, The weather generation game: a review of stochastic weather models, Progress in Physical Geography, 23 (1999) 329‐357. [17] Gueymard, C., The sun’s total and spectral irradiance for solar energy applications and solar radiation models, Solar Energy 76 (2004) 423‐453.
26
[18] Gueymard, C., Interdisciplinary applications of a versatile spectral solar irradiance model: A review, Energy 30 (2005) 1551‐1576. [19] Gueymard, C., Prediction and validation of cloudless shortwave solar spectra incident on horizontal, tilted, or tracking surfaces, Solar Energy 82 (2008) 260‐271. [20] Burrows, W., CART regression models for predicting UV radiation at the ground in the presence of cloud and other environmental factors, Journal of Applied Meteorology 36 (1997) 531‐544. [21] Herman, G., M.‐L. C. Wu, W.T. Johnson, The effect of clouds on the earth’s solar and infrared radiation budgets, Journal of the Atmospheric Sciences 37:6 (1980) 1251‐1261. [22] Raschke, E., A. Ohmura, W.B. Rossow, B.E. Carlson, Y.‐C. Zhang, C. Stubenrauch, M. Kottek, M. Wild, Cloud effects on the radiation budget based on ISCCP data (1991‐1995), International Journal of Climatology 25 (2004) 1103‐1125. [23] Staiger, H., P.N. den Outer, A.F. Bais, U. Feister, B. Johnsen, L. Vuilleumier, L., Hourly resolved cloud modifications in the ultraviolet, Atmos. Chem. Phys. 8 (2008). 2493‐2508. [24] Holland, P. W., R. E. Welsch, Robust regression using iteratively reweighted least‐squares, Communications in Statistics: Theory and Methods A6 (1977) 813‐827. [25] Street, J. O., R. J. Carroll, D. Ruppert, A note on computing robust regression estimates via iteratively reweighted least squares, The American Statistician 42 (1988) 152‐154. [26] Calbó, J., D. Pagès, J.‐A. González, Empirical studies of cloud effects on UV radiation: A review, Review of Geophysics 43 (2005) RG2002/2005. [27] Toronto Hydro, 2008. Accessed September 15, 2008. http://www.torontohydro.com/electricsystem/understanding_your_bill/bill_breakdown/index.cfm [28] Independent Electricity System Operator. Hourly Ontario Electricity Price (HOEP). Accessed September 8, 2008. http://www.ieso.ca/imoweb/marketData/marketData.asp [29] Independent Electricity System Operator. Accessed September 10, 2008. http://www.ieso.ca/imoweb/infoCentre/ic_index.asp [30] Cartea A., M.G. Figueroa, Pricing in electricity markets: a mean reverting jump diffusion model with seasonality, Applied Mathematical Finance 12:4 (2005). [31] Hoppe, H., Sariciftci, N.S., Morphology of polymer/fullerence bulk heterojunction solar cells, J. Mater. Chem. 16 (2006) 45‐61 [32] Hagemann, O., M. Bjerring, N.C. Nielsen, F.C. Krebs, All solution processed tandem polymer solar cells based on thermocleavable materials Solar Energy Materials and Solar Cells 92 (2008) 1327‐1335 [33] Canada’s National Climate and Weather Data Archive. http://climate.weatheroffice.ec.gc.ca/ Accessed September, 2008.
27
CHAPTER 3
USING TEEOS TO DETERMINE PAYBACK PERIODS FOR ORGANIC SOLAR CELLS
28
3 Using TEEOS to determine payback periods for organic solar cells
3.1 Initial Background
This chapter is taken from a submitted journal article entitled, Using a Financial Model to Determine
Payback Periods for Organic Solar Cells. This is a companion paper to the previous chapter which
detailed the methodology for the TEEOS model and included a small demonstration using solar hats
as an example. This chapter includes a full results section with financial indicators and sensitivity
analysis on these factors for domestic electricity generation in Toronto, Ontario, Canada. Using the
concept of organic solar cells as roof shingles and electricity generators at the same time, a host of
financial indicators are calculated according to the model described previously.
Two different cells are used for this model. The first cell, known as a dye‐sensitized cell, uses dyes to
absorb certain wavelengths of light; it has a higher efficiency and is closer to (and in some cases, at)
full commercialization than the strict organic solar cell. The second cell, an organic/polymer cell is
the more traditional organic solar cell. It has a low efficiency and an absorptive wavelength range
that has the potential to be quite large, depending on the exact material used.
3.2 Abstract
The TEEOS model is used to evaluate payback periods of common organic solar cells, namely dye‐
sensitized solar cells (DSSCs) and organic solar cells (OSCs), for domestic solar energy production in
Toronto, Ontario, Canada. The results show that the DSSC produces a payback period between 8 and
26 years. The OSC, with a lower efficiency and narrower absorptive wavelength range, has a very
high payback period, as well as a negative NPV. The financial indicators dramatically improve when
taking advantage of a government feed‐in tariff program offered in the province of Ontario. A
sensitivity analysis shows that the OSC, using the current technological characteristics with varying
electricity prices and initial costs, will never have a positive NPV. Different combinations of materials
are explored to produce wavelength ranges and efficiencies that produce financially viable solar cells.
29
3.3 Introduction
Photovoltaic (PV) technologies have been used for many years in a variety of applications, but these
are often for specific and cost‐insensitive purposes, such as generating power in space. Only
recently, with governments looking to reduce greenhouse gas emissions in light of climate change
have PV technologies seen an increase in domestic and commercial use. Applications include urban
and rural domestic electricity generation, utility peak load reduction, and direct applications such as
highway telephones and billboards, parking meters, water pumping, and municipal lighting fixtures
[1].
The Canadian Solar Industries Association (CSIA) states that it is feasible to install a total of over
3000 MW of PV on single‐detached homes in Ontario for domestic electricity generation, and up to
14000 MW installed for all buildings by 2025 [2], which represents over half of current peak demand
in Ontario. While this may be significantly optimistic, these generation goals could be attainable
through the use of government incentives, such as the Standard Offer Contract (SOC) now in place in
Ontario, and mandatory building regulations, such as LEED certification for use of renewable energy
in buildings. The SOC guarantees a price to a renewable energy supplier and was introduced by the
province of Ontario in Canada in an effort to boost the share of renewable energy in its power supply.
The standard offer was recently increased and now offers up to $0.802/kWh for solar energy,
depending on the size and type of installation. The Government of Ontario has estimated that the SOC
would help develop about 1000 MW of renewable energy over the next ten years, but in only one
year, applications for the SOC surpassed this goal [3]. This expansion in PV technology, however,
does depend on the availability of cost‐effective products for consumers.
The highest efficiency reported for an organic solar cell (OSC) is 6.5% [4], but the active area for this
cell was quite small. Bundgaard and Krebs [5] state that the efficiencies of OSCs with a small active
area are much higher than the best reported efficiencies for a large active area device with the same
polymer. It is shown in some experimental observations that as the active area of an OSC increases,
the efficiency decreases, but this may be more due to differences in morphology than active area [5].
30
Nonetheless, this presents a problem with the scale‐up needed for OSCs to compete against already
commercialized inorganic solar cells, which have much higher efficiency.
In order to improve the efficiency of OSCs, it has been asserted that low band gap polymers could be
used because they have a better overlap with the solar spectrum [4‐6], that is, they harvest more
photons to turn into electrons or electricity. Low band gap polymers in a bulk‐heterojunction device
have the possibility for even higher efficiencies, also due to the absorption of the acceptor [4, 7].
In order to completely meet the electricity needs of a typical North American home, an array of
inorganic solar cells would have to be over 15 m2 depending on the location and type of solar cell. An
organic solar array of this size is currently unheard of experimentally, let alone in commercial use.
Indeed, most reports on the production and testing of OSCs deal with cells with very small active
areas (on the order of mm2). There have, however, been slow advances in increasing the scale of the
active areas of organic solar cells [8‐16]. Nonetheless, it is conceivable that careful design of the
geometry of the organic solar array could result in minimal losses. This would allow for a large
number of these small area cells to be used in parallel to take advantage of their relatively high
efficiency.
The objective of this paper is to evaluate the financial viability of organic solar cell technology from a
consumer (retail) perspective in order to better direct future technological research efforts.
Domestic solar energy production in Toronto, Ontario, Canada, using both a dye‐sensitized solar cell
(DSSC) and an OSC are used as the evaluator. The Technical and Economic Evaluator for Organic
Solar (TEEOS) [17] model will compare cells that can be used to offset the need for electricity from
the grid for a home. This paper will also look at the economics of potential cells that can absorb
energy in lower band gap solar spectral regions and possibly provide a goal for researchers in order
to produce an economically feasible, large‐area OSC.
3.4 Model Development
The TEEOS model methodology is described in detail in Powell et al. [17]. This paper will present the
financial indicators that are estimated by the model and discuss some of the implications of the
31
results. Figure 3.1 shows a flow chart that outlines the inputs and outputs of various stages of
TEEOS.
There is no broadband spectral irradiance data for Toronto, Ontario, outside the ultraviolet (UV) and
visible range, so the SMARTS2 model will be utilized to estimate spectral irradiance.
Figure 3.1 Flow Chart of TEEOS Process [17]
3.4.1 Weather Conditions and Irradiance Data
Hourly weather conditions and UVB irradiance data were obtained from Environment Canada at the
Downsview weather station, in northwest Toronto. As noted above, the SMARTS2 model [18‐20] is
used to determine the clear‐sky broadband (280 – 4000 nm) spectral irradiance. The inputs for
Toronto, Ontario are given in Appendix A.
32
3.4.2 Cloud Modification Factor (CMF)
The cloud modification factor (CMF) is used to modify clear‐sky irradiance data to better estimate
actual weather conditions in absence of actual broad‐spectrum irradiance data. CMF is defined in
TEEOS as the ratio of the irradiance under a specific weather condition to the irradiance under clear
sky. The CMF is calculated using the method outlined in Powell et al. [17]. The average monthly CMFs
are shown in Table 3.1. For clear conditions, the average CMF is approximately 1 for both 2006 and
2007. For cloudy conditions, the average CMF is 0.3324 and 0.5724 for 2006 and 2007, respectively.
The CMFs have a relatively large standard variation, but are consistent with estimates used in other
literature. See Calbó et al. [21] for a review of empirical CMFs for overcast skies.
Table 3.1 Average monthly CMF for Toronto, ON based on 2006 and 2007 irradiance data and weather
Month 2006 2007
Cloudy Clear Cloudy Clear
January 0.3016 1.0377 0.7202 1.0332
February 0.5785 0.9758 0.6861 1.0719
March 0.3756 0.9765 0.2779 1.0541
April 0.1530 1.0630 0.3597 1.0405
May 0.3004 0.9698 0.8458 0.9637
June 0.3502 0.9848 0.6284 1.0036
July 0.2051 0.9726 0.8356 0.9817
August 0.3348 1.0361 0.2801 0.9427
September 0.3482 0.9593 0.8355 1.0583
October 0.2443 1.0289 0.4447 1.0526
November 0.4036 1.0037 0.5055 1.1039
December 0.3937 1.0679 0.4489 1.1260
Mean of Monthly CMF
0.3324 1.0056 0.5724 1.0360
St. Dev. of Monthly CMF 0.1087 0.0393 0.2142 0.0548
33
3.4.3 Modified Solar Irradiance
Figure 3.2 represents the modified solar irradiance in Toronto, Canada for each day in 2006 using the
default wavelength range 280 – 4000 nm. This is the amount of hourly solar irradiance that is
available to the solar cell over the course of one year using 2006 weather conditions.
Figure 3.2 Integrated irradiance over one year in Toronto, Canada
3.4.4 Electricity Pricing Data
Electricity prices for Toronto, Ontario are used here to determine the avoided cost of electricity and
calculate the financial indicators. There are two components in the electricity price: volumetric
charges and fixed charges. The volumetric charges used here consist of the wholesale price in
Ontario for a given hour, known as the hourly Ontario electricity price (HOEP), as well as other
charges, such as a distribution charge. These volumetric charges are based on the quantity of
electricity purchased from the grid and are described in more detail in Powell et al. [17]. Fixed
34
charges do not vary with electricity consumption; they are not included in the avoided cost of
electricity because a consumer must still pay them while connected to the grid. The volumetric
charges, for Toronto Hydro, the local utility, amount to $0.0377/kWh and are detailed in Table 3.2.
Table 3.2 Volumetric Electricity Charges other than the HOEP from Toronto Hydro [7]
Description Volumetric Charge Distribution Charge $0.0146/kWhTransmission $0.0099/kWhWholesale Market Charges $0.0062/kWhDebt Retirement Charge $0.007/kWhTOTAL additional charge $0.0377/kWh
3.4.5 Technological Characteristics
Two cells were chosen as a demonstration of the model. The first cell is a DSSC and is described by
Grtätzel [23, 24]. The DSSC has an efficiency that is well above the highest achieved by an OSC. This
cell is a titanium oxide (Ti‐O2) DSSC, which can be manufactured in a very similar manner to organic
cells (outlined in Krebs [11]). The cost of manufacturing DSSCs has been evaluated by both Smestad
et al. [25] and Meyer [26] and a range of these costs are used in this model (there is a low and high
cost per area, depending on different substrate costs and author). The lifetime of the DSSC used in
this model is estimated at 5 years. The second cell type, developed primarily by Alan Heeger, is an
OSC and is described further in Gnes et al. [27]. The cost of this particular OSC has been evaluated by
both Krebs et al. [12] and Kalowkamo & Baker [28]. Kalowkamo & Baker [28] provide a low and high
cost per area for an OSC, which will be used in this model. The estimate by Krebs et al. [12] falls
within this range. The lifetime of the OSC is estimated at five years, also used by Kalowkamo & Baker
[28] which is higher than the current record for operational lifetime shown by Bundgaard & Krebs
[5] for an OSC, but is consistent with estimates of lifetime in the future (see Curtright et al. [29]) The
model input parameters are given in Table 3.3. Changes in the model inputs will be evaluated in the
sensitivity analysis.
Table 3.3 Technological Characteristics for Two Sample Organic Solar Cells
Characteristic DSSC[23,24] OSC[8]
Wavelength Range 280 – 700 nm 280‐620 nm Efficiency 11.2% 3.4%
35
Cell Lifetime 5 years 5 years Cell Area 10 m2 10 m2
Cost of Cell $37/m2 ‐ 158/m2 $49/m2 ‐ $139/m2
3.4.6 Ancillary Costs
The ancillary costs or balance of system (BOS) costs are made up of costs including support
structures, inverters, wiring, power conditioning, installation, and transportation. The BOS costs for
photovoltaic arrays are discussed in a BES [30]. Kalowekamo & Baker [28] estimate the BOS costs at
$75/m2 based on the long‐term goal for traditional silicon solar arrays in BES [30]. However, the
same report gives a much lower number for BOS costs for low efficiency third‐generation (ie.
organic) solar cell arrays at the efficiency of the OSC used in this paper. Here we will use a range of
$40/m2 to $75/m2. The balance of system equipment is assumed to have a longer lifetime than the
DSSC and OSC because it is an established technology; the lifetime here is assumed to be 15 years.
Therefore, the total initial cost of the system to the end user for a five year project lifetime (one solar
array and one set of ancillary equipment) is between $770 and $2330 for a 10 m2 DSSC array and
between $890 and $2140 for a 10 m2 OSC array. The total initial cost of the system to the end user for
a 15‐year project lifetime (three solar arrays purchased at years 0, 5 and 10 and one set of ancillary
equipment) is between $1288 and $4538 for a 10 m2 DSSC array and between $1575 and $4082 for a
10 m2 OSC array, with a discount factor of 5%. These costs are summarized in Table 3.4.
Table 3.4 Summary of solar cell cost, balance of system costs, and total initial cost for the two example cells. A low and high range is given for each according to the ranges determined in the previous section.
Solar Cell Project Lifetime (years)
Solar Cell Cost ($/m2)
Balance of System Costs ($/m2)
Total Initial Cost ($/m2)
Low High Low High Low High
DSSC 5 37 158 40 75 77 233
OSC 5 49 139 40 75 89 214
36
3.4.7 Financial Indicators
The financial indicators used in the TEEOS model are payback period and net present value (NPV), as
described in Powell et al. [17]. Simple payback period is not frequently used as an economic
evaluator of projects because it fails to include the time value of money and is too simplistic to make
decisions on these projects. NPV is more popular because it uses discount factors for cash flows in
the future and is easy to understand and calculate. Payback period is used in this model, however,
because of its simplicity and ease of comprehension for users of the model. A potentially more
accurate, flexible, but less adopted measure for economic decision‐making is a real option valuation.
However, this calculation is not widely used in industry and is not as intuitive as payback period or
NPV to a potential customer.
3.5 Results and Discussion
For the payback period and NPV calculation in this section, the annual savings calculated for years
2006 and 2007 is assumed to be constant for each year over the lifetime of the cell. When a value is
given for the year 2006, this means that the annual savings in 2006 is used as the annual savings over
the five year life of the cell in question.
3.5.1 WavelengthCost Distribution
It is important to see which wavelength range produces the greatest savings. Figure 3.3 shows the
contributions of the different wavelength ranges to the annual savings of a possible solar cell that can
absorb energy across the full wavelength range (280‐4000 nm) with a power conversion efficiency of
11.2%, the same as the DSSC. The difference between the two years is the weather conditions and
electricity prices in Toronto for each year; this is why the relative contributions are not consistent
from year to year. Each section represents the different ranges between 280 and 4000 nm: UVB
(280‐315 nm), UVA (315‐400 nm), visible (400‐750 nm), near infrared (IR) (750 – 1000 nm) and mid
IR (1000‐4000 nm).
37
Figure 3.3 Contribution of each wavelength range to annual savings in Toronto, Canada
The contribution from the UVB range is low (<0.01%) and is not displayed. The largest contribution
is from the visible range, which comprises between 48 and 50% of the annual savings for each year.
This is mainly due to the fact that the visible range represents about 50% of the total irradiance
available across the full spectrum.
3.5.2 Cell Comparison
The model results for 2006 and 2007 for the two example solar cells are shown in this section. In
Table 3.5, the annual savings and payback period are given. In Table 3.6, the NPV is given. Two
project lifetimes are used: five years and fifteen years. A five‐year project lifetime consists of
purchasing one solar cell array and one set of ancillary equipment at year 0. A fifteen‐year project
lifetime consists of purchasing three solar arrays (one every five years, at year 0, 5, and 10) and one
set of ancillary equipment. The annual savings are in 2007 dollars using the consumer price index
(CPI) for 2006 relative to 2007 of 0.972. The discount rate for NPV is 5%. The values are calculated at
the low and high range of the initial costs shown in Table 3.4.
315‐400 nm $9.17315‐400 nm $10.40
400‐750 nm$84.80
400‐750 nm$96.33
750‐1000 nm$33.22
750‐1000 nm$42.83
1000 ‐4000 nm$42.06
1000 ‐4000 nm$47.78
$0
$20
$40
$60
$80
$100
$120
$140
$160
$180
$200
2006 2007
Annual Savings
Year
38
Table 3.5 Annual Savings and Payback Period for 2006 and 2007 for the two example solar cells. Brackets indicate negative values. The two values for payback period reflect the low and high range of initial costs given in the previous section for a five year project lifetime.
Cell DSSC OSC Years 2006 2007 2006 2007
Annual Savings (2007 dollars)
$86.81 $95.84 $20.58 $22.71
Payback Period (years) 8.87‐26.84 8.03‐24.31 43.25‐104.00
39.19‐94.23
Table 3.6 Net Present Value for 2006 and 2007 for the two example solar cells with five and fifteen year project lifetimes. Brackets indicate negative values. The two values reflect the low and high range of initial costs given in the previous section.
Cell DSSC OSC Years 2006 2007 2006 2007
NPV – 5 year lifetime
$(1954.14) ‐ $(394.14)
$(1915.08) ‐$(355.08)
$(2050.92) ‐$(800.92)
$(2041.68)‐$(791.68)
NPV – 15 year lifetime
$(3636.85) ‐ $(385.95)
$(3543.20) ‐$(292.30)
$(3868.87) ‐$(1361.18)
$(3846.72) ‐$(1339.03)
The payback period (as defined in Powell et al. [17]) range for the DSSC cell, is much lower than for
the OSC because of the higher efficiency and a wider absorption range of the DSSC. Consistent with
the payback period analysis, the DSSC has a higher NPV than the OSC. Table 3.6 shows that for the
higher range of the estimate of initial costs, the NPV is higher for the five‐year projects. This is mainly
due to the dominance of the higher initial costs. Using the lower limit of the initial costs for both
cells, the fifteen‐year project has a higher NPV for the DSSC and a lower NPV for the OSC. This is
because, although the initial costs are lower for both cells, the annual savings is much higher for the
DSSC because of a higher efficiency.
There is also variability between each year for the cells due to changes in weather and electricity
prices. The CMF for cloudy hours was significantly higher in 2007 than for 2006, resulting in a higher
modified irradiance which translates into more solar energy available for the solar cells. Also, the
mean electricity price for 2007 was higher than 2006, resulting in a higher avoided electricity cost.
3.5.3 Comparison with Inorganic Solar cells
The low range payback period estimates for the DSSC presented above are consistent with
commercially available silicon solar cells. Ucar and Incalli [31] have estimated payback periods for
solar heating systems in Turkey between 13 and 35 years, but much lower payback periods have
39
been seen for this technology in other locales. For a combination of solar heating and electricity
system in Hong Kong, payback periods of fourteen years have been presented [32]. In Saudi Arabia,
very favourable payback periods have been estimated for large‐scale photovoltaic installations, at
fewer than 10 years [1]. In Rehman et al. [1], however, the avoided cost of electricity used is
$0.50/kWh (with an electricity cost escalation rate of 4%), which is almost five times the average
cost of electricity in Ontario.
3.5.4 Standard Offer Contract
The TEEOS model can be used to determine financial indicators for PV systems that take advantage
of government incentives, such as feed‐in tariffs for renewable energy. In Ontario, a domestic
renewable energy system is eligible for the Standard Offer Contract (SOC) described above. Those
using the SOC must pay a one‐time $100 licensing fee which includes the cost of the meter that allows
two‐way flow of electricity; this is added to the initial cost.
The concept for this section is a domestic photovoltaic system that is used solely to generate power
to send back to the grid, not supplementing the electricity use of that particular home. This
drastically improves the economics of the organic solar cell system. The payback periods and NPV,
using the SOC, are shown in Table 3.7.
Table 3.7 Annual Savings, Payback Period, and NPV for the two example cells using the Standard Offer Contract. The payback period is calculated using the initial cost for a fiveyear project lifetime. The two values for each reflect the low and high range of in the previous section.
Financial Indicator DSSC OSC 2006 2007 2006 2007
Annual Savings (2007 $)
$742.75 $807.96 $175.96 $191.38
Payback Period (years) using a 5 year project
lifetime
1.17‐3.27 1.08‐3.01 5.79‐13.10 5.17‐11.70
NPV (2007 $)
5 year lifetime
$785.72 ‐$2345.72
$1068.04 ‐$2628.04
$(1499.52) ‐ $(249.52)
$(1411.41) ‐ $(161.41)
15 year lifetime
$3071.55 ‐$6322.45
$3748.40 ‐$6999.30
$(2407.18) ‐ $100.51
$(2195.95) ‐ $311.74
The payback period for the DSSC is well below the five‐year lifetime of the system, while the low
range estimate for the OSC is just above the five‐year lifetime. The NPV for the DSSC system is very
40
attractive; these results show that is more beneficial to use a fifteen‐year lifetime when making this
investment. The OSC system produces a negative NPV for the five‐year lifetime systems, but using the
lower range of the initial costs, it is possible to obtain a positive NPV using a fifteen‐year lifetime. The
use of these government incentives drastically improves the viability of the two example solar cells,
but the DSSC outperforms the OSC and is the better investment.
3.6 Sensitivity Analysis
There were three main assumptions in this paper. First, the CMF used to determine the effect of
weather on irradiance was assumed to be constant across the full wavelength range of the cells, but
was calibrated based solely on the UVB range irradiance data. There are no actual irradiance data
outside the UVB for Toronto; therefore, it is assumed that the regression of actual versus modeled
irradiance is similar across the full wavelength range. Second, the electricity prices used in this
model are assumed to vary by hour but be the same for the same hour a year later. While this is not
strictly true, past data indicate that the mean of the electricity price has not changed by more than
10%. However, with more unregulated hourly pricing values may see greater volatility in the future.
The exact hourly price will change, but the data for 2006 and 2007 are used because they match the
weather data provided by Environment Canada. Third, the initial cost has been assumed to be a
function of the cost of the cell and the ancillary costs which have a wide. A sensitivity analysis on the
NPV and payback period for each of these assumptions will be performed in this section.
3.6.1 CMF Sensitivity
The CMF is assumed to be applicable across the full wavelength range of the solar cell. As mentioned
before, the reason for this assumption is the lack of available broadband spectral irradiance data for
Toronto, Ontario. Figure 3.4 shows the change in payback period and NPV with the change in CMF
from 0 to 1 for the DSSC using 2006 electricity prices and weather conditions and the lower initial
cost range for a fifteen year project lifetime ($1287 (2007 $) using 5% discount factor) . The vertical
line at 0.3324 on the horizontal axis represents the original calculated CMF in 2006. A negative CMF
is not valid because a negative irradiance value is not possible. A CMF above 1, sometimes seen in
41
literature due to the reflecting and amplifying effects of snow, would only improve the financial
indicators produced in the model and is thus ignored.
Figure 3.4 Sensitivity of payback period and NPV to CMF for DSSC in 2006 using the lower initial cost value of $1287. The vertical line represents the point of the current CMF for 2006 – 0.3324.
It is shown that with a CMF close to 1, the payback period for the DSSC, shown on the right axis, is
just under the assumed lifetime of the project. Even if the cloudy CMF is zero, there are still “clear”
hours that will produce electricity savings and the payback period is just above the assumed lifetime
of the system. As the CMF increases, the payback period decreases. If there is more solar radiation
available, then less electricity needs to be used from the grid: this increases annual savings, thereby
decreasing payback period. On the left axis, the NPV is shown to be negative for all values of the CMF,
which shows that under current weather conditions and electricity prices in Toronto, no matter the
CMF, this cell is not financially viable. A variance in the CMF only produces a relatively small change
in the financial indicators and is not the most significant driver of sensitivity.
3.6.2 Electricity Price Sensitivity
Although electricity prices are the largest factor determining the annual savings in this model, they
are not necessarily a source of uncertainty because they are fixed by a government body in Ontario.
Nonetheless, there is a possibility that they can rise or fall depending on a variety of factors, such as
0
2
4
6
8
10
12
14
16
18
‐600
‐500
‐400
‐300
‐200
‐100
0
0 0.2 0.4 0.6 0.8 1
Payback Period (years)
NPV (2007 $)
CMF
NPV
Payback Period
42
demand, supply, temperature, weather, and source of fuel. Figure 3.5 demonstrates the wide
variation in payback period and NPV based on changing the electricity price. A change of ‐100%
represents an electricity price of zero, which is unrealistic, but is shown.
Figure 3.5 Payback Period and NPV sensitivity to electricity price changes for the OSC in 2006 using the lower range of the fiveyear lifetime project initial cost.
Figure 3.5 also shows an exponential curve for the payback period. This is because as the electricity
price approaches zero, the annual savings would approach zero, requiring the payback period to
approach infinity. The payback period will never reach zero because of the initial cost of $1288.
While it is quite unrealistic to expect a tripling of electricity prices (300% change in electricity
prices) in quick succession, it does show that it is virtually impossible for the OSC to be a financially
viable investment, with the NPV never becoming positive.
3.6.3 Initial Cost Sensitivity
Figure 3.6shows the effect of changes in the initial cost on payback period and NPV. The OSC is used
as an example cell, using 2006 weather conditions and electricity prices, and a 5% discount factor.
The initial cost includes the cost of the solar cells and BOS costs. As expected, an increase in the initial
cost of the project produces a decrease in NPV and an increase in payback period. The vertical line
between $1000 and $2000 represents the initial cost for the OSC using the lower estimate for the
‐$1,000.00
‐$900.00
‐$800.00
‐$700.00
‐$600.00
‐$500.00
‐$400.00
‐$300.00
‐$200.00
‐$100.00
$0.00
0
50
100
150
200
250
‐100 ‐50 0 50 100 150 200 250 300
NPV (2007 $)
Payback Period (years)
% Change in Electricity Prices
Payback Period NPV
43
fifteen year project lifetime of $1575 (2007 dollars). There is a significant effect on both financial
indicators with a change in initial costs.
Figure 3.6 Payback Period and NPV sensitivity to Initial Costs. The vertical line represents the current initial cost estimate of $1575.
The NPV of the OSC system becomes positive when the initial cost is lower than $87. We can assume
the BOS costs are fixed at $40/m2 (the lower estimate for an OSC array). Therefore, it is safe to
assume that the NPV of the OSC will never become positive no matter the cost of the cells themselves.
The payback period of the DSC becomes lower than the assumed lifetime of the project when initial
costs are under $300. Again, assuming the BOS costs are fixed, the payback period of the DSC will
never actually be below the assumed project lifetime.
3.6.4 Wavelength Range and Efficiency
Curtright et al. [29] state that dye‐sensitized and organic solar cells may achieve a certain cost per
peak watt in 2030 that could make them a financially viable solar technology. An important question
in the organic solar cell field is whether it is better to increase the absorptive wavelength range of a
cell or to increase the efficiency of the cell and what affect a change of one has on the other. The
0
50
100
150
200
250
300
‐6000
‐5000
‐4000
‐3000
‐2000
‐1000
0
1000
0 1000 2000 3000 4000 5000
Payback Period (years)
Net Present Value ($2007)
Intial Cost ($2007)
NPV Payback Period
44
TEEOS model can help answer this first question because it determines financial indicators for a
variety of solar cells using cell efficiency and absorptive wavelength range. Indeed, it can be used to
make comparisons between different cells before they are developed in order to determine their
economic viability. The second question was addressed partly by Smestad et al., [25], by predicting
that an increase in wavelength range results in an increase in efficiency. These interactions won’t be
taken into account in this paper. Changes in wavelength range and efficiency are done in isolation
and are shown simply to determine payback periods for different combinations of wavelength ranges
and efficiencies in order to determine a combination that is financially viable.
In this section, five different wavelength ranges, UVA, UVB, visible, near IR and mid IR, (denoted as 1,
2, 3, 4, and 5, respectively) are combined to form potential cells that can absorb in a specific
wavelength range. Also, we can consider the effect of changing cell efficiency. Figures 3.7 – 3.10
represent the different combinations of wavelength ranges mentioned above and their payback
periods at different efficiencies: 5, 10, 15, and 20%, respectively. In these figures, a threshold
payback period of fifteen years is used; any cell with a payback period above fifteen years is shown as
a full bar. The different wavelength ranges for each number (1, 2, 3, etc. on the horizontal axis of the
graph) are shown in Table 3.8. This analysis was completed assuming the initial cost of all the cells
was constant, using the lowest price range for the OSC at $890 for a 10 m2 array. While this
assumption may be unrealistic, especially since different materials must be used to capture the
energy in the different spectral ranges, the exercise shows the dependency of efficiency and
absorptive range on the payback period. The physical possibility of these cells depends on the
development of specific materials and/or a combination of materials that absorb in that wavelength
range.
Table 3.8 The range of each numbered wavelength range category shown in Figures 7 10.
Numbered Category Wavelength Range (nm) 1 – UVB 280 – 315 2 – UVA 315 – 400 3 – Visible 400 – 750 4 – near IR 750 ‐ 1000 5 – mid IR 1000 ‐ 4000
45
Figure 3.7 Payback periods of organic solar cells with different combinations of wavelength ranges at 5% efficiency.
Figure 3.8 Payback periods of organic solar cells with different combinations of wavelength ranges at 10% efficiency
0123456789101112131415
2 23 234
2345 24 245 25 3 34 345 35 4 45 5
Payback Period (years)
Wavelength Range Combination
5 % Efficiency
0123456789101112131415
2 23 234
2345 24 245 25 3 34 345 35 4 45 5
Payback Period (years)
Wavelength Range Combination
10% Efficiency
46
Figure 3.9 Payback periods of organic solar cells with different combinations of wavelength ranges at 15% efficiency.
Figure 3.10 Payback periods of organic solar cells with different combinations of wavelength ranges at 20% efficiency
It was found that any wavelength range combination that included the first wavelength range (280‐
315 nm) had a very similar payback period to the combination without it. For example, the
combination of 1, 2 and 3 (total wavelength range 280 – 750 nm) had essentially the same payback
0123456789101112131415
2 23 234
2345 24 245 25 3 34 345 35 4 45 5
Payback Period (years)
Wavelength Range Combination
15% Efficiency
0123456789101112131415
2 23 234
2345 24 245 25 3 34 345 35 4 45 5
Payback Period (years)
Wavelength Range Combinations
20% Efficiency
47
period as the combination of 2 and 3 (total wavelength range 315 – 750 nm). The first wavelength
range represents a very small part of total irradiance and does not contribute much to the payback
period, thus, these results were not shown on the graph.
By looking at the progression of efficiencies on the graphs, from 5% to 20%, the number of
wavelength combinations that have payback periods under 15 years increases from 2 (at 5%) to 12
(at 20%) out of 14. At 5% (close to the current maximum reported efficiency of OSC as of writing),
there are two combinations which yield payback periods lower than fifteen years: cells with
wavelength ranges between 315 – 4000 nm and 400 – 1000 nm. The two cells are quite similar, with
the latter not including the UVA wavelength range. The latter cell has a higher payback period
because it is not absorbing photons with wavelengths between 315 and 400 nm. In Figures 3.8 – 3.10,
it is evident that as new wavelength ranges are added to the cells (i.e. from 23 to 234), the payback
period decreases. This is intuitive as well because more light energy being available means a higher
annual savings which leads to a reduced payback period. Furthermore, it is evident that certain
wavelength ranges contribute more to a reduction in payback period than others. For example, in
Error! Reference source not found., one can see that a cell absorbing in the range 400 – 700 nm
(indicated by a ‘3’) has a payback period less than 12 years. If you add a material that absorbs in the
UVA range (indicated by ‘23’), it reduces the payback period by approximately one year. However, if
you add a material that absorbs in the visible range (indicated by a ‘34’), the payback period is
reduced by over three years. This is simply because a larger amount of solar energy is available in
the visible range.
Table 3.9 shows two examples of groupings of different combinations of wavelength range and Table 3.9 shows two examples of groupings of different combinations of wavelength range and
efficiency that have similar payback periods.
Table 3.9 Wavelength range and efficiency of cells that have similar payback periods
Example Cell Wavelength Combination
(1,2,3,4,and/or 5)
Absorptive Wavelength Range
(nm)
Cell Efficiency Payback Period (years)
1 2,3 315 ‐ 750 20% 5.302,3,4 315 – 1000 15% 5.222,3,4,5 315 – 4000 10% 5.89
2 2,4,5 315‐400 and
750‐4000 10% 11.80
48
3 400‐750 10% 11.75
It is evident here in the first example, that a short wavelength range (but one that has the highest
percentage of available irradiance) and a high efficiency produce a very favorable payback period.
However, by increasing the wavelength range and reducing the efficiency, the same payback period is
achieved.
An OSC with an absorptive wavelength range of 315 – 750 nm and an efficiency of 10% has a payback
period of just over 10 years in Toronto, Canada. In order to improve the payback period, there would
be multiple options as shown in Table 3.9: increase the efficiency to 20% and keep the wavelength
range the same, or increase both the wavelength range and efficiency. Each would produce a similar
reduction in payback period. In the second example, the wavelength ranges do not overlap, but the
cells have the same efficiency and still produce the same payback period. This has different
implications for cell engineering and material choice. Although the TEEOS model uses the same cell
costs for both hypothetical cells, it could be possible to engineer a cell using a material that is
inexpensive and only absorbs in the third wavelength range – lowering the payback period.
Moreover, the second example has 10% efficiency; this is considered the next benchmark for organic
solar cells. The TEEOS model shows the two cells above have relatively low payback periods at that
efficiency and there are a number of other wavelength range combinations that have payback
periods well below the presumed cell lifetime. As shown in Figure 3.7, even cells with an efficiency of
5% can achieve payback periods of less than fifteen years. Even still, this would be improved greatly
in a locale with more sunshine hours than Toronto.
Ignored in this analysis, however, is the interaction between efficiency and wavelength range. When
increasing the absorptive wavelength range, the efficiency (defined as the ratio of energy out to
energy in from the sub) will undoubtedly increase. As shown by Smestad et al., [25], a DSSC that
absorbs in the 400 – 650 nm range has an efficiency of 7‐8%, using predicted data and IPCE = 1.
When the higher range is increased to 750 nm, the efficiency increases to 10‐11% and again when
the higher range is increased to 800 nm, the efficiency increased to 12‐13%.
49
Also present is the challenge of manufacturing a cell for a specific wavelength range or a specific
efficiency. Specific materials that absorb light in certain wavelength ranges may not perform the
same way in close contact with other materials that absorb light in other wavelength ranges. If a
number of stacked materials are used, the amount of light reaching the lowest stack may be
significantly reduced and may not produce the predicted amount of energy from the stack cell. These
are all issues that must be explored, but are currently outside the scope of this paper.
3.7 Conclusions
The TEEOS methodology is applied to two sample solar cells, a DSSC and an OSC. The inputs used
include weather conditions and wholesale electricity prices from Toronto, Canada, and the specific
cell characteristics. Due to a lack of irradiance data in Toronto outside the UV range, the broadband
irradiance is estimated using the SMARTS2 model. The initial costs used in this paper represent the
most significant source of variation of the inputs and have been estimated using previous literature
estimates. The wide range represents different cost accounting methodologies, as well as different
manufacturing methodologies using materials that vary in cost.
Using different project lifetimes, the DSSC is shown to perform better in terms of payback period and
NPV than the OSC because it has a higher efficiency and wider wavelength range. It is shown that the
DSSC has a payback period between eight and 25 years depending on the initial cost estimate. The
OSC has a payback period at well over 40 years. When taking into account government incentives,
the financial indicators drastically improve. While the DSSC has payback periods between one and
three years, and a range of NPVs that are highly positive, the OSC still has a negative NPV, but is much
more attractive than without using the government incentives.
The sensitivity analysis performed on the CMF, electricity prices, and the initial cost of the solar cells,
shows that the OSC can never have a positive NPV using current characteristics.
The model is shown to be able to help guide decision making, particularly with respect to increasing
efficiency or increasing absorptive wavelength range. The results cannot decisively say that one
approach is better financially than the other and indeed do not address the interactions between the
two approaches; however, research in both areas is worthwhile in reducing the cost of OSC. An
50
efficiency milestone of 10% is certainly achievable in the next ten years; the TEEOS model shows that
these cells, with certain absorptive wavelength ranges, are indeed financially viable, with low
payback periods and positive NPVs. Even at 5% efficiency, there are hypothetical cells that can be
made financially viable in more northern climates like Toronto, and indeed, perform better in
sunnier locales. The question remains how to find the different materials that absorb in certain
wavelength ranges and to minimize interference between the materials.
51
3.8 References
[1] Rehman, S., M.A. Bader, S.A. Al‐Moallem, Cost of solar energy generated using PV panels, Renewable & Sustainable Energy Reviews 11 (2007) 1843‐1857. [2] Canadian Solar Industry Association. “The Potential of Solar PV in Ontario”. 2006 [3] Government of Ontario. http://www.powerauthority.on.ca/sop/ Accessed March 1, 2009. [4] Kim, J.Y., Lee, K., Coates, N.E., Moses, D., Nguyen, T.‐Q., Dante, M., Heeger, A.J., Efficient tandem polymer solar cells fabricated by all‐solution processing, Science 317 (2007) 222‐225. [5] Bundgaard, E., Krebs, F.C., Low band gap polymers for organic photovoltaics, Solar Energy Materials and Solar Cells 91 (2007) 954‐985. [6] Dhanabalan, A., J.K.J. van Duren, P.A. van Hal, J.L.J. van Dongen, R.A.J. Janssen, Synthesis and characterization of a low bandgap conjugated polymer for bulk heterojunction photovoltaic cells, Advanced Functional Materials 11 (2001) 255‐262 [7] Colladet, K., M. Nicolas, L. Goris, L. Lutsen, D. Vanderzande, Low‐band gap polymers for photovoltaic applications, Thin Solid Films 451‐452 (2004) 7‐11 [8] Krebs, F. C., J. Alstrup, H. Spanggaard, K. Larsen, E. Kold, Production of large‐area polymer solar cells by industrial silk screen printing, lifetime considerations and lamination with polyethyleneterephthalate Solar Energy Materials & Solar Cells 83 (2004) 293‐300. [9] Suemori, K., Y. Matsumura, M. Yokoyama, M. Hiramoto, Large area organic solar cells with thick and transparent protection layers, Japanese Journal of Applied Physics 45 (2006) L472‐L474 [10] Krebs, F. C., H. Spangaard, T. Kjær, M. Biancardo, J. Alstrup, Large area plastic solar cell modules, Materials Science and Engineering B 138 (2007) 106‐111 [11] Krebs, F.C., Alternative PV: large scale organic photovoltaics Refocus 6 (2005) 38‐39 [12] Krebs, C.F., Jørgensen, M., Norrman, K., Hagemann, O., Alstrup, J., Nielsen, D.T., Fyenbo, J., Larsen, K., Kristensen, J., A complete process for production of flexible large area polymer solar cells entirely using screen printing – First public demonstration, Solar Energy Materials and Solar Cells 93 (2009) 422‐441. [13] Lungenschmied, C., G. Dennler, H. Neugebauer, S. N. Sariciftci, M. Glatthaar, T. Meyer, A. Meyer, Flexible, long‐lived, large‐area, organic solar cells Solar Energy Materials and Solar Cells 91 (2007) 379‐384 [14] Brabec, C. J., N.S. Sariciftci, J.C. Hummelen, Plastic solar cells, Advanced Functional Materials 11 (2001) 15‐26 [15] Al‐Ibrahim, M., H.‐K. Roth, U. Zhokhavets, G. Gobsch, S. Sensfuss, Flexible large area polymer solar cells based on poly(3‐hexylthiophene)/fullerene, Solar Energy Materials and Solar Cells 85 (2005) 13‐20 [16] Bundgaard, E., F.C. Krebs, Large‐area photovoltaics based on low band gap copolymers of thiophene and benzothidiazole or benzo‐bis(thiadiazole), Solar Energy Materials and Solar Cells 91 (2007) 1019‐1025
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[17] Powell, C., T. Bender, Y. Lawryshyn,A model to determine financial indicators for organic solar cells, Solar Energy 83 (2009) 1977‐1984. [18] Gueymard, C., The sun’s total and spectral irradiance for solar energy applications and solar radiation models, Solar Energy 76 (2004) 423‐453. [19] Gueymard, C., Interdisciplinary applications of a versatile spectral solar irradiance model: A review, Energy 30 (2005) 1551‐1576. [20] Gueymard, C., Prediction and validation of cloudless shortwave solar spectra incident on horizontal, tilted, or tracking surfaces, Solar Energy 82 (2008) 260‐271. [21] Calbó, J., D. Pagès, J.‐A. González, Empirical studies of cloud effects on UV radiation: A review, Review of Geophysics 43 (2005) RG2002/2005. [22] Toronto Hydro, http://www.torontohydro.com/electricsystem/understanding_your_bill/bill_breakdown/index.cfm Accessed October 23, 2008. [23] Grätzel, M., The advent of mesoscopic injection solar cells, Prog. Photovolt: Res. Appl. 14 (2006) 429‐442. [24] Gratzel, M., Solar energy conversion by dye‐sensitized photovoltaic cells, Inorg. Chem. 44 (2005) 6841‐6851. [25] Smestad, G., C. Bignozzi, R. Argazzi, Testing of dye‐sensitized TiO2 solar cells I: Experimental photocurrent output and conversion efficiencies, Solar Energy Materials and Solar Cells 32 (1994) 259‐272. [26] Meyer, T., 1996. Solid state nanocrystalline titanium oxide photovoltaic cells, Thèse N° 1542, École Polytechnique Fédérale de Lausanne. [27] Gnes, S., H. Neugebauer, N.S. Sariciftci, Conjugated polymer‐based organic solar cells, Chem. Rev. 107 (2007) 1324‐1338. [28] Kawlowekamo, J. and E. Baker, Estimating the manufacturing cost of purely organic solar cells, Solar Energy 83:8 (2009) 1224‐1231. [29] Curtright, A.E., M.G. Morgan, D.W. Keith, Expert assessments of future photovoltaic technologies, Environ. Sci. Technol., 42(24) (2008) 9031–9038 [30] Basic Research Needs for Solar Energy Utilization, 2005. Report of the Basic Energy Sciences Workshop on Solar Energy Utilization. US Department of Energy. [31] Ucar, A., M. Incalli, A thermo‐economical optimization of a domestic solar heating plant with seasonal storage, Applied Thermal Engineering 27 (2007) 450‐456. [32] Chow, T.T., A.L.S. Chan, K.F. Fong, Z. Lin, W. He, J. Li, Annual performance of building‐integrated photovoltaic/water‐heating system for warm climate application, Applied Energy 86 (2009) 689‐696.
53
CHAPTER 4
USING STOCHASTIC MODELS TO DETERMINE FINANCIAL
INDICATORS AND TECHNICAL OBJECTIVES FOR ORGANIC SOLAR
CELLS
54
4 Using stochastic models in TEEOS to determine financial indicators and technical objectives for organic solar cells
4.1 Abstract
Two stochastic models are used to develop synthetic electricity price and weather series for the
TEEOS model. A mean –reverting jump diffusion model was used to model the synthetic electricity
price and a third‐order Markov‐chain cloud occurrence model was used to generate the synthetic
weather series. The payback period and NPV of both example cells, a dye‐sensitized solar cell (DSSC)
and an organic solar cell (OSC) are consistent with previous results for the same cells and
geographical location. It is shown that current costs for the OSCs make it impossible to achieve a
reasonable payback period and a positive NPV in Toronto, Canada. A target of $7/m2 for the OSC
would produce a positive NPV but this target well below current estimates. The near‐term efficiency
for large‐scale manufactured organic solar cells needs to be increased quickly, with similar
advancements in wavelength absorption range.
4.2 Introduction
The TEEOS model provides a framework to allow researchers to determine financial indicators,
limited to payback period, net present value (NPV) and internal rate of return (IRR) here, for organic
solar cells (OSCs) in different locations [1]. The results in Powell et al., [2] show that a dye‐sensitized
solar cell (DSSC) can have a payback period and NPV that are favourable, depending on the initial
cost estimation, and the OSC can never have a positive NPV or payback period shorter than 25 years,
no matter the initial cost or electricity price escalation. However, this analysis used only two years of
weather and electricity price data assuming the weather and electricity price for one year was
constant for all years over the lifetime of the cell. This is not necessarily a valid assumption; this
paper will seek to improve the accuracy of the results provided in [2] using stochastic modeling
techniques for both the electricity price and weather data.
To facilitate this, two stochastic models will be developed and implemented in the TEEOS model. The
first model will be used to predict hourly electricity prices. A mean‐reverting jump diffusion
55
electricity price model [3] will be used owing to the tendency for electricity prices to revert to a mean
price level and have random jumps. The second model, a stochastic weather model, known as a
rainfall occurrence model [4‐6], will be adopted to fit the needs of the TEEOS model. This model uses
Markov probabilities to predict the future state of a process, in this case, whether it is cloudy or clear,
based on the current and previous states. This process will be correlated to the electricity price
model under the assumption that the weather influences the electricity price on that day; this
correlation will be simplified for this model. The model will also be tested without this correlation.
After describing each stochastic model, this paper will present results analogous to the results
presented by Powell et al. [2] and these results will be compared to those in that paper.
4.3 Electricity prices
Electricity prices are used in TEEOS to determine the cost savings associated with the displacement
of electricity from the grid with electricity from solar cells [1].
In this section, the procedure for the generation of electricity prices over the lifetime of the cell will
be outlined. The electricity price data currently available for Toronto will be used to predict
electricity prices over a much longer period than the data itself and will provide a more robust
estimate of the annual savings on electricity bills compared to just using the same one year of
electricity prices for the entire cell lifetime. As well, the stochastic electricity price model allows us to
look at the sensitivity of different parameters, such as volatility, mean reversion rates, and price
escalation rates.
Electricity prices typically exhibit specific characteristics such as mean‐reversion, jumps, seasonality,
and time‐varying volatility structures [3, 7‐8]. The dynamics of daily average electricity prices are
important in describing market conditions and serve as a base for option contracts for electricity
prices and their derivatives. These characteristics are accounted for in the variety of models that are
used to model electricity prices, especially in explaining the dynamics of daily average prices. These
models, however, only capture the average daily performance of the electricity price market and
provide no indication of the dynamics on a smaller scale. For example, the daily electricity price
model cannot accurately describe a specific hourly mean reversion price level. Indeed, a daily price
56
model assumes that the mean reversion level is the same over the course of the day, but this is not
necessarily an accurate assessment of the price dynamics. Other factors must be included to account
for hourly price changes in a time series.
An overview of different wholesale price models is described by Weron [9] and Escribano et al. [10].
While a great deal of literature is available on daily average electricity price models, there are fewer
studies available on hourly‐specific electricity price models. Some authors have recognized the gap
between daily average models and the hourly‐specific models, such as Borenstein et al. [11]. Li and
Flynn [12] consider the change in hourly volatility in fourteen electricity markets. Wolak [13]
studies the hourly price dynamics of day‐ahead markets in a number of jurisdictions and studied the
intra‐day correlation between error terms using various electricity models. Knittel & Roberts [14]
use traditional and atypical electricity price models to forecast real‐time California electricity prices
and find that the traditional models do not accurately address the unique characteristics of electricity
prices. Others use a neural network approach to forecast the electricity price in various markets,
while also looking at other factors such as demand, capacity shortfalls, and outages [15‐16]. Huisman
et al. [4] models hourly electricity prices for day‐ahead markets using the idea that these prices do
not follow time‐series dynamics, but can be described by treating the information set as panel data.
It is shown that hourly electricity prices exhibit hourly‐specific characteristics, such as mean‐
reversion level and mean price. Nogales et al. [17] use time series models, in contrast to [4], to
forecase hourly electricity prices using different models. Cartea & Figueroa [3] use a mean‐reverting
jump diffusion model to explain the dynamics of spot electricity prices in Wales and England, while
Hikspoors and Jaimungal [18] used a similar model to forecast the spot price for oil. A similar model
will be used to forecast electricity prices for the TEEOS model in this paper but adjusted to account
for diurnal and monthly variations in price.
It is important for the TEEOS model to use hourly electricity price dynamics to provide an accurate
estimate of the annual savings, not just an average estimate. Electricity prices have a typical
variability during the day according to demand and supply patterns, among other things. These are
ignored when using an average daily estimate. Furthermore, an hourly microstructure for weather is
57
used because describing the ‘mean’ weather over the course of a day as ‘sunny’ or ‘cloudy’ is not
necessarily accurate and, therefore, the same logic is applied to electricity prices.
4.3.1 Electricity price model
The model proposed to predict electricity prices using past performance is a mean‐reverting jump
diffusion model. The model is shown here and is similar to that used by Cartea and Figueroa [4]:
(4.1)
In eqn. (4.1), St [$/MW‐h] is the electricity price process in, κt [h‐1] is the hourly speed of mean‐
reversion for a given month, [$/MW‐h] is the hourly mean‐reversion level for a given month, σt
[$/MW‐h]is the hourly volatility for a given month, dZt is the increment of the standard Brownian
motion, J is the jump size, and dqt is a Poisson process such that
(4.2)
where l is the intensity or frequency of the process. J, dqt, and dZt are independent and the jump size,
J, is normally distributed, i.e. J ~ N(μJ, σj2).
The interpretation of eqn. (4.1) is as follows. Most of the time, dqt = 0, so the process is simply the
mean‐reverting diffusion process. At random Poisson‐distributed times however, St will jump from
the previous value to a higher level, based on the level of J.
In order to account for escalation in electricity prices, an annual escalation factor, ε, will also be
included. This is not explicitly part of the equation but is included in the hourly mean‐reversion level
for a given month, . Each year, the mean‐reversion level will be increased by (1+ ε) to account for
increases in electricity prices due to structural changes such as cost of fuel and other maintenance
that may arise.
4.3.1.1 Parameter Estimation
The electricity price data, as noted before, is the Hourly Ontario Energy Price (HOEP) published by
the Independent Electricity System Operator (IESO) in Ontario, Canada in units of $/MW‐h. The
dataset consists of wholesale hourly spot prices between May 1, 2002 at 12:00 am and June 30, 2009
at 11:00 pm. There are 62784 observations. Over 99.5% of the electricity prices fall within the range
58
of $0.00/MW‐h and $200.00/MW‐h. Table 4.1 includes the electricity price summary statistics for the
dataset.
Table 4.1 Electricity price summary statistics for HOEP between May 1, 2002 and June 30, 2009. The mean, maximum and minimum have units of $/MWh (CAD).
Mean 51.17Standard Deviation 33.86
Maximum 1891.14 Minimum ‐52.08 Skewness 7.21Kurtosis 209.17
The following parameters must be determined using the underlying data: J, λ, κt, θt ,σt,
4.3.1.2 Jump Size, J
A jump is defined here as a price that is higher than the mean plus or minus three times the standard
deviation of the price series. The jumps in the dataset were extracted using the methods described by
Cartea & Figueora [4] and Clewlow & Strickland [16]. A numerical algorithm filters the defined jumps
from the price series. After the first iteration, the mean and standard deviation of the remaining
series (without jumps) is calculated; prices defined as jumps according to the new mean and
standard deviation are filtered again. The process is repeated until no further prices can be filtered.
This algorithm facilitates the estimation of the size of jumps, J and also the frequency of jumps, l. As
noted previously, J, is normally distributed with a mean and variance estimated from the jumps that
were filtered on the final iteration. The mean jump size and standard deviation are shown in Table
4.2.
4.3.1.3 Poisson parameter, λ
The Poisson parameter is defined as the time between a jump in the price series and is estimated
using the filtered jump prices. The jumps are isolated from the original price series, but the time
series is maintained in order to determine the time between jumps. The time between jumps is
assumed to be Poisson‐distributed and the Poisson parameter, λ, is calculated from the distribution
of the times between jumps. This distribution is shown in Figure 4.1 and shows a clear Poisson
distribution (weighting toward the left side of the graph). The vast majority of time between jumps is
below 200 hours, or 8 days, with only six times greater than 200 hours. The Poisson parameter, λ,
59
shown in Table 4.2, means that there is, on average, one jump in price every day, or just over 24
hours. The inverse of the Poisson parameter represents the frequency of a jump.
Figure 4.1 Histogram of time between jumps for historical electricity price series
Table 4.2 Jump Parameters Mean, Standard Deviation, and Poisson Parameter of the HOEP between May 1, 2002 and June 30, 2009
μJ ($/MW‐h) σj λ (hours)
154.85 81.13 24.26
4.3.1.4 Mean Reversion Speed, κt ,Mean Reversion Level, θt, and Volatility, σt
These parameters are typically estimated using a linear regression [4, 16]. In this case, the price
series was filtered of jumps and to maintain the time series, the filtered values were replaced with
prices that were interpolated between the non‐jump values. St+1 were regressed against St because of
the discretization of the mean‐reverting model in eqn. (4.3):
1 ∆ ∆ (4.3)
where ε represents the error of the regression. The slope is the used to find the mean‐reversion
speed, κt, which was estimated for each hour of the day for a given month, with a Δt = 1. The
intercept was used to find the mean reversion level, θt, and was also calculated each hour of the day
0
200
400
600
800
1000
1200
1400
1600
1800Frequency
λ ‐ Time Between Jumps (h)
60
for a given month. The standard deviation of the residuals of this regression was used to find the
volatility, σt, on the same time basis, as well. Appendix B shows the mean reversion speed, level, and
volatility for each hour of the day in each month in the historical price series.
4.3.1.4.1 Seasonality
Many daily electricity price models use Fourier series or polynomial fits to determine functions of
seasonality [3]. Furthermore, seasonality is removed from the original price series in order to
calibrate the model. Hourly diurnality is much more complex, however, and methods are not well
established. In order to maintain the different levels of seasonality present in the spot price dataset,
the mean reverting levels and volatility are calculated separately for each hour in a day and for each
month in a year.
4.3.1.5 Verification of Model
The electricity price simulation was run 15000 times for one simulated year and, for verification, the
parameters of the synthetic price sets were determined using the same method of obtaining the
parameters from the historical price dataset, as described above. Appendix C shows the average
hourly and monthly mean reversion speeds, κt, and also the average jump parameters, μj, σj and λ for
the synthetic prices. There is a slight difference in the parameters for the synthetic electricity price
than for the historical dataset. Each monthly synthetic mean reversion speed is slightly larger that its
corresponding historical mean‐reversion speed. This means that the synthetic prices take a shorter
time to return to their respective means than the historical prices. This effect is most powerful after a
jump. The difference between the synthetic and historical parameters does not significantly affect the
price dynamics, however, possibly because the time between jumps in the synthetic price series is
higher as well. The mean jump in the synthetic electricity price is slightly lower than the historical
dataset, with a higher standard deviation, implying a higher volatility than the historical dataset. The
time between jumps in the synthetic data set is considerably longer than the historical dataset. This
can be explained by the large number of jumps that occur in succession in the historical dataset– if
three jumps (as defined by the procedure above) occur in succession, the time between jumps is one
hour. If a lot of the jumps in the dataset are in succession, it reduces the Poisson parameter. When
61
using this parameter to generate a synthetic dataset, the randomly generated numbers will not
behave exactly the same as the historical process because it is not exactly a Poisson process, just an
estimation of the distribution of the times between jumps. Nonetheless, true verification of the
synthetic electricity price series can be found in the results section where it can be seen whether the
financial indicators are close to the estimates found in Powell et al. [2].
4.4 Markov Chain Weather Patterns
A stochastic weather model uses a fixed set of data to statistically model outputs that behave like the
fixed set of data over a longer period of time. It is expected that this model can give a more robust
weather pattern than the methods described in Powell et al. [1]. Wilks and Wilby [20] provide an
extensive review of stochastic weather models from simple analyses of runs of consecutive weather
conditions to models of daily precipitation. These models have been used extensively in agricultural,
ecosystem, and hydrological impact studies in order to provide synthetic weather series based on
weather station records. It is important to make the distinction between these stochastic weather
models and weather forecasting algorithms, such as those used for weather broadcasts. The
synthetic weather series developed here are never expected to occur in reality and are used as a
means to model trends from past data sets.
The precipitation occurrence model, a Markov chain model, has been used extensively in literature
[4, 20, 21]. A first‐order Markov chain is a random process by which the information about the future
state of a random process is only dependent on the current state, not the past – it is essentially a
memoryless process. A rainfall occurrence model, which uses Markov chain probabilities, determines
the future state of a weather process (that is, raining or not raining) depending only on the current
weather conditions.
In the studies above, a Markov chain model is used to determine runs of consecutive rainy or dry
days or the probability of rainfall on a future day depending on the weather on the current day (or
current sequence of days). Because information about rainfall is not specifically needed for the
TEEOS model, the rainfall occurrence model will be modified so that the two possible states are
cloudy or clear; therefore, this model will be called the cloud occurrence model. Even though a day
62
can be dry and cloudy, this is a simplification that is necessary. Indeed, it follows other simplifications
of the weather observations made in Powell et al. [1] where all precipitation events are considered
“cloudy” and all conditions described as “mostly cloudy”, “partly cloudy”, and “clear” are considered
“clear”.
While most studies using the rainfall occurrence model use daily weather data, for the purposes of
the TEEOS model, hourly weather data will be used. As mentioned previously, hourly precision is
used because weather does change throughout the day and a daily average of weather conditions is
rarely accurate. It is believed that hourly weather conditions can better reflect reality and provide
more accurate estimates of the financial indicators. Hourly rainfall occurrence models have not been
used as extensively as daily models because they typically require higher‐order Markov models and
are not of interest to most researchers.
Most daily models assume a first‐order Markov chain for precipitation occurrence [5,6,21], such that
the probability of rainfall on the following day depends only on whether the current day is wet or
dry. However, Pattison [5] notes that this assumption is inadequate when describing dry periods
between wet events. Nkemdirim [6] concludes that a first‐order Markov chain is inappropriate for
modeling the transition between wet and dry hours, especially humid climates, and uses eighth‐order
hourly dependence. This means that the probability that an hour, t+1, is dry or wet depends on the
sequence of states of the process for the hour, t, through to hour, t7, i.e., the preceding eight hours.
These higher‐order models describe the hourly weather series evolution more accurately than a first
order model because it is unrealistic to expect that the weather at one hour only depends on the
previous hour.
4.4.1 Modifications for TEEOS
The cloud occurrence model used in TEEOS is a modification of the rainfall occurrence model. In
addition, this stochastic weather model will be modified to account for a correlation to the electricity
price model, described previously. Typically, it is assumed that the electricity price is partly
dependent on the weather, such as clear or cloudy days and air or water temperature [14, 22, 23]. In
this model, for simplicity, the electricity price is assumed to be the independent variable, and the
synthetic weather dataset is driven by changes in the electricity price. Either assumption is valid
63
when dealing with synthetic datasets, the approach taken here is the easier of the two. The following
section will explain the procedure for determining the synthetic weather dataset using the electricity
price correlation.
4.4.2 Parameter Estimation
The underlying weather data for the stochastic weather model is from Environment Canada. The
cloudy and clear designations are from observational data at the Downsview weather station in
Toronto, Ontario, Canada. There were 62112 hourly observations used, which represent the
observations from May 1, 2002 at 12:00 am to May 30, 2009 at 11:00 pm. As mentioned previously,
the data was adjusted to only include two weather observations, cloudy and clear.
4.4.2.1 Determining Markov chain order
Both Pattison [5] and Nkemdirim [6] use a graphing technique to determine the order of the Markov
chain. Other methods to determine Markov chain order include the Bayesian (BIC) and Akaike’s
(AIC) information criterion [4, 24] and the efficient determination criterion (EDC) [25]. These
methods have also shown that in the routine fitting of precipitation data, first‐order Markov chains
are inadequate [4, 24]. The graphing method will be used in this paper.
To implement the graphing technique for Markov order determination, the probability that a current
cloudy hour t+1 is preceded by t = 1, 2, 3, … , n clear hours, must be found. Each combination of
sequences is classified into a different group: a cloudy hour, t+1, preceded by 1 clear hour, a cloudy
hour, t+1, preceded by 2 clear hours, or more generally, a cloudy hour, t+1, preceded by n clear hours,
are in different groups [5,6]. The conditional probability for each of these classifications is
determined as follows:
Ffclearhcloudyh itt n/}|Pr{ 1 (4.4)
where fi is the frequency with which a cloudy hour t+1 is preceded by clear hours tn = 1, 2, …, n and F
is the sum of the frequencies. These probabilities were calculated on a monthly basis (over the full
data set) and are shown below in Figure 4.2.
64
Figure 4.2 Probability estimates for cloud occurrence model for Toronto, Canada
One can see stabilization, albeit with some monthly variability, of the conditional probabilities
around the sixth hour, which signifies that the state of hour t+1 is dependent only on the sequence of
states from hour t through to hour t5. There is also a noticeable seasonality in the conditional
probability curves, which means that when determining the conditional probability of each of the
sequences, the probabilities used should be monthly, if not at least seasonal.
According to the graphing method, the cloud occurrence model for Toronto, Canada, is a sixth‐order
Markov process. However, it is shown in Table 4.3 that there is no significant difference between
using a sixth‐order model and using a second‐, third‐, fourth‐, or fifth‐order Markov model with
respect to the annual savings. This analysis was done using the dye‐sensitized solar cell (DSSC),
(characteristics are outlined in [2]) using 1000 one‐year iterations of the annual savings.
Table 4.3 Significance test for difference of mean of Annual Savings with respect to Markov order for weather model
Markov Order Mean Annual Savings ($)
T‐stat (6th order control)
Critical Value Significant Difference?
2 85.79 ‐0.3228 1.646 N3 85.81 0.0356 1.646 N4 85.80 ‐0.2001 1.646 N5 85.75 ‐1.372 1.646 N6 85.81 ‐ ‐ ‐
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 5 10 15 20
Probability that Hour, t+1, is
Cloudy
Number of Preceding Clear Hours
January
February
March
April
May
June
July
August
September
October
November
December
65
As a result, a third‐order model will be chosen for this paper due to convenience. A second‐order
model may be too simplistic and a higher‐order model is too data‐ and time‐intensive.
4.4.2.2 Determining Markov parameters
The approach to determine conditional probabilities used in this paper is different than that
described by Pattison [5] and Nkemdirim [6]; here, it is adapted in order to model the occurrence of
cloudy or clear skies, not the amount of rainfall, and also to account for the dependence with
electricity prices. The correlation will be between the departure of the price of electricity from its
mean at hour t+1 and the weather condition at hour t+1. A distribution will be found that describes
the probability that, given a certain ΔS in the electricity price, the hour t+1 will be cloudy or clear.
This distribution is scaled by the frequency of the occurrence of this sequence of weather states and
then a conditional probability is determined. The parameters to be found are the frequency, fi, of each
of the possible sequences of weather, the deviation from the mean, ΔSθ, of each price in the series,
and five parameters for fitting distributions, w1, μ1, μ2, σ1, σ2.
4.4.2.2.1 Frequency fi
For a third‐order Markov chain, the frequency of a given three hour sequence of weather (from t2 to
t) preceding either a cloudy or clear hour, t+1, must be found in order to determine the state at t+1
for a synthetic dataset. Therefore, to generate a synthetic weather series, four hour sequences of real
data must be used; as a result, there are 16 (24) possible combinations of states. Table 4.4 shows an
example of the frequency for the 0000 and 0001 sequences in January.
Table 4.4 Frequency of the historical weather series in January in the underlying dataset. ‘0’ represents a cloudy hour and ‘1’ represents a clear hour.
Time t2 t1 t t+1 fi
Sequence 1 0 0 0 0 1879
Sequence 2 0 0 0 1 170
66
Sequence 1 happens much more frequently than Sequence 2; this means that, independent of the
electricity price, there is a higher probability that a cloudy hour will follow three cloudy hours, than a
clear hour following three cloudy hours.
4.4.2.2.2 Deviation from the mean ΔSθ
For each electricity price in the historical data, the value ΔSθ must be calculated according to the
following equation:
∆ (4.5)
where St represents the historical electricity price for a given hour, t, θm represents the mean
reversion level of the electricity price at the same time and was calculated previously in this paper.
ΔSθ is the deviation from the average monthly electricity price of each electricity price. For each ΔSθ,
determine the state of the historical weather for that hour and the three preceding hours; this
separates each ΔSθ into one of 16 possible sequences for one of 12 possible months.
4.4.2.2.3 Distribution Fitting w1, μ1, μ2, σ1, σ2
For each vector of ΔSθ, a specific distribution will be fit using the method described below for the
sequence 0100 in January. The histogram for ΔSθ for the weather sequence 0100 in January is shown
in Figure 4.3.
Figure 4.3 Histogram of ΔS parameter for weather sequence 0100 in January
0
2
4
6
8
10
12
14
16
18
20
‐40 ‐30 ‐20 ‐10 0 10 20 30 40 50 More
Frequency
ΔSθ ‐ Deviation from the Mean
67
While the histogram shows a distribution close to that of Gaussian, it is evident that this is not exactly
the case. This happens throughout the dataset and as a result, the distributions of each vector of ΔSθ
will be fit to some combination of two normal distributions in order to provide greater accuracy
when determining the probability density. The function used for fitting is the following:
, , (4.6)
where w1 and w2 are the weightings of the first and second normal distribution, respectively, with
the restraint:
1 (4.7)
Φ represents the cumulative distribution function (CDF), μ1 and μ2 represent the means of the first
and second normal distribution, respectively, and σ1 and σ2 represent the standard deviations of the
first and second normal distribution, respectively. The fit was performed in Matlab using an
algorithm that minimizes the total error of the fitted function through numerous iterations. Figure
4.4 shows the sorted ΔSθ vector for the sequence 0100 in January normalized by the length of the
vector in blue and the fitted function P in red.
Figure 4.4 CDF and the function Φ from eqn. (4.6) of the ΔS distribution of the Sequence 0100 in January. The Rsquared value for the fit is 0.9957.
R2 = 0.9957
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This fitting was done for each of the 16 possible weather sequences for each month in the historical
dataset, giving five parameters, w1, w2, μ1, μ2, σ1 and σ2 for each and 960 in total. These parameters
and the frequency of each weather sequence in the historical dataset are used in the next section to
generate a synthetic weather dataset that is correlated to the synthetic electricity price.
4.4.2.3 Synthetic Weather Series Generation
The following procedure is used to generate the synthetic weather series:
1. Generate the synthetic electricity price series using the methods previously mentioned in
this paper.
2. Calculate the ΔSθ parameter for each datapoint in the synthetic price series, using the
synthetic prices for St and the historical electricity price series for θm and eqn. (4.5).
3. Begin by assuming that the first three hours in the synthetic weather series are 000, that is,
they are all clear (any sequence will work, but 000 is the simplest).
4. Calculate the probability density, p0, for the sequence 0000, given the ΔSθ for that next hour
using the five distribution fit parameters for the sequence 0000 in the required month as
follows:
, , 1 , , (4.8)
Where ϕ represents the probability density function PDF and ϕ ΔSθ, μ, σ) is the PDF of a
distribution with mean, μ, and standard deviation, σ, evaluated at the value ΔSθ.
5. Calculate the probability density, p1, that the state of the next hour will be 1, given the same
ΔSθ, but using the five distribution fit parameters for the sequence 0001 in the required
month, according to eqn. (4.8).
6. Scale the two probabilities, p0 and p1, by multiplying them by their respective proportional
frequencies calculated previously as such:
(4.8)
(4.9)
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7. Calculate the probability, P0:
(4.10)
8. Generate a uniformly random number. If it is less than P0, then the state of hour t+1 is 0 or
clear, if it is greater, the state of hour t+1 is 1 or cloudy.
9. Move to the next hour and repeat starting at Step 4 for the duration of the required time
period.
4.4.3 Verification of the Synthetic Dataset
Verification of rainfall occurrence models typically involves comparing the length of dry and wet
spells and comparing the distribution of rainfall amounts during wet spells [4,26]. In this paper, the
verification will involve comparing the length of cloudy and clear spells and the overall ratio of
cloudy and clear hours between the historical weather series and the synthetic weather series.
4.4.3.1 Cloudy/Clear Spell Lengths
Table 4.5 shows the probability of cloudy spell lengths and clear spell lengths for both the synthetic
and historical weather series. Table 4.6 shows the regression analysis between the historical and
synthetic weather series for cloudy and clear spell lengths.
Table 4.5 Cloudy and clear spell probabilities of various lengths for historical and synthetic weather series. Synthetic weather series completed with 1000 iterations.
Run Length (Hours)
Cloudy Spell Length Probabilities Clear Spell Length ProbabilitiesHistorical Synthetic Historical Synthetic
1 0.3348 0.3187 0.3500 0.37302 0.1528 0.2547 0.1667 0.25943 0.0989 0.2427 0.1000 0.05884 0.0742 0.0693 0.0833 0.03745 0.0427 0.0307 0.0500 0.02946 0.0382 0.0173 0.0333 0.02407 0.0270 0.0120 0.0333 0.02018 0.0247 0.0080 0.0333 0.01749 0.0202 0.0067 0.0167 0.014710 0.0157 0.0053 0.0167 0.0120
Over 10 0.1708 0.0347 0.1167 0.1658
Table 4.6 Regression analysis between wet spells and probabilities of various sequence length for historical and synthetic weather series. Synthetic weather series completed with 1000 iterations.
Weather Correlation R2 Standard Error of
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Condition Coefficient Estimate Cloudy 0.7995 0.6391 0.0754 Clear 0.9551 0.9122 0.0380
Table 4.6 shows that there is a stronger correlation between the historical and synthetic weather
series for clear spell lengths than cloudy spell lengths. In the historical dataset, there are
considerably longer spells of cloudy hours than in the synthetic weather series; over 17% of cloudy
spells in the historical weather series are over 10 hours, while only 3% are greater than 10 hours in
the synthetic dataset. This occurs most likely because of the dependence of the synthetic weather
series on the electricity price series. This assumption may affect the accuracy of the probability of a
longer cloudy spell length in the synthetic weather series because of the lower frequency of cloudy
hours would affect the accuracy of fitting the cloudy state/electricity price distributions. The clear
spell lengths seem to agree quite well, however, with a high correlation coefficient and R2 between
the historical and synthetic weather series.
4.4.3.2 Ratio of Cloudy to Clear Days
In the historical weather series, there are 21496 (35%) cloudy hours and 40616 (65%) clear hours.
In the synthetic weather series, with 1000 iterations, there are on average 2981 (34%) cloudy hours
and 5779 (66%) clear hours, per year. Although this does not necessarily show the distribution of
cloudy or clear days over the course of a year, especially during certain time periods, it shows that
the stochastic model chosen does follow the general trend of the historical weather series.
4.5 Results
The TEEOS model used in this paper is essentially the same as in Powell et al. [2], except it now
includes the stochastic models developed in the previous chapter, instead of the single year of
electricity price and weather data. In addition, the cloud modification factor (CMF) is randomly
generated. For each iteration, a normally distributed value between 0 and 1 is generated based on
the mean and standard deviation of the CMFs discussed in [2]. In this random number generation,
any number below zero is made zero and any number greater than 1 is made 1. The CMF does
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greatly affect the value of annual savings [2], so in an attempt to increase the randomness of the
simulation, the CMF is included as a random variable with a known distribution.
The example solar cells used in this paper are a dye‐sensitized solar cell (DSSC) and an organic solar
cell (OSC), similar to that of [2]. A summary of their technological characteristics is shown in Table
4.9.
Table 4.7 Technolgoical Characteristics of Two Example Cells
Characteristic DSSC[27.28] OSC[29]
Wavelength Range 280 – 700 nm 280 ‐ 620 nm Efficiency 11.2% 3.4%Cell Lifetime 5 years 5 years Cell Area 10 m2 10 m2
Cost of Cell $37/m2 ‐ 158/m2 $49/m2 ‐ $139/m2 Balance of System (BOS) Costs $40/m2 ‐ 75/m2 $40/m2 ‐ 75/m2
The main output of the model is the annual savings accrued based on the avoided cost of electricity
by using OSCs for domestic electricity use. Using the annual savings, the simple payback period and
net present value (NPV) are calculated. The histogram for the annual savings for the DSSC and the
OSC is shown in Figure 4.5 and Figure 4.6, respectively.
Figure 4.5 Histogram of Annual Savings of DSSC using 1000 iterations
0
50
100
150
200
250
60 65 70 75 80 85 90 95 100 105 110 More
Frequency
Annual Savings ($2007)
72
Figure 4.6 Histogram of Annual Savings of OSC using 1000 iterations
The annual savings for each of the cells is $83.64 (± 0.57) and $19.75 (± 0.14) for the DSSC and OSC,
respectively, using 1000 iterations. These results, using synthetic data, are within 5% of the results in
Powell et al. [2] which used historical data.
The results for the two sample cells are shown for a varying initial cost, according to the ranges
provided in Table 4.7. The different initial costs used in this analysis are shown in Table 4.8 for a 10
m2 array of each example cell.
Table 4.8 Different ranges of solar cell and BOS costs
Cell Cost
DSSC OSC
BOS Costs ($)
Cell Cost ($)
Total Cost ($)
BOS Costs ($)
Cell Cost ($) Total Cost ($)
Cost 1 400 370 770 400 490 890 Cost 2 750 370 1120 750 490 1340 Cost 3 400 1580 1980 400 1390 1790 Cost 4 750 1580 2330 750 1390 2140
The different costs represent the high and low range of initial costs, including two mid‐level prices.
This will give an indication of the range of payback period and NPV that results when the large range
0
50
100
150
200
250
300
350
400
15 17 19 21 23 25 More
Frequency
Annual Savings ($2007)
73
of the estimate of the initial costs is taken into account. Figure 4.7 and Figure 4.8 show the payback
period and NPV for the DSSC and OSC, respectively.
Figure 4.7 Payback Period and NPV for DSSC
Figure 4.8 Payback Period and NPV for OSC
‐$404.57
‐$1,614.40
‐$755.25
‐$1,961.60
9.23
23.73
13.45
27.72
0.00
20.00
40.00
60.00
80.00
100.00
120.00
‐$2,500.00
‐$2,000.00
‐$1,500.00
‐$1,000.00
‐$500.00
$0.00
$0.00 $500.00 $1,000.00 $1,500.00 $2,000.00 $2,500.00
Payback Period (years)
NPV ($2007)
Intial Cost
‐$803.71
‐$1,152.60
‐$1,702.90
‐$2,053.30
45.20
62.15
90.06
108.09
0.00
20.00
40.00
60.00
80.00
100.00
120.00
‐$2,500.00
‐$2,000.00
‐$1,500.00
‐$1,000.00
‐$500.00
$0.00
$0.00 $500.00 $1,000.00 $1,500.00 $2,000.00 $2,500.00
Payback Period (years)
NPV ($2007)
Initial Cost ($)
74
The range of payback period and NPV is consistent with the results from Powell et al. [2] using
historical data. The difference between the two results may be due to the CMF randomness in the
stochastic analysis. The lowest payback period for the DSSC is just under 10 years, while for the OSC,
the lowest payback period is approximately 45 years. The NPV is negative for both cells within the
range of initial costs used.
4.6 Sensitivity Analysis
A sensitivity analysis will be performed on a number of variables. Similar to the analysis performed
in Powell et al. [2], this paper will also look at changes in the CMF, the difference between a
correlated weather series and a non‐correlated weather series and changes in the jump frequency in
the electricity prices.
4.6.1 CMF
In this paper, the CMF was chosen as a random variable, with a range between zero and 1 and a
known mean and standard deviation as calculated in Powell et al [2]. This was done because of the
small amount of data available for analysis of the CMF. Figure 4.9 shows the sensitivity of NPV to
changes in CMF for a DSSC. Both the historical [2] and synthetic results are shown. The synthetic
results were run 1000 times with a random CMF. There is significant deviation in the NPV of the
DSSC when the CMF runs from zero to 1.
75
Figure 4.9 Sensitivity of NPV to CMF in the DSSC
A CMF of zero means that when it is cloudy, no irradiation from the sun reaches the solar cell; a CMF
of one means that when it is cloudy all of the irradiation from the sun reaches the solar cell. As the
CMF increases, the NPV increases, because an increasing CMF means that there is more solar
radiation reaching the solar cell [2]. Figure 4.9 shows that the synthetic results have a higher slope
than the historical results. This difference occurs because of the variability present in the synthetic
electricity price series and the larger number of simulations performed. More detailed analysis on
the CMF is needed to improve the accuracy of these results. A similar trend occurs for the other
example cell and is expected for different materials as well.
4.6.2 Correlation of Weather to Electricity Prices
A further validation of the weather series is to determine the difference in the financial indicators
when using a correlated weather series or a non‐correlated weather series. Previously in this paper,
the method for determining the weather series was outlined. This included using a distribution of
electricity prices pertaining to a specific weather series to determine the weather condition of a
specific hour. This is what is referred to as the correlated weather series. For the non‐correlated
weather series, electricity prices do not factor into the determination of a specific hourly weather
condition. The process for generating such a weather series depends solely on past weather
‐600
‐500
‐400
‐300
‐200
‐100
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1NPV ($2007)
Cloud Modification Factor (CMF)
Stochastic Analysis Powell et al, 2009 [2]
76
condition data. The procedure for this has been outlined by both Pattison [5] and Nkemdirim [6] and
will not be detailed here.
Table 4.9 shows the mean annual savings with 95% confidence intervals for 1000 iterations using
the correlated weather series and the non‐correlated weather series.
Table 4.9 Annual Savings of DSSC and OSC using correlated weather series and noncorrelated weather series.
DSSC OSC Correlated Non‐
Correlated Correlated Non‐Correlated
Annual Savings $83.64 ± 0.57 $76.15 ± 0.75 $19.75 ± 0.14 $17.98 ± 0.17
The non‐correlated annual savings is considerably less than the correlated annual savings.
Furthermore, the non‐correlated results deviate even more from the results from Powell et al. [2]
than the correlated results. This suggests that the correlated weather series makes the synthetic
results more precise and reflects the historical data better than the non‐correlated weather series
results.
4.6.3 Modelling without Jumps
In this section, the analysis was performed using a different stochastic electricity price model that
does not include model the jumps in the historical data, as defined earlier. The equation used for this
model is shown here:
(4.12)
The same parameters used in the original model are used here, a jump never occurs. It can be
deduced here that the average and standard deviation of the electricity prices will be lower, thus
making the annual savings lower, leading to a longer payback period and a lower NPV. The annual
savings, payback period and NPV are shown in Table 4.10 for the DSSC and OSC. An electricity‐price
correlated weather series is used for this simulation.
77
Table 4.10 Financial Indicators for DSSC and OSC without modelling jumps in electricity prices
Financial Indicator DSSC OSC Annual Savings ($2007) $79.69 ± 0.55 $18.97 ± 0.13 Payback Period (years) 9.60 ± 0.06 – 28.87 ± 0.18 46.75 ± 0.23 – 112.10 ±
0.74 NPV ($2007) (1976.90) ± 2.23 – (419.17) ±
2.33 (2056.60) ± 0.54 – (806.75) ± 0.52
A lower annual savings does result when the electricity price is modelled without jumps. This results
in a higher payback period and lower NPV. While this is expected, it is not significantly different than
the synthetic results presented earlier. It does, however, differ more than the historical results
presented in Powell et al. [2], which means that the mean‐reverting jump diffusion model does
predict the historical behaviour of the electricity price more precisely.
4.6.4 Averaged Data
It is interesting to look at the difference between using monthly average values for electricity prices
and irradiance and using hourly values determined by the TEEOS model. In a previously
unpublished work [28], average electricity prices and irradiance were used to calculate the annual
savings accrued from a 10 m2 OSC installation. Instead of a CMF, in this paper, the author used a
solar scaling factor (SSF). A SSF is a linear scaling factor that allows adjusts given UVB data and
calculates the amount of solar irradiance available across the full spectrum (280‐4000 nm). Using 5%
efficiency across the whole spectrum, the annual savings was $98.59 (using 2006 electricity prices).
Using TEEOS with synthetic electricity prices and a synthetic weather series for the same cell, the
annual savings was calculated at $75.29 ± 0.23. Using TEEOS again, but with monthly average
electricity prices and irradiance, the annual savings was calculated at $68.06 ± 0.21. Using monthly
average values should even out any spikes in electricity prices, especially those that happen at times
of peak irradiance; this should lead to a lower annual savings. And between the two TEEOS
estimates, this does occur. Therefore, the previous author’s work should be lower than the TEEOS
estimates. However, it is much higher. As a result, the difference must be in the irradiance
estimation. The previous author used a solar scaling factor that seemed to predict significantly more
irradiance than the CMF, inferred from the higher annual savings. This work should be reviewed and
78
a possibly more robust scaling factor or a CMF like that used in Powell et al. [1,2] and in this paper,
should be used to estimate the total available irradiance.
4.7 Technical Objectives
The technical objectives outlined here are based on the current analysis for Toronto, Canada, but
these recommendations are certainly applicable all over the world. Analyses will be made on the
major inputs of the TEEOS model: initial costs of an organic solar array, including solar cell costs, the
efficiency and wavelength range of the cells, and the lifetime of the cells.
4.7.1 Initial Costs
The results in Powell et al. [2] show that a higher efficiency can provide similar financial benefits as
an increase in absorptive wavelength range. This analysis was done assuming that initial costs would
be the same for each. The results presented in this paper show a range of initial costs, consisting of a
range of solar cell costs and a range of BOS costs.
In Figure 4.8 if the line of NPV were extended to zero, this would correspond to an initial cost for the
OSC of approximately $143 for a 10 m2 array. Using the same proportion of cell costs to BOS costs
(55% ‐ 65% of the initial costs are cell costs on the lower and higher price estimate, respectively),
and assuming that the BOS costs could be reduced proportionally, the OSC itself (not including the
BOS costs) would need to cost below $7/m2 in order to have a positive NPV. This also assumes that
efficiency and wavelength range are the same as in Table 4.8 for an OSC. It is conceivable that if the
price for the OSC does reach $7/m2, the other characteristics, such as efficiency and wavelength
range, would have improved as well.
The detailed costing of organic solar cells is still very preliminary, as most cells have not yet been
developed for commercialization. Krebs et al. [29] have developed a detailed cost analysis for a
PEDOT‐PSS OSC but only with a limited printing. Kawlowekamo & Baker [30] provide an updated
estimate of the same OSC and these estimated are used in this paper. There is high potential for OSCs
to be a very low‐cost power generating system; low‐cost printing press production techniques, non‐
commodity organic materials, and adaptability and flexibility, make OSCs an attractive technology for
energy harvesting. At $7/m2, an OSC array in Toronto would provide a positive NPV and be a
79
promising investment. Krebs et al. [31] notes that the price of organic solar cells may be under
$2/m2 within the next ten years. Indeed, it is safe to assume that if the OSC were tested in a locale
closer to the equator and with more sunny days, the price point for OSC to achieve a positive NPV
would be higher.
4.7.2 Wavelength Range and Efficiency
As evidenced by Powell et al. [2], a higher efficiency does certainly provide a higher annual savings,
payback period, and NPV. Indeed, this same research shows that a larger absorptive wavelength
range will also produce more favourable financial indicators. The OSC examined in this paper
produces a payback period over 45 years and an NPV lower than ‐$800, with an efficiency of 3.4%. If
this efficiency is increased to 6% and wavelength range is kept constant, the payback period
decreases to 25 years and the NPV is increased by 9%. The same reduction in payback period occurs
if the upper absorptive wavelength range is increased from 620 nm to 800 nm, with the efficiency
kept constant at 3.4%. In order to be financially viable, a positive NPV is desired for most business
projects. This will not happen for an OSC at the current cost for any reasonable wavelength range or
efficiency for an OSC. This reinforces the previous paragraph on cell cost; without a decrease in cell
costs, the OSC will not become an inexpensive renewable energy technology. It will be important to
focus on expanding the wavelength range and increasing the efficiency of the OSCs; if this done in
tandem with a reduction in manufacturing costs, the target cell price could be increased from $7/m2
for a climate similar to that of Toronto.
In the near‐term, it will be important for the efficiency of organic solar cells produced using large‐
scale manufacturing techniques to be increased. The efficiencies used in this paper are for cells
produced on a very small scale. The highest efficiency for large‐scale manufactured cells is
approximately 1% [1]; in order to compete with traditional solar technologies, a much higher target
should be set for the next five years for large‐scale production of organic solar cells. It is difficult to
determine a specific target, with the previous paragraphs detailing that there are many factors that
can help improve the financial indicators of the OSCs.
80
4.7.3 Cell Lifetime
In the results presented here, the TEEOS model assumes that the lifetime of the OSCs is 5 years. This
is slightly larger than the best lifetime for an OSC, estimated at 20, 000 h [33], but more realistic
lifetimes of small cells have reached 10, 000 h, or just over one year, have been achieved in standard
conditions [34]. If the OSCs have longer stability, the NPV will undoubtedly improve with more time
to save money from not having to use electricity from the grid. Considerable effort should be made to
increase the stability of the organic solar cell; this is one of the barriers to commercialization, as
discussed in Powell et al. [1] and Brabec [35]. A long cell lifetime is most desirable to homeowners; if
one desires a fifteen year project lifetime, currently, the purchase of three solar cell systems would
be required, reducing the NPV considerably. As more research is done on the stability of polymers
and plastics in different outdoor conditions, a variety of cells can be developed that suit different
climates and locations, but in order to compete with traditional solar technologies, a much higher
lifetime is necessary.
4.8 Conclusions
In this paper, two stochastic models were used to develop synthetic electricity price and weather
series for the TEEOS model. A mean–reverting jump diffusion model was used to model the
electricity price of Toronto, Canada, and while some parameters of the synthetic electricity price
series deviated from that of the historical price series, the final results show that the electricity price
model was adequate. A third‐order Markov‐chain cloud occurrence model was used for the
generating the synthetic weather series. This stochastic model was correlated to the electricity price
model by assuming the synthetic weather series was dependent on the synthetic electricity price
series. This model showed correlation with the clear hour occurrence in the historical weather
series, but did not fare as well predicting cloudy hour occurrences.
By implementing these models, it was assumed that a more accurate estimation of the financial
indicators, annual savings, payback period and NPV could be found. It was shown that without the
correlated weather, the annual savings achieved was not consistent with the estimates in Powell et
al. [2], which were produced using historical data. It is unclear, however, whether the results shown
here are more accurate, due to the absence of experimental testing.
81
Using the results from this paper and from previous papers [1,2], an cell cost target for organic solar
cells was provided. However, this target is only for Toronto, Canada. It is expected that in a climate
with more sunlight, the target price per m2 for OSC to have a positive NPV would be higher, and in a
climate with less sunlight, the target would be lower. It is shown that the cost of the organic solar
cells themselves can be up to 50% of the total initial costs of an organic solar array and that while
research in reducing the cost of the organic solar cells is still important, research should be started to
also reduce the costs of the ancillary costs associated with a solar array. This is especially important
for those in the developing world where the initial cost of a solar array is the biggest barrier to
adoption.
It is evident that increasing either efficiency or wavelength range will improve financial indicators,
but more specific research is needed to determine the correlation between the two factors. In
Toronto, Canada, using the given costs in this paper, it was impossible for an OSC to have a positive
NPV even with a reasonable efficiency and wavelength range for an OSC. Further research into
different climates will provide more information on this. Nonetheless, it is suggested that materials
that absorb wider wavelength ranges, or a combination of materials that absorb in wider wavelength
ranges, specifically the near‐ and mid‐IR, should be used in the development of organic solar cells.
The lifetime of an OSC is currently below two years, under ideal conditions. It is important to
increase this, but, in the meantime, research can be done to adapt cells to different weather climates;
an OSC in Vancouver, Canada may not have the same lifetime as an OSC in Miami, USA and this should
be considered an advantage. These cells have the potential to be so inexpensive that they can be
manufactured for specific locales depending on local weather conditions.
82
4.9 References
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CHAPTER 5
CONCLUSIONS
85
5 Final Conclusions This section will provide a general overview of the three papers used in this study and address some
of the limitations of the TEEOS model and how to improve upon them. It will also discuss future
work that is proposed by the author.
5.1 Limitations of the model
In this section, some limitations of the model will be discussed; specifically, sensitivity to CMF and
the initial cost estimates.
5.1.1 CMF
The annual savings calculated in the model are highly sensitive to CMF as shown in Figure 3.4 and
Figure 4.9. The CMF calculations presented in the first figure are based only on two years of UV data,
while the second is based on synthetic data. The results from both are consistent. It would be ideal to
have much greater accuracy for these numbers; there is a relatively high standard deviation around
the monthly CMFs and this affects the accuracy of the calculations. Two assumptions made in this
model produce limitations with respect to the CMF.
First, the CMF for Toronto is calculated only using UV data and is applied across the full spectrum,
from 280 nm to 4000 nm. Historical irradiance data outside the UV spectrum is not publically
available for Toronto, so this assumption must be made at this time. The development of organic
solar technologies makes the case for broad‐spectrum irradiance data collection for urban centres;
this is done in some locations but most are rural around low population centres.
And secondly, notwithstanding the lack of broad‐spectrum irradiance data, the UV data for Toronto is
only available for two years at a time. Numerous years of data could produce a more robust CMF
estimate for these simulations.
86
5.1.2 Cost Data
A concrete cost for fully commercialized organic/plastic solar cells is not available due to the infancy
of the technology. Therefore, a range of published costs was used. While research is being done on
costing the cells [1‐2], the range is still quite wide and the cells have not been tested outside a
laboratory. An expert assessment of where the cost of these cells will be in the future is shown by
Curtwright et al. [3], but this is not definitive as it is based on the opinions of researchers, not
necessarily hard evidence. It is one of the objectives of this paper to determine a target for the cost of
fully commercialized OSCs in order for them to be financially viable for domestic electricity
generation. While this target is location specific, this paper does provide the methodology for
determining targets for other locales, which is encouraging.
However, the range used in this paper shows that the solar cells may represent as low as 50% of the
total initial costs of a solar array. Therefore, any reduction in the cost of the cells would be aided by a
reduction in the BOS costs, which are not discussed in this paper. Indeed, research should also focus
on developing low cost inverters and smart meters that would complement the use of inexpensive
OSCs, especially for markets in developing countries where traditional solar technologies will always
be too expensive for large‐scale adoption.
5.2 Future Work
In an effort to streamline the process of determining financial indicators for organic solar cells, the
TEEOS model should be made more user‐friendly. The current method uses a combination of
Microsoft Excel and Matlab for the various calculations, as well as SMARTS2 for the irradiance
modeling. It would be easier for researchers to use TEEOS if it was contained in one program, and a
program that is readily available, such as Microsoft Excel. A ‘dashboard’‐like interface will be
developed that will take inputs, such as wavelength range, efficiency, cell cost and discount factor,
and output the necessary financial indicators, using a Microsoft Excel macro. Initially, the program
will only be for Toronto, Canada, but it will be important for the dashboard to include a link to the
SMARTS2 program so that other locations can be used as well.
TEEOS will also be applied to different locations in the near future. This was not completed for this
thesis due to time constraints and the need to find suitable data. Certain locations of interest are
87
those in contrast to Toronto’s climate, such as Arizona, Hawaii, and the Northwest Territories, and
those similar to Toronto’s climate, such as New York state.
Additionally, more broadband spectral irradiance data will be sought for various locations, including
Toronto. Broadband spectral irradiance data will help improve the CMF estimations and improve the
accuracy of the financial indicators. This will also help increase the demand for such data, which is
very much limited to satellite data and rural ground stations.
Indeed, as the original price series parameters did not match that well with the synthetic price series,
a different model and approach to calibration should be attempted.
As well, the TEEOS results will be verified by real organic solar arrays when they are ready for
deployment in Toronto.
5.3 Conclusions
In this paper, a model called TEEOS was developed that enables researchers to determine financial
indicators, such payback period, net present value, and internal rate of return, for OSCs, using inputs
such as location, electricity prices, solar cell efficiency, absorptive wavelength range, and initial costs.
First, TEEOS was used to determine financial indicators for a solar hat. This was proven
uneconomical, but even the researchers who performed this test admitted it was not meant to be an
economic exercise, just a demonstration of the technology. The third chapter of this paper was
meant to fully demonstrate the TEEOS model. First, a DSSC and an OSC were evaluated using the
TEEOS model. The OSC was shown to not be able to obtain a positive NPV in Toronto. In the fourth
chapter, two stochastic models were introduced to generate synthetic electricity price and weather
series. The purpose of this was two‐fold: to improve accuracy of results and to allow for varying of
parameters to address sensitivity to various changes. A mean‐reverting jump diffusion electricity
price model was correlated to a third‐order Markov chain weather model and implemented in the
TEEOS model. The results for both cells were very close to that of the initial estimate in Chapter 3.
Chapter 4 also included technical objectives, based on the financial indicators, which should be
targets for those developing organic solar cell technologies. First, a concerted effort is needed to
88
reduce not only the cost of the solar cells, but also the cost of the ancillary equipment, such as
inverters and meters. Secondly, when dealing with financial indicators, a near‐term goal for
efficiency of OSCs is highly dependent on climate, but need to be drastically increased regardless of
climate. While research on small‐area OSCs is still important, to move forward in OSC research more
work needs to be done on large‐scale manufacturing techniques so that when high‐efficiency OSCs
are available, the methodologies are there for those interested in developing the technology.
Similarly, more research is needed in materials that absorb light in the near‐ and mid‐IR wavelength
ranges. This is not only important because it improves financial indicators, but also because it opens
the door for a host of new materials to be used for purposes other than OSCs, such as camouflage for
military uses. Although current organic solar cells last, on average, up to one year, a longer term is
needed to reach the larger market. If cell lifetime is kept short, OSCs will remain a nice technology; a
longer lifetime will certainly make it more accessible to companies looking to expand their
photovoltaic business and customers looking to “go green.” Future directions for the TEEOS model
including developing a dashboard model that makes using the model more streamlined and more
results will be calculated for different locations and comparing them to the analysis for Toronto,
Canada.
89
5.4 References
[1] Krebs, C.F., Jørgensen, M., Norrman, K., Hagemann, O., Alstrup, J., Nielsen, D.T., Fyenbo, J., Larsen, K., Kristensen, J., A complete process for production of flexible large area polymer solar cells entirely using screen priting – First public demonstration, Solar Energy Materials and Solar Cells 93 (2009) 422‐441. [2] Kawlowekamo, J. and E. Baker, Estimating the manufacturing cost of purely organic solar cells, Solar Energy 83:8 (2009) 1224‐1231. [3] Curtright, A.E., M.G. Morgan, D.W. Keith, Expert assessments of future photovoltaic technologies, Environ. Sci. Technol., 42(24) (2008) 9031–9038
90
Appendix A Table A1 ‐ The inputs used for the SMARTS2 modelling software
Model Characteristic Input Value(s)Site pressure (calculated from latitude, altitude, and height of site)
Altitude (at ground) 0.108 km Height (above ground) 0.09 km Latitude 43°78’ N
Atmosphere (Reference Atmosphere) “Mid Latitude Winter” from October to March and “Mid Latitude Summer” from April to September, inclusive
Water Vapour calculated from reference atmosphere and altitude
Columnar Ozone Abundance default from reference atmosphereGaseous Absorption and Pollution default from reference atmosphereCarbon Dioxide Concentration 370 ppmExtraterrestrial Spectrum Gueymard 2002 (synthetic) Aerosol Model Shettle & Fenn – Urban Turbidity 0.084, specified as Aerosol Optical
Depth at 500 nm Albedo Spectral albedo data file = Manmade
amterials – concrete slab Default Spectral Range 280‐4000 nmSolar Constant 1367 W/m2Solar Position
Year 2006 and 2007 Month 1 through 12 Hour depends on sunrise/sunset hours Latitude 43.78 N Longitude 79.5 W Time Zone ‐5:00 UTC Desired Output
Direct Horizontal Irradiance (W m‐2
nm‐1)
91
Appendix B
Table 5.1 Historical Hourly Volatility for each month Electricity Prices
Hour/Month
January February March April May June July August September October November December
1
2.8690 2.8401 2.8849 2.2682 1.9727 2.1812 2.6367 2.6022 2.1012 1.7726 2.2191 3.0467
2
2.7898 2.9782 2.8166 1.9983 2.0233 2.1344 2.1920 2.3927 1.9777 1.9092 1.9696 2.6799
3
2.6337 2.9703 2.7232 2.0582 1.8690 1.8002 2.1883 2.2797 1.9049 1.9311 1.8960 2.6939
4
2.7257 2.5250 2.6535 2.3736 1.7203 1.9618 2.1307 1.9639 2.0675 1.8295 1.4750 2.8429
5
2.6597 2.3138 2.6258 1.9088 1.6742 1.9152 1.8015 1.9886 2.0941 1.6571 1.3625 2.4732
6
2.6233 2.5909 2.7677 2.6761 2.0457 2.1149 1.9838 2.4336 2.3182 2.2806 1.6495 2.1331
7
3.1226 3.1083 3.4094 3.6722 3.2549 2.6261 2.3193 2.6010 3.3913 3.8996 2.4033 2.5174
8
4.2915 3.9580 3.8961 3.8344 3.5423 3.1531 3.1051 3.3943 3.4879 3.6791 3.5716 3.6316
9
4.4531 4.1165 4.0744 3.7039 3.5403 3.5421 3.8032 3.6795 3.3042 3.6248 3.6873 4.0138
10
4.1764 3.9927 3.9190 3.6673 3.6720 4.1758 4.0499 3.7115 3.5157 3.4201 3.6415 3.8196
11
4.1892 3.9266 3.9307 3.8614 3.9171 4.2426 4.1088 3.8198 3.3331 3.4540 3.4677 3.9189
12
4.0137 3.6395 3.8500 3.6388 4.0348 4.3810 4.1833 3.9961 3.2281 3.1774 3.3909 4.1800
13
3.8457 3.3358 3.6729 3.6899 3.8720 4.2949 4.1187 3.9359 3.5596 3.2176 3.2897 4.0181
14
3.8464 3.3169 3.8427 3.8461 4.0089 4.1312 4.0881 3.8634 3.4600 3.2702 3.0311 3.7169
15
3.5291 3.2649 3.8990 3.8762 3.9103 4.0350 3.9544 3.8804 3.4428 3.3803 3.0014 3.6174
16
3.4592 3.0857 3.6010 3.9452 4.0380 4.5487 3.9962 3.7139 3.5034 3.4741 2.7424 2.9219
17
3.3224 3.1969 3.5409 3.7155 3.7610 3.9878 4.1099 3.7820 3.6246 3.5522 2.9086 3.2033
18
3.6758 3.2092 3.7351 3.4047 3.0303 3.7112 3.9943 3.7043 3.0891 3.4830 3.3714 3.9119
19
3.7256 3.4523 3.7157 3.4961 2.8899 3.3731 3.8189 3.5482 3.3243 3.3574 3.4041 3.9414
20
3.5700 3.6104 3.7102 3.7040 2.9447 3.2485 3.7880 3.7789 3.2917 3.2877 3.2770 3.9087
21
3.5720 3.5647 3.5831 3.2856 3.0847 3.6090 3.8185 3.6335 3.1115 2.6901 2.9134 3.8456
22
3.2016 3.3536 3.0237 2.3450 2.0027 2.8549 3.4391 2.8472 2.8079 2.2844 2.6379 3.4898
23
2.5929 2.4888 2.5948 2.4646 2.0845 2.6463 3.0663 2.7890 2.2896 1.9512 1.9346 3.1440
24
2.7484 2.7622 2.6480 0.8472 1.6640 2.3733 2.7236 2.8696 1.8573 1.5310 2.2267 2.8686
92
Table 5.2 Historical Mean Reversion Rates Electricity Price
January February March April May June July August September October November December
1 0.6248 0.7509 0.3437 0.2871 0.4480 0.4312 0.6009 0.4706 0.4694 0.5922 0.5528 0.4238
2 0.5635 0.7578 0.4297 0.2152 0.4565 0.4221 0.5877 0.4861 0.4025 0.6015 0.6086 0.4045
3 0.5778 0.6395 0.3703 0.2327 0.3830 0.3517 0.5796 0.4545 0.3628 0.6389 0.5858 0.3499
4 0.5434 0.6454 0.3794 0.3254 0.3382 0.3939 0.4598 0.4551 0.3942 0.5954 0.4313 0.4350
5 0.5367 0.7222 0.4479 0.2087 0.2997 0.4474 0.4816 0.4883 0.5115 0.5376 0.3918 0.3966
6 0.5054 0.6606 0.5377 0.3691 0.4460 0.4453 0.5028 0.5893 0.4322 0.5415 0.4992 0.3149
7 0.5902 0.6877 0.6511 0.4735 0.6005 0.5049 0.4741 0.4316 0.5947 0.7136 0.6097 0.3481
8 0.6097 0.6364 0.6667 0.5065 0.6411 0.5574 0.4696 0.5271 0.6720 0.7336 0.7561 0.4599
9 0.6210 0.6773 0.6012 0.5986 0.6223 0.5380 0.5999 0.5740 0.5969 0.7542 0.7931 0.5263
10 0.6311 0.7404 0.6254 0.6267 0.5818 0.5422 0.6135 0.5598 0.5877 0.6911 0.9293 0.5187
11 0.6282 0.8088 0.6365 0.6207 0.6343 0.4933 0.5955 0.6394 0.5428 0.8062 0.8525 0.5414
12 0.5732 0.7602 0.6079 0.5431 0.6433 0.4669 0.6054 0.6903 0.4595 0.6049 0.8320 0.6079
13 0.5672 0.6401 0.5868 0.5190 0.5261 0.4333 0.6121 0.7331 0.5226 0.5854 0.7427 0.6425
14 0.5479 0.7011 0.5520 0.5041 0.5613 0.4017 0.5564 0.6350 0.4238 0.5328 0.6688 0.5584
15 0.5058 0.6159 0.5499 0.5048 0.5767 0.3583 0.4909 0.6833 0.4098 0.5760 0.7838 0.6423
16 0.5236 0.6833 0.5467 0.5401 0.6017 0.4795 0.5476 0.6387 0.4368 0.7076 0.8452 0.4961
17 0.5096 0.7697 0.5456 0.5120 0.5657 0.3593 0.5262 0.6254 0.4791 0.6241 0.8081 0.4042
18 0.5399 0.5858 0.6051 0.4735 0.5487 0.3188 0.4619 0.5977 0.3811 0.5231 0.6668 0.4319
19 0.5375 0.4914 0.4728 0.4830 0.5675 0.3519 0.4267 0.5873 0.4406 0.6707 0.6626 0.4139
20 0.4834 0.5723 0.4248 0.4798 0.5519 0.3482 0.4836 0.6310 0.4760 0.7219 0.6388 0.3939
21 0.4991 0.5997 0.4187 0.4911 0.5650 0.3779 0.4818 0.6103 0.4649 0.5886 0.5810 0.4088
22 0.4568 0.7724 0.3467 0.3472 0.3919 0.3285 0.5152 0.4464 0.4488 0.4805 0.5898 0.4131
23 0.4145 0.7567 0.2973 0.3441 0.4033 0.3632 0.4470 0.4250 0.4397 0.4986 0.7103 0.4800
24 0.5119 0.7942 0.2895 0.3453 0.2692 0.3933 0.4625 0.5358 0.3440 0.4702 0.6223 0.3562
93
Table 5.3 Historical Mean Reversion Levels Electricity Prices
January February March April May June July August September October November December
1 41.22092 45.08356 42.60215 27.65581 24.57105 27.83289 32.29562 35.16667 32.9681 32.67313 38.35839 40.53835
2 39.06745 42.06355 37.8614 25.18763 22.42783 24.84067 28.78196 30.93075 29.40447 30.17473 33.66522 35.67284
3 37.78427 41.17571 37.63888 25.1101 21.16359 22.84844 27.05163 29.09262 27.9701 28.47489 31.83629 34.33099
4 37.93486 39.54355 36.79597 25.72293 21.2802 22.54835 26.59157 28.47362 28.32014 29.20323 30.64466 33.18625
5 37.46686 39.17463 37.98942 27.99675 22.505 22.87822 25.80302 30.06907 30.90556 32.58328 31.01989 32.4466
6 39.17825 41.92002 42.62229 34.49722 26.2597 24.32133 26.25205 32.16924 37.15383 41.00833 33.11419 33.35844
7 45.09197 48.81474 47.09859 44.43049 35.91177 31.16596 31.84996 37.07038 44.77429 53.20552 39.50062 38.91877
8 56.70357 56.73112 54.0267 51.68802 42.7051 39.60892 41.98826 46.46698 50.21029 56.70979 49.29896 48.42673
9 57.27405 57.71493 58.19184 53.50238 45.78768 45.56748 50.20963 53.30919 54.77362 57.1516 53.31463 50.71003
10 55.9994 57.30649 59.44313 56.375 49.63253 53.57326 59.10101 60.51993 59.26213 60.83184 55.96975 51.50103
11 57.40707 58.22205 60.28562 58.04078 52.56847 57.23546 63.52082 64.52267 61.14505 61.93388 57.89664 53.52567
12 57.63965 56.88827 59.50781 55.8772 52.97864 59.7613 67.73277 69.66395 63.08314 61.05073 58.1319 54.56699
13 58.07853 55.19899 55.9559 55.56275 53.60819 60.58357 69.72381 70.36471 65.0838 59.56456 55.34151 52.72015
14 58.20517 55.06081 55.49992 53.91228 52.12418 60.78821 68.70677 68.25602 62.58235 57.09491 52.4771 51.06185
15 54.17228 51.86102 52.58981 53.39943 49.85495 57.86844 66.99039 65.70475 61.39077 55.19816 49.90378 46.74887
16 50.80282 47.88234 51.50953 54.44798 50.63446 59.26313 68.02268 67.56731 63.06112 57.63437 45.18237 41.74267
17 53.6519 48.95098 52.86334 52.48832 49.84938 57.08121 69.04515 67.49701 63.84905 57.73423 52.16914 50.87239
18 69.75683 54.45925 50.53827 45.88578 43.76414 51.58531 64.2674 62.93509 56.68412 59.5346 73.39158 69.40933
19 72.16932 69.14643 57.97345 43.43629 40.36824 45.89105 59.97262 58.58813 61.52422 63.62516 72.47247 70.05935
20 65.99127 68.34502 62.08252 56.17383 45.6308 45.99136 58.09043 64.2772 62.51575 60.1306 64.87705 64.65974
21 60.95208 63.30667 59.27193 51.31235 48.52709 51.2579 62.54218 63.44383 54.54232 51.55637 59.2768 61.9451
22 53.68629 56.38003 49.90655 38.38186 37.09085 42.00824 51.98487 49.1361 45.03263 43.24422 51.37051 55.94411
23 46.13956 45.49434 43.72353 33.37655 30.52211 36.19125 44.47734 43.07945 40.58946 37.97394 41.83721 47.28848
24 42.53753 44.54635 42.00333 34.46194 24.49163 29.8103 36.37823 38.28651 34.98857 33.71571 39.67728 41.58069
94
Appendix C
Table 5.4 Synthetic Mean Reversion Levels 1000 iterations
January February March April May June July August September October November December
1 41.6853 45.3414 44.8251 30.9405 25.4847 28.9033 32.8003 36.1372 33.7736 33.0426 38.9015 41.9570
2 39.6780 42.3250 39.1706 32.5415 23.3138 25.9498 29.2455 31.7728 30.6111 30.5444 34.0288 37.1679
3 38.3488 41.6230 39.4926 30.9449 22.5284 24.4858 27.5438 30.0718 29.4747 28.7976 32.2388 36.5114
4 38.6145 39.9259 38.5112 28.1278 23.0882 23.8660 27.5058 29.3633 29.6240 29.5745 31.4142 34.4875
5 38.1537 39.4029 39.0959 35.9909 24.9436 23.7688 26.5171 30.7829 31.5695 33.0403 31.9685 33.9387
6 39.9878 42.2647 43.3140 36.3149 27.2131 25.3089 26.9264 32.6522 38.2461 41.5832 33.6780 35.7879
7 45.6722 49.1648 47.5640 45.6480 36.5362 31.9857 32.7236 38.2922 45.4089 53.6090 39.9249 40.9665
8 57.3914 57.2941 54.5352 52.7601 43.2552 40.3408 43.0933 47.3275 50.6395 57.0671 49.6240 49.7080
9 57.9474 58.1818 58.8747 54.1327 46.3766 46.4195 50.8669 54.0543 55.3649 57.4593 53.5829 51.6969
10 56.6111 57.6721 60.0229 56.9363 50.3523 54.5069 59.7492 61.2916 59.8899 61.2374 56.1229 52.4858
11 58.0269 58.4589 60.8299 58.6158 53.1404 58.4095 64.2188 65.0256 61.8913 62.1559 58.0675 54.4230
12 58.4060 57.2102 60.1454 56.6796 53.5566 61.1055 68.3987 70.0985 64.1456 61.5819 58.3187 55.2613
13 58.8362 55.6689 56.6232 56.4738 54.5350 62.1694 70.3405 70.7228 65.9255 60.1518 55.6456 53.2706
14 59.0463 55.4073 56.3286 54.9317 52.9723 62.6124 69.4958 68.7917 63.9762 57.8823 52.8557 51.8620
15 55.1166 52.3739 53.4412 54.4351 50.5964 60.2793 68.0905 66.1201 62.9197 55.8616 50.1380 47.2870
16 51.7065 48.2530 52.3084 55.3482 51.3236 60.6000 68.8432 68.0864 64.3911 58.0043 45.3334 42.6568
17 54.5582 49.2121 53.6720 53.4549 50.6300 59.4636 69.9899 68.0477 64.9323 58.2788 52.3695 52.4648
18 70.5397 55.0299 51.1772 47.0376 44.4772 54.6803 65.5522 63.5432 58.3958 60.3928 73.7861 70.8516
19 72.9356 70.0563 59.1369 44.5304 41.0207 48.2594 61.5086 59.2460 62.7771 64.0439 72.8745 71.6344
20 67.0155 68.9726 63.6047 57.3041 46.3156 48.3593 59.2222 64.8105 63.5520 60.4501 65.3251 66.4733
21 61.9019 63.8604 60.7686 52.2841 49.1642 53.3313 63.6668 64.0235 55.6016 52.0698 59.8226 63.6586
22 54.8557 56.6811 52.1327 40.2964 38.3514 44.5247 52.8960 50.2602 46.1469 44.0545 51.8718 57.5058
23 47.4312 45.6945 46.7550 35.4658 31.7782 38.0961 45.6980 44.3551 41.5883 38.6189 42.0622 48.3376
24 43.3388 44.7609 45.3193 35.3873 27.8044 31.3158 37.4464 39.0451 36.6629 34.3592 40.0616 43.6786
95
Table 5.5 Synthetic Mean Reversion Rates Electricity Prices
January February March April May June July August September October November December
1 0.5578 0.6493 0.2143 0.1376 0.3297 0.3023 0.5302 0.3645 0.3476 0.5083 0.4509 0.3114
2 0.4850 0.6590 0.3164 0.0790 0.3405 0.2887 0.5073 0.3792 0.2613 0.5211 0.5163 0.2817
3 0.4963 0.5317 0.2436 0.0884 0.2430 0.1990 0.4957 0.3417 0.2154 0.5651 0.4869 0.2157
4 0.4545 0.5343 0.2508 0.1788 0.1875 0.2488 0.3462 0.3362 0.2536 0.5146 0.2885 0.3228
5 0.4475 0.6177 0.3364 0.0748 0.1451 0.3188 0.3654 0.3818 0.3960 0.4378 0.2407 0.2699
6 0.4069 0.5549 0.4510 0.2340 0.3282 0.3154 0.3973 0.5086 0.3032 0.4526 0.3791 0.1680
7 0.5190 0.5830 0.5929 0.3660 0.5293 0.3950 0.3657 0.3183 0.5068 0.6659 0.5198 0.2125
8 0.5466 0.5345 0.6088 0.4057 0.5771 0.4627 0.3675 0.4443 0.5990 0.6869 0.6892 0.3639
9 0.5614 0.5795 0.5349 0.5166 0.5542 0.4444 0.5331 0.4994 0.5110 0.7084 0.7284 0.4472
10 0.5730 0.6407 0.5647 0.5473 0.5082 0.4523 0.5470 0.4853 0.5033 0.6375 0.8734 0.4348
11 0.5661 0.7157 0.5774 0.5396 0.5732 0.3951 0.5285 0.5849 0.4507 0.7687 0.7906 0.4615
12 0.5029 0.6587 0.5409 0.4539 0.5826 0.3656 0.5410 0.6401 0.3514 0.5340 0.7717 0.5492
13 0.4924 0.5310 0.5162 0.4246 0.4455 0.3284 0.5522 0.6915 0.4268 0.5121 0.6787 0.5844
14 0.4703 0.5956 0.4756 0.4072 0.4866 0.2907 0.4840 0.5753 0.3105 0.4503 0.5916 0.4806
15 0.4196 0.5106 0.4718 0.4081 0.5061 0.2408 0.4032 0.6303 0.2927 0.5028 0.7195 0.5806
16 0.4388 0.5799 0.4682 0.4476 0.5363 0.3787 0.4745 0.5792 0.3269 0.6564 0.7854 0.4004
17 0.4213 0.6690 0.4635 0.4158 0.4911 0.2382 0.4503 0.5624 0.3767 0.5598 0.7473 0.2910
18 0.4611 0.4729 0.5378 0.3670 0.4645 0.1922 0.3679 0.5319 0.2546 0.4365 0.5964 0.3348
19 0.4615 0.3713 0.3783 0.3773 0.4825 0.2214 0.3253 0.5175 0.3290 0.6147 0.5879 0.3130
20 0.3922 0.4652 0.3212 0.3746 0.4701 0.2174 0.3912 0.5725 0.3688 0.6711 0.5613 0.2859
21 0.4126 0.4906 0.3145 0.3865 0.4850 0.2544 0.3924 0.5464 0.3527 0.5113 0.4909 0.3012
22 0.3557 0.6731 0.2202 0.2051 0.2590 0.1900 0.4316 0.3371 0.3313 0.3746 0.4990 0.3050
23 0.2945 0.6546 0.1587 0.2032 0.2743 0.2231 0.3367 0.3127 0.3129 0.3945 0.6365 0.3835
24 0.4182 0.6929 0.1521 0.1702 0.1156 0.2560 0.3560 0.4497 0.1943 0.3482 0.5316 0.2285
96
Table 5.6 Synthetic Volatility Electricity Prices 1000 simulations
January February March April May June July August September October November December
1 2.9260 2.7771 3.1649 2.6653 2.0503 2.2528 2.6930 2.7123 2.1455 1.8044 2.2458 3.2237
2 2.8621 2.9201 2.9695 3.4945 2.1008 2.2076 2.2332 2.4834 2.0649 1.9419 1.9844 2.8519
3 2.6976 2.9281 2.9283 3.0589 1.9906 1.9287 2.2399 2.3730 2.0157 1.9575 1.9113 2.9488
4 2.8081 2.4832 2.8460 2.6139 1.8916 2.0513 2.2162 2.0436 2.1638 1.8586 1.5143 2.9919
5 2.7356 2.2631 2.7460 3.6163 1.9490 1.9665 1.8566 2.0547 2.1291 1.6941 1.4157 2.6292
6 2.7199 2.5323 2.8458 2.8431 2.1337 2.1781 2.0518 2.4873 2.3972 2.3426 1.6759 2.4060
7 3.2065 3.0472 3.4656 3.7865 3.3382 2.6875 2.4055 2.7371 3.4358 3.9584 2.4290 2.7473
8 4.4025 3.8952 3.9691 3.9404 3.6113 3.2022 3.2417 3.5126 3.5054 3.7332 3.5851 3.8070
9 4.5607 4.0452 4.1775 3.7587 3.6212 3.6227 3.9088 3.7827 3.3518 3.6685 3.6829 4.1485
10 4.2720 3.9227 4.0100 3.7112 3.7711 4.2578 4.1527 3.8320 3.5748 3.4745 3.6420 3.9729
11 4.2765 3.8288 4.0152 3.9022 4.0034 4.3748 4.2341 3.9026 3.4026 3.4606 3.4316 4.0464
12 4.1412 3.5754 3.9502 3.7109 4.1177 4.5435 4.2890 4.0603 3.3384 3.2544 3.3662 4.2890
13 3.9657 3.2841 3.7631 3.7741 4.0052 4.4909 4.2214 3.9842 3.6475 3.2992 3.2946 4.1026
14 3.9780 3.2488 3.9768 3.9519 4.1369 4.3510 4.2186 3.9470 3.6270 3.3830 3.0475 3.8317
15 3.6619 3.2189 4.0348 3.9884 4.0218 4.3378 4.1344 3.9396 3.6187 3.4715 2.9993 3.6945
16 3.5818 3.0237 3.7158 4.0423 4.1372 4.7083 4.1323 3.7901 3.6532 3.5165 2.7151 3.0331
17 3.4526 3.1418 3.6684 3.8167 3.8740 4.2912 4.2588 3.8635 3.7359 3.6371 2.9016 3.4108
18 3.7971 3.1664 3.8265 3.5089 3.1197 4.1205 4.2042 3.8070 3.2736 3.6114 3.3972 4.1437
19 3.8482 3.4454 3.8889 3.5880 2.9666 3.6413 4.0418 3.6468 3.4541 3.4119 3.4181 4.2040
20 3.7363 3.5700 3.9398 3.8306 3.0367 3.5146 3.9602 3.8605 3.4019 3.3278 3.3016 4.1912
21 3.7143 3.5107 3.8031 3.3702 3.1756 3.8466 3.9930 3.7255 3.2183 2.7566 2.9552 4.1127
22 3.3649 3.2934 3.3164 2.5276 2.1234 3.1384 3.5688 2.9911 2.9074 2.3731 2.6757 3.7050
23 2.7453 2.4298 3.0028 2.6674 2.2092 2.8274 3.2148 2.9401 2.3675 2.0182 1.9341 3.2818
24 2.8503 2.7056 3.0989 0.9066 2.1064 2.4950 2.8454 2.9570 1.9951 1.5805 2.2408 3.1233