Chapter 1 Introduction

Chapter 1

Introduction

The development of solar energy conversion systems (solar power) is reaching the commercialisation phase. The growing focus at international conferences such as those of the International Solar Energy Society and Solar Power and Chemical Energy Systems (PACES) has been on large scale projects, system developments, political trends and timelines for economically competitive systems. Therefore in developing solar power, the major challenges are not in understanding the fundamental physics of the processes, but rather, in the engineering required to build such systems to create solar power that can compete economically against more established forms of power generation. Australia is blessed with an abundant solar resource. However, the possibility of large scale implementation of solar energy without government intervention is remote. This is due to the fact that Australia has some of the cheapest electricity in the world. Eighty per cent of all electricity is produced by the burning of coal (a fossil fuel) from the vast reserves which are situated close to densely populated areas. It is very difficult for renewable energy to compete against this source of base load electricity. With the growth of understanding within the civic community of the threat to the global environment from the burning of fossil fuels and the increasing awareness of the need for governments to react, the possibility of implementing solar power is now increasing. Large scale concentrating solar power systems exploit economies of scale to increase their economic viability. To commercialise such systems, some fundamental technical problems still need to be overcome. A significant aspect of this is devising an optical system that can achieve the high fluxes in the absorber plane required for such systems. This is the central focus of this thesis, which describes a computational framework to better model and solve this aspect of large scale concentration solar power systems.

1 CHAPTER 1. INTRODUCTION 2

1.1 Energy trends and the global environment

Before addressing some of the logistical problems of implementing large scale solar power, it is necessary to identify the need for renewable energy, and in particular, solar power within the global community.

1.1.1 Findings of the IPCC

Several international agencies have been created to monitor, advise and react to the perceived changes in the global climate. The most significant and widely accepted agency is the Intergovernmental Panel on Climate Change (IPCC) established from the (United Nations’) World Meteorological Or- ganisation and the United Nations Environmental Programme. The IPCC’s specific tasks are to assess scientific, technical and socio-economic information relevant to the understanding of climate change, its potential impacts and options for adaptation and mitigation1.

Having recently published their major septennial report, the general con- sensus of the IPCC was that there would be an increase in average surface temperature of our planet, of between 1.4 Kelvin and 5.8 Kelvin (depending on levels of economic growth, commitment to implementing renewable energy alternatives and uncertainties between global climate models) through the years of 1994 to 2100 due to an increase in levels of greenhouse gases in the atmosphere. The IPCC also concludes that:

In the light of new evidence and taking into account the remaining uncertainties, most of the observed warming over the last 50 years is likely to have been due to the increase in the greenhouse gas concentrations. IPCC2.

Greenhouse gases are those atmospheric gases that are responsible for trap- ping thermal radiation, driving the temperature of the planet towards warmer conditions (positive radiative forcing). The three most responsible gases for 3 the enhanced greenhouse eﬀect are methane (CH4), nitrous oxide (N2O) and

1Mitigation is deﬁned here as an anthropogenic intervention to reduce the sources of greenhouse gases or enhance their sinks 2Summary for Policymakers (2001), Intergovernmental Panel on Climate Change, p 10 3 Enhanced refers to anthropogenic sources only. Water vapour (H2O) is the greatest absorber of thermal energy. CHAPTER 1. INTRODUCTION 3

4 carbon dioxide (CO2). Of this list, the gas most responsible for the measured increase in the average surface temperature is carbon dioxide. This is not because carbon dioxide is the greatest absorber of solar radiation, but due to the fact that carbon dioxide levels in the atmosphere have increased by 35% between 1750 and the present. Projections are that the total increase from 1750 to the year 2100 will be at least 260% leading to the conclusion of the IPCC that:

Emissions of carbon dioxide due to fossil fuel burning are virtually certain to be the dominant inﬂuence on trends in atmospheric carbon dioxide levels in the 21st century. IPCC5

Anthropogenic sources of these greenhouse gases are motor vehicles run by the burning of crude oil distillates (oil), industry, heating, cooking, agricul- ture, land clearing6 and electricity generation from burning oil, natural gas and more extensively coal. In general though, the creation of greenhouse gases responsible for global warming can be directly attributed to anthropogenic energy consumption.

1.1.2 Global energy trends

The International Energy Agency (IEA) is a working group of the Organiza- tion for Economic Cooperation and Development (OECD), set up to monitor global energy trends and advise OECD countries on energy policy. One of the statistics it monitors is fuel shares of total ﬁnal consumption (FSTFC), which is literally, the consumption of energy by the diﬀerent end-use sectors (Figure 1.17).

It is important to point out that the FSTFC reproduced in Figure 1.1 do not directly represent greenhouse gas emissions. Electricity generation from the combustion of oil, gas and coal has a typical average eﬃciency of approximately 33%. Systems using combustible renewables and waste burn fuel with lesser eﬃciency to produce an equivalent amount of end-use energy.

4 Sulfur dioxide SO2 is also inﬂuential in global climate models as a negative radiative forcer. 5Summary for Policymakers (2001), Intergovernmental Panel on Climate Change, p 11 6While land clearing is not directly responsible for greenhouse gases it is responsible for an increase in the equilibrium levels of carbon dioxide in the atmosphere. 7The units in Figure 1.1 represent millions of tonnes of oil equivalent (Mtoe) each of which represents 4.2 x 104 TJ. CHAPTER 1. INTRODUCTION 4

Figure 1.1: International Energy Agency: Fuel shares of total final consumption 1973 - 2001 (not in primary energy equivalent terms, reproduced with permission from the IEA (2003)) 1 Mtoe = 4.2 104 TJ. * Prior to 1994 the effects of combustible renew×able & waste final consumption has been estimated on the total primary energy supply. ** Others include geothermal, solar, wind, heat etc.

Figure 1.2: International Energy Agency: Fuel shares of electricity generation 1973 - 2001, excluding pumped storage (reproduced with permission from the IEA (2003)). ** Others include geothermal, solar, wind, combustible renewables & waste. CHAPTER 1. INTRODUCTION 5

The statistics of FSTFC do however, represent accurate trends in the relative demand for consumable energy.

Categories of end-use sectors are: oil - now almost exclusively used for transport; gas - used for cooking and processes heat; coal and combustible renewables for process heat; other: consisting of renewable thermal energy; and electricity.

Investigating trends in the demand for energy, over the 28 year period beginning with the creation of the OECD, global consumption increased by approximately 54%, representing an average growth rate of 1.5% per annum. Of the categories of end-use sectors, the greatest change in the proportion of the FSTFC was the global demand for electricity, which increased at a rate of more than double that of the global demand for consumable energy, at 3.3% per annum.

Focusing on the break up of generated electricity in 2001 (Figure 1.2), approximately 64% can presently be attributed to the burning of fossil fuels producing carbon dioxide. While considerable, this does represent a marked decrease from the dependence of generated electricity on the burning of fossil fuels in 1973. The reasons for this reduction are that: nuclear power has made a significant impact on the global market, there has been the replace- ment of low efficiency oil burning power stations with higher efficiency gas turbines and there has been an increase in the efficiency of both coal and gas turbine generators.

Electricity generated from renewable resources contributed 18.4% of the total global generation 2001. Hydroelectricity produces the greatest proportion of this, being 2569 TWh (1TWh = 3600 TJ), rising from 1285 TWh in 1973 . The remainder of electricity generated from renewable resources is from the burning of biomass (1.6%), with the remainder, in order of output, generated from: wind, solar thermal, geothermal and then solar photovoltaic.

The IEA predicts that over the next 20 years the electricity generated from renewable resources will increase by 53 percent. The majority of this increase in capacity can be directly attributed to large-scale hydroelectric dam projects in the developing world, particularly Asia, where China8, India and other nations such as Malaysia, Nepal, and Vietnam, are already building or planning to build hydro projects that each exceed 1,000 MW of capacity.

8China is building the Three Gorges Dam project, the largest hydroelectricity plant and man-made structure built. It will produce 84 TWh annually. CHAPTER 1. INTRODUCTION 6

No noticeable trend can yet be registered that infers the proportion or even capacity of renewable energy from non-hydro sources that will make a considerable impact on the global market, in the near future. These sentiments were also acknowledged by the International Atomic Energy Agency (IAEA), stating that:

“... we should be under no illusion that in the short or medium term these [renewable energy] sources will bring us the huge quantities of energy that will be demanded [by the global market].” Dr Hans Blix, former Director General of the IAEA9

Of equal importance are the regions of the world that are responsible for consuming the electricity that is generated. The OECD nations which make up only 18% of the world’s population consume 61.3% of global electricity (Figure 1.3). Similarly, of all the world’s anthropogenic carbon dioxide dumped into the atmosphere, the OECD is responsible for 52.8% of total emissions (Figure 1.4). This is in stark contrast to China, home to 19.4% of the worlds population, yet responsible for consuming only 9.7% of the global generated electricity, or the continent of Africa, which contains 10.5% of the world’s population, yet is responsible for consuming only 3.1% of the generated electricity.

The regional share of both carbon dioxide and electricity consumption over the 28 year period from the foundation of the OECD has changed considerably. In 1973, the OECD was responsible for 72.8% of total global electricity production, while China and Africa were responsible for 2.8% and 1.8% respectively. In terms of megatons of carbon dioxide emitted (Figure 1.4), the global rate of increase of emissions has been 1.5% annually since 1973, while the OECD’s contribution has only grown at a rate of 0.7% annually during that same period.

To summarise these trends in the global energy market:

1. Currently, the developed world is responsible for emitting more than 50% of the world’s anthropogenic carbon dioxide, but this percentage has been decreasing over the last 28 years.

9Joint IAEA/CNNC Seminar on 21st Century Nuclear Energy Development in Beijing China, 23rd May, 1997. CHAPTER 1. INTRODUCTION 7

Figure 1.3: International Energy Agency: Regional shares of electricity gener- ations 1973 - 2001, excluding pumped storage. (Reproduced with permission from the IEA (2003)). ** Asia excludes China.

Figure 1.4: International Energy Agency: Regions share of CO2 emissions 1973 - 2001: Calculated using IEA’s Energy Balance Tables and the Revised 1996 IPCC Guidelines. CO2 emissions are from fuel combustion only. (Re- produced with permission from the IEA (2003)). ** Asia excludes China. CHAPTER 1. INTRODUCTION 8

2. The annual increase in carbon dioxide emissions of the developing world is twice that of the developed world.

3. The global demand for electricity is increasing at a rate double that of its demand for total consumable energy.

4. There is no evidence to infer that the contribution of electricity generated from renewable resources will make a signiﬁcant impact on the global market in the short to mid-term (approximately 20 years).

1.1.3 The impact on the developing world

It is important to highlight some of the reasons that underlie the trends in the global energy market presented in the summary of Section 1.1.2. Firstly, the reason for such a large percentage of energy being consumed by the developed nations is put forward by quoting a prominent United States Senator:

“[The United States of ] America should be celebrating the strength of our [its] system and our [its] economy, which provides a standard of living that is the envy of the world. ... A fundamental element of our [United States of America’s] economic strength has been cheap, reliable energy. Without energy, the U.S. economy would collapse.” Pete Domenici10 U.S. senator for New Mexico.

While Pete Domenici may be confusing envy with resentment, he has made a valid point, that part of the economic strength of the United States of America, as with most of the OECD countries, is due to the reliable supplies of cheap electricity. A higher quality of living is synonymous with high electricity usage, and similarly, poverty is prevalent mostly in those regions of the world without suﬃcient energy sources and services.

The continuation of relatively higher increases in the global demand for electricity to that of the global energy demand can be attributed to the increase in the economic and social standards of the world. This drives a general shift to centralised energy production from an increase in domestic use of grid electricity in countries when it becomes available and the direct correlation between electricity generation and domestic economic prosperity. Over the

10Pete Domenici, (2002),Powering the future. The Bridge (National Academy of Engi- neering) 32(2):18-22. CHAPTER 1. INTRODUCTION 9 next 20 years the driving mechanism for this trend will be the developing world.

As countries like China and India continue to open their doors to western culture, it would be fair to assume that these trends will continue for decades. As it has now become apparent, the impact of global warming and climate change can be directly attributed to anthropogenic sources of greenhouse gases. The cheapest form of electricity is generated from the burning of fossil fuels producing greenhouse gasses. Societal development is directly linked to economic growth which is inevitably linked to electricity (energy) consumption.

If the developing world is going to be allowed the same levels of growth and standards of living as developed nations, then it should be allowed the economic advantages of cheap electricity. To eﬀectively address these apparent trends in the global energy market, the global community must acknowledge the increasing demand on world resources, the impact that the burning of fossil fuels has on the global environment and consider alternatives to the burning of fossil fuels to supply the ever increasing demand for consumable energy. This must all be done while still allowing the developing world the same beneﬁts that the developed world has exploited since the Industrial Revolution.

1.2 Alternatives to burning fossil fuels

The United Nations Development Programme (UNDP) has been established to address some of the issues of environmental global governance such as the broad issues of energy and development as discussed in the previous section. The UNDP states that the technical options for meeting the objectives of increasing energy services while decreasing the environmental, social and economic impact of such policies are:

1. More eﬃcient use of energy, especially at the point of end- use in buildings, electric appliances, vehicles, and production processes. 2. Increased reliance on renewable energy sources. 3. Accelerated development and deployment of new energy technologies, particularly next-generation fossil fuel technologies CHAPTER 1. INTRODUCTION 10

that release almost no harmful emissions into the atmosphere - but also nuclear technologies, if the problems associated with nuclear energy can be resolved.UNDP11

The ﬁrst of the technical options set out by the UNDP, that of implementing increased energy eﬃciency, will only slow the demand for consumable energy. Therefore the increased generation of electricity according to the UNDP must be supplied by renewable energy sources, next-generation fossil fuels and possibly nuclear technologies. In light of the limited development of next- generation fossil fuels, and the growing public opposition to nuclear power, especially after the Chernobyl disaster which has slowed its expansion in many countries and removed it as an option in others, there appears to be a large disparity between the projected global energy demands (Section 1.1.2) and the ability of renewable energy resources to provide this potential.

The UNDP goes on to point out that:

Energy scenarios suggest that a combination of these [three] approaches can satisfy the energy demands of the growing world population (expected to reach 10 billion people by mid-century) while also meeting sustainability concerns - and with lower capital investments than implied by current trends. Indeed, developing the energy technologies needed seem to present less of a challenge than mustering the political will and developing the human capacity to employ them eﬀectively. This will require changes of policies related to energy for sustainable development that go far beyond the energy sector. In fact, none of these scenarios will come about without changes in the policy environment. The new policies will have to be designed and implemented in the broader context of overall global development. UNDP12

If this political revolution does occur in the near future and cost-eﬀective renewables were to be implemented, the current options to generate electricity from renewable resources are, in order of presently installed capacity: hydroelectricity, wind generation, solar thermal, geothermal and then solar photovoltaic. Grossand et al. (2003) reviewed these key technological options and their market developments. He concluded that no one option was

11Energy for Sustainable development (2002), UNDP, p 4. 12Energy for Sustainable development (2002), UNDP, p 4. CHAPTER 1. INTRODUCTION 11 the solution to replacng the baseload13 power for the ever increasing global demand for energy. The load must be shared by combinations of all of the technolgies where they are available. Solar thermal energy is one of these developing technologies.

1.3 Problems with large scale solar power

Typical large scale solar power systems can be either: mid concentration - trough and line focus systems; modular high concentration - dish systems and small central receiver plants; and high concentration plants - power tower and large central receivers systems. The School of Physics at the University of Sydney is the home of the Solar Energy Group (SEG). In general, the group’s research interests are in large scale (modular) combined photovoltaic thermal systems. These are speciﬁcally line focus systems and central tower receivers. The SEG investigates both optical and thermal design aspects of concentrating solar power.

Regardless of which aspects of concentrating solar power are being considered, there are typically two fundamental problems. The first is finding a cost efficient method of storing solar energy. The second is the ability to get the most energy out of the flux distribution in the focal region of the optical system. Mean olar fluxes are of the order of 20 - 80 suns14 for linear Fresnel and trough systems, below 1000 suns for power tower and central receiver systems and 1000-2000 suns for dish systems.

In common with the development of all large scale power systems, the cost of experimental research is expensive and increasing. To adequately test such systems, scale models do not provide conﬁdence to investors, and large scale development systems require signiﬁcant research money. An alternative and complement to experimental research for optimising the performance of solar power systems is to use computer simulations. To do this accurately, a precise knowledge of all of the components of the system must be obtained. If the model accurately replicates the actual system, then the resulting design will greatly reduce both the cost of experimental research and increase the performance that can be achieved from such systems.

For optical simulations, there are a large number of computer resources, some freely available others commercial software, that have been created to

13Baseload refers to dependable continuous grid protential. 141 sun is one times the incident direct beam solar flux. CHAPTER 1. INTRODUCTION 12 model both individual components as well as entire systems, for solar power applications. These include Helios, Fiat Lux, delsol, wdelsol, Solver, SCT and CIRCE being the more common packages15. Generally, these resources vary in their structure, programming language, and relevance to a specific task which hinders their usability for broader applications. Some of these applications are modern, developed during the course of writing this body of work, others have been around for 20 years. All of the packages though, use assumption to simplify the large computation time required for such simulations. One of these assumptions was the spatial energy distribution of the incoming solar radiation. Obviously, if simulations of the spatial flux distribution in the focal region of a concentrating system is required, an accurate model of the terrestrial solar energy distribution has to be developed. Three components are required to create this model: the position of the sun in the sky, its spectral and spatial energy distribution and the broadening of the spatial energy distribution after its reflection off non-ideal mirrored surfaces. Combining these components completely characterises the reflected beam. More over, if all of the components of the terrestrial solar beam can be modelled accurately, then one of the major problems associated with large scale solar power systems, that of the flux distribution in the focal region, can be adequately addressed, removing this problem and facilitating the path to a renewable energy future.

1.4 Solar energy distribution

If the Sun is viewed from the surface of the earth’s moon, the radial energy distribution of the incident solar radiation would be no wider than the angular size of the sun. The colour of the Sun would represent the average of the spectrum of the radiation given out from the Sun’s surface. When standing on the Earth’s surface however, the sunlight must pass through a certain thickness of the Earth’s atmosphere. The atmosphere interferes with the spatial and spectral energy distribution of the incident solar radiation. Two scattering processes are predominant for radiation passing through the atmosphere. Rayleigh scattering occurs when radiation interacts with particles or molecules much smaller in diameter than the wavelength of the radiation. Typically, atmospheric particles such as air molecules are responsible, where the size of the particles are of the order of sub-nanometer scale. 15Examiners comment CHAPTER 1. INTRODUCTION 13

During the process of Rayleigh scattering, solar radiation undergoes absorption and immediate re-emission, creating an isotropic distribution (Fig- ure 1.5). The degree of Rayleigh scattering is inversely proportional to the fourth power of the wavelength and mainly occurs in the upper 4 to 5 km of the atmosphere. As a result, short wavelength radiation tends to be scattered much more than long-wave radiation hence forming the isotropic blue sky.

Where the incoming radiation encounters particles that are relatively large in comparison to that of the wavelength of the propagating light Mie scattering (Mie, 1908) occurs. This type of scattering takes place in the lower atmosphere, smoke and dust particles being the dominant sources. Mie scattering is predominantly small angle forward scattering (Figure 1.5) transforming some of the incident solar radiation from within the conﬁnes of the solar disc, forming the solar aureole.

Figure 1.5: Schematic diagram representing the relative angular response of diﬀerent processes occurring in atmospheric scattering. (Diagram produced by Georgia State University).

The combination of these two scattering processes creates a specific spatial and spectral energy distribution that is dependent on the local atmospheric conditions. Characteristic solar profiles can be used to determine the performance of specific concentrating systems. To create an optimised optical design, or to evaluate an existing optical design for a specific site, a reasonable statistical database of the sunshapes must be created. This alone proves not only time consuming but expensive and it gives no surety that the design will prove optimal in other locations. What is required is a sunshape model that replicates real systems and is not site dependent. This is one of the major themes addressed in this thesis. CHAPTER 1. INTRODUCTION 14

1.5 Circumsolar ratio

One method of classifying the spatial energy distribution of the sun is by using a property of the terrestrial distribution termed the circumsolar ratio (CSR). The CSR (χ) is defined as the radiant flux contained within the circumsolar region of the sky Φcs , divided by the incident radiant flux from the direct beam and aureole Φi ,

Φ χ = cs . (1.1) Φi

In this thesis we define the direct beam radiation as the radiant flux collected from within the extent of the solar disc only. This differs from some uses of this term in the past. Some workers have not drawn the distinction between the radiant flux in the normal incident beam, measured using a pyrheliometer or active cavity radiometer (ACR), and that contained within the solar disc alone. If we assume the radial solar spatial energy distribtion φ(θ) has radial symmetry, which is a reasonable assumption for the greater proportion of solar distributions (Schubnell, 1992a; Neumann et al., 2002), the radiant flux from each region of the solar image can be calculated using,

θ∆ Φi,cs = 2π φ(θ)sin(θ)dθ , (1.2) !0,θδ θ∆ 2π φ(θ)θdθ , (1.3) ≈ !0,θδ

for small θ, where θ is the radial angular displacement and θδ & θ∆ are the radial angular bounds of the direct beam and circumsolar regions respectively.

There are a large range of pyrheliometers and ACRs on the market for measuring the normal incident beam. The acceptance angle of these devices ranges between 5◦ and 7◦ depending on the brand and model, or alternatively expressed as a radial displacement (half angle) of between 2.5◦ (43.6 milliradians (mrad)) and 3.5◦ (61.1 mrad). For this work we have deﬁned the angular extent of the aureole θ∆ to have a radial displacement of 43.6 mrad. There is evidence to suggest that the circumsolar region extends further than this. A large proportion of the experimental work used for comparison in this work have been taken by the Lawrence Berkley Laboratories (LBL). The LBL CHAPTER 1. INTRODUCTION 15

used an ACR with a ﬁeld of view of 5◦ (half angle of 2.6◦ or 43.6 mrad), so the limit was set so the calculations were consistent with their experimental results. The inner limit of the circumsolar region is the edge of the solar disc and varies due to the Earth’s position in its elliptical orbit about the sun. Neu- mann et al. (2002) pointed out that standardising the edge of the solar disc (in Equations 1.1-1.3) for a full year does not necessarily represent the actual CSR. However, to obtain a generic model of the sun, it is essential that we accept one value for the edge of the solar disc, and allow a small systematic error to be introduced into the calculations. The limit accepted for θδ is a radial displacement of 4.65 mrad or 0.266◦, as calculated by Puliaev et al. (2000) at the Observatorio Nacional de Brazil.

1.6 Overview of this thesis

To summarise this chapter, there exists a disparity between the requirement for non-polluting sources of electricity to take the load of the ever increasing demand for consumable energy, and both the willingness of the civic community to make the sacrifices required to implement such systems and the development of the technology to provide this renewable energy. A detailed understanding of the flux distribution in the focal plane of the optical system is required for the development of large scale solar power. Experimental research is expensive, and even with adequate funding, gaining an understanding of the inter-dependent components of high solar fluxes experimentally proves a challenging task. However, computer simulations provide the infrastructure to not only model but optimise the optical characteristics of any concentrating system. To obtain this though, an accurate model of all of the components of an optical system need to be obtained. A key component of this is the spatial energy distribution of the terrestrial solar beam. Following this introduction, an investigation of the degree to which circumsolar radiation contributes to the collection efficiency of imaging concentrators is presented and how it should be taken into consideration in the design phase of concentrating systems is described. By tracing averaged solar data taken from the Reduced Data Base gathered by the Lawrence Berkley Laboratories the influence of the amount of circumsolar radiation on the acceptance angle of an arbitary absorber was calculated. CHAPTER 1. INTRODUCTION 16

Chapter 3 goes on to present an overview of all of the resources that have been gathered with regards to the terrestrial solar beam; primarily the Lawrence Berkley Laboratories Reduced Data Base and the German Aerospace Cen- ter’s solar proﬁles. In doing so the energy distribution of the sun is quantiﬁed and an algorithm developed that shows a strong correlation to observed data. The algorithm is invariant to a change in location being dependent on only one variable, the circumsolar ratio. The sunshapes generated using the algorithm demonstrate a much greater level of agreement with observed data than the previous models.

Using this algorithm that describes the spatial energy distribution of the sun, Chapter 4 investigates the effect of reflecting a solar beam off a mirrored surface. A virtual solar cone is defined, whose principle axis is aligned with the solar vector16, which has a radial angular displacement containing a proportion of the terrestrial beam. By simulating the reflection of these solar cones off a set of non-ideal mirrored surfaces, accounting for non-specular reflection and mirror shape errors, the combined effect of sunshape and mirror properties on the solar image is obtained. Trends are presented that show the dependence of the effective size of the solar image on the accuracy of a mirrored surface for different sunshapes. This model can then be applied during the design phase of a solar concentrating project to accurately determine the size of the absorber plane.

Combining all of the information gathered about the direct solar beam, Chap- ter 5 presents two libraries of functions for the purpose of creating an abstract solar concentrating system. The first, solar.h, contains all of the components of the terrestrial solar image including a solar vector calculator, a spectral and spatial energy distribution simulator and the broadening of the solar image caused from reflection off a non-ideal mirrored surface. The second, vecmath, builds the infrastructure of a Euclidean space together with all the the rules of vector mechanics, along with the mechanism to facilitate a ray-tracing algorithm.

With the tools of the solar algorithm and vector class established, Chapter 6 simulates the performance of thin films in optical systems. Considering the performance of silicon cells where the incident solar radiation is normal to the surface and one that simulates a silicon solar cell under a single day, thin film layers are optimised so that the amount of power is maximised. It is illustrated that normal insolation is sufficient to simulate real conditions. The 16A vector representing the position of the sun in the sky from the local position. CHAPTER 1. INTRODUCTION 17 algorithm to simulate thin films is integrated into the solar class, completing the optical library.

Chapter 7 simulates an abstract linear Fresnel concentrator system for the purpose of demonstration and to extract optical characteristics of this particular design. A direct comparison is made between different field design to establish trade-offs in performance. Also complex absorber designs are created to investigate the absorption characteristics of such systems.

Finally, the ﬂux distribution is addressed in the case of a large scale solar concentrating system, illustrating methods to create uniform ﬂux distributions and identifying detailed analysis of the optical characteristics of the absorber plane. Chapter 2

The eﬀect of circumsolar radiation on a solar concentrating system

As introduced in Section 1.4, the terrestrial spatial solar energy distribution (sunshape) is an important characteristic in determining the resultant solar ﬂux distribution in the imaging plane of concentrating systems. Empirical models for the energy distribution of the sun, as seen after atmospheric scattering, show a strong correlation on an annual or month-by-month basis to observed data. When applied to cases where the requirement is for a real-time solar energy distribution, such as in the optimisation of the ﬂux distributions in imaging concentrators, these models prove inappropriate.

In this chapter, results are presented that illustrate trends in observed solar profiles that are invariant to changes in location. This implies that there exists a more accurate model of the terrestrial spatial energy distribution of the sun than is currently available. Using this new model, a more complete understanding of the effect that variations in the spatial energy distribution of the sun has on the spatial solar flux distribution in the absorber plane of a linear Fresnel concentrating collector. The results justify the need for a more accurate model of the sun, where the small variations in the spatial solar energy distributions are considered.

The results from this chapter were published in Solar Energy by Buie and Monger (2003), which was selected from the papers presented at the Interna- tional Solar Energy Society’s Solar World Congress held in Adelaide in 2001 (Buie and Monger, 2001).

18 CHAPTER 2. THE EFFECT OF CIRCUMSOLAR RADIATION 19

2.1 Introduction

In the mid to late 1970s the Lawrence berkeley Laboratories (LBL) recorded and analysed almost 200 000 terrestrial solar proﬁles (called the reduced data base RDB) using purpose-built solar telescopes. Eleven individual sites were chosen across the United States to collect this data, where each site had individual atmospheric characteristics (i.e. low humidity, high altitude, proximity to heavy industry etc.), and a broad cross-section of observed data were obtained. The data can be accessed via a public ftp site referenced by Noring et al. (1991). This data set is discussed totally in Chapter 3.

What was clear from the research conducted by the LBL’s was that the amount and character of the sunshape proﬁles varied widely with geographic location, climate, season, time of day and the observing wavelength. Also, the amount of energy contained in the solar aureole (circumsolar region) varied across locations and the energy could have an eﬀect on the performance of solar concentrating systems.

This chapter investigates the variations in the instantaneous flux distribution created by small variations in the circumsolar energy. Firstly, apparent trends in terrestrial solar energy distributions are identified. Then by simulating a solar concentrating system and using average sunshapes acquired by the LBL, the influence that the amount of circumsolar radiation has on the acceptance angle of the absorber in the imaging plane is illustrated.

2.2 Apparent trends in solar proﬁles

Two distinct regions exist when examining a single sunshape profile (Fig- ure 2.1): the distribution within the solar disc, and that of the circumsolar region. They are distinct because different processes create them. The solar disc profile is created from the processes of solar limb darkening, atmospheric extinction and atmospheric scattering, whereas the circumsolar regions is simply created from the small angle scattering of the incoming beam off large particles in the troposphere (discussed further in Chapter 3.

Immediately evident from the proﬁles in Figure 2.1, is the linear relationship between the intensity of the circumsolar region of the sky to that of the radial distribution in log-log space (for most proﬁles). This relationship can CHAPTER 2. THE EFFECT OF CIRCUMSOLAR RADIATION 20

Figure 2.1: An example of 20 filtered solar profiles collected by the LBL plotted in log-log space. The two regions are shown split by an angular displacement of 4.65 mrad. be defined by a power law where relative solar intensity φ(θ) can be expressed by the angular displacement to the power of the gradient γ,

φ(θ) = eκθγ , (2.1) where κ is a scaling factor determined by the intercept of the curve in log-log space (the factor of e is included to simplify the intercept factor, as the curve is only linear in log-log space).

As previously mentioned in Section 1.5, an effective method to classify individual sunshapes is by their circumsolar ratio. For each of the 11 sites where solar profiles were acquired, the gradient γ and intercept κ are determined for each of the sunshape curves. This allows not only the characteristics of an individual site to be identified but also makes it possible to examine global characteristics illustrating commonality between locations.

The initial results of the linear relationship of the circumsolar region determined by each proﬁles gradient γ and intercept κ are plotted against their CHAPTER 2. THE EFFECT OF CIRCUMSOLAR RADIATION 21

Figure 2.2: Plot illustrating the correlation between the circumsolar ratio and the gradient γ of the solar proﬁles of the Reduced Data Base in the circumsolar region for all 11 sites across the United States.

Figure 2.3: Plot illustrating the correlation between the circumsolar ratio and the intercept κ of the solar profile of the Reduced Data Base in the circumsolar region for all 11 sites across the United States. CHAPTER 2. THE EFFECT OF CIRCUMSOLAR RADIATION 22 corresponding CSR (Figures 2.2 & 2.3). Clearly, a strong correlation for all of the 11 sites can be seen across all of the CSRs. This result implies that, although various atmospheric conditions give rise to different amounts of scattering, the profiles themselves, when grouped according to their CSR, on average possess similar characteristics. This suggests that a terrestrial solar model based on the CSR can be developed.

Previously, when conducting solar simulations, the sun shape was simulated using a Gaussian or a pillbox distribution. Alternatively, real profiles could be acquired using a Charged Coupled Device (CCD) camera, and the resulting distribution digitised. Using the premise that solar profiles with similar CSR exhibit similar solar profiles, the sunshapes within the RDB were averaged according to their CSR. This allows sunshapes with various CSRs to be recreated and a direct comparison made to the performance of a solar collector for each averaged CSR profile. This allows a direct comparison of the performance of solar collectors on different solar distributions.

It was shown by Neumann et al. (2002) that individual locations have trends in the solar insolation with regards to its CSR. For example, one site may experience 80% of its solar irradiation with a CSR below 0.08, while another may experience the same percentage of its radiation with a CSR below 0.15. Therefore, based on a particular location mean or average CSR, a typical sunshape created from this averaged LBL data, would be an accurate tool to model the performance of a plant built there.

2.3 The eﬀect of variations in sunshapes

Given that an individual mirror design and imaging system gives a characteristic flux distribution in the imaging plane, we investigate the effect of various sunshapes on simulations of a line-focus system similar to those discussed by Mills and Morrison (2000). Here, 20 parabolic linear Fresnel mirrors were placed about a central linear thermal absorber line that is horizontal and parallel to the mirror rows. Each mirror has an aperture of 1.6 m, and only a section of the absorber was considered that was completely illuminated from the entire reflector field. A more general analysis of linear Fresnel systems is given in Chapter 7.

Using a three dimensional representation of various averaged sunshape profiles generated from the LBL, and thus representing real solar images, the CHAPTER 2. THE EFFECT OF CIRCUMSOLAR RADIATION 23 spatial energy distribution was traced through the optical system of the Fres- nel mirrored array. A high-resolution flux image (developed in Chapter 5) was then generated in the absorber plane to investigate a design’s optical efficiency based on a given CSR (Figure 2.4).

Figure 2.4: Resultant ﬂux in the imaging plane of a line-focus concentrator throughout an entire day of operation.

An absorber’s angular acceptance can be defined where the size of the absorber corresponds to a viewing angle of the solar distribution. For example, a 2◦ viewing angle relates to an absorber width where rays 1◦ either side of the solar vector would strike the imaging plane. Further, a percentage can be defined such that an absorber acceptance angle of 2◦ could contain some percentage of the total incident radiation. Because we defined and normalised all of the solar images with a given CSR to have their energy contained within 2.5◦ about the solar vector, any absorber with an acceptance angle of

5◦ would contain 100% of the terrestrial normal incident radiation.

Figure 2.5 illustrates the relationship between the optical eﬃciency (percentage of total energy) and the size of the absorber (acceptance angle) in our speciﬁc line-focus Fresnel concentrator for various CSRs. As the acceptance CHAPTER 2. THE EFFECT OF CIRCUMSOLAR RADIATION 24

angle of the absorber increase to 5◦ all of the profiles approach 100% of the incoming radiation. As previously stated different sites exhibit different solar profiles. Sunshapes with CSRs between 0.02 and 0.2 were modelled. Firstly considering the CSR 0.02 profile, 98% of the energy is contained within an absorber acceptance angle of approximately 0.5◦, as expected, as this defines the limit of the solar disc. As the CSR of the sunshapes increase, the size of the absorber must also increase to capture the same proportion of incident energy.

Figure 2.5: The overall optical eﬃciency of an ideal line-focus imaging concentrator plotted against the acceptance angle of the absorber for various circumsolar ratios (zero tracking and surface errors).

If a site that was acceptable for a solar power station had 80% of its sunshapes with a CSR below 0.15, then the absorber would need to have an acceptance angle of 2 degrees to be able to accept 98% of the solar insolation (not including tracking and surface errors). The corollary to this is it would be unnecessary to increase the acceptance angle of the absorber greater than this amount (reducing thermal and radiative losses) if the plant was to be built in this environment. CHAPTER 2. THE EFFECT OF CIRCUMSOLAR RADIATION 25

2.4 Conclusion

Although a physical model of the results has not been provided, it is clear that the profile of the circumsolar region of the sun’s radial distribution is linear in log-log space. The actual distributions when averaged over a large database show invariance to location or atmospheric conditions when pro- filed against their corresponding circumsolar ratios. Also, it is clear that the amount of circumsolar radiation must be taken into account when determining the ultimate flux distribution in the imaging plane of a generic concentrator. Using this assumption, it is then possible to identify the conditions at a particular site with respect to the average circumsolar ratio, and optimise the acceptance angle of the absorber, reducing radiative heat losses and improving performance.

In the following chapter we look more closely at the structure of the LBL RDB and other sunshape data, and using these assumptions, create a model for this terrestrial spatial energy distribution based on the CSR, that has a high correlation to observed data. Chapter 3

The terrestrial spatial solar energy distribution

Individual sunshapes are created by the small angle forward scattering of sunlight oﬀ aerosols in the troposphere, having the eﬀect of transferring some part of the solar energy from within the solar disc to the circumsolar aureole. As shown in the previous chapter, the energy distribution in the focal plane of an optical system is dependent on this radial distribution (sunshape) of the incident solar energy and clear trends exist in the these sunshapes when grouped according to each distribution’s circumsolar ratio.

Using the Lawrence Berkeley Laboratory’s vast circumsolar database, collected from 11 sites across the United States in the late 1970s and early 1980s, and the more recent sunshapes from the German Aerospace Centre (DLR) correlating three European sites, this chapter describes a sunshape model that is independent of geographic location. Further, this chapter illustrates that, on average, the circumsolar ratio deﬁnes the spatial energy distribution across the solar disc and aureole, and presents an algorithm that can be used to model these distributions for the majority of atmospheric conditions.

The results and the majority of this chapter were published in Solar Energy (Buie et al., 2003c).

26 CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 27

3.1 Literature review

As light propagates through the Earth’s atmosphere, the solar beam is broadened due to its interaction with atmospheric particulates. Where the radius of the particles are large (between 0.1 µm and 1 µm, Junge, 1963) in comparison to the wavelength of the propagating light, small angle forward scattering occurs, forming the solar aureole (Mie, 1908). The amount of energy in this circumsolar region (solar aureole) is important for two reasons. Firstly, depending on the acceptance angle of a solar concentrating system, overestimation of the power output may occur if all of the power is assumed to fall within the solar disc (Noring et al., 1991). Secondly, the radial energy profile (sunshape) can play a non-negligible role in determining the overall flux distribution in the focal plane of concentrating systems inferring that as the solar profile changes, so does the flux distribution. Large particles in the troposphere are generally created by two methods: firstly, direct detachment from the Earth’s surface either by wind or the surface activity of water (Bagnold, 1941; Monger, 1996), and secondly, anthropogenic sources of aerosol particles, such as those from large industry. These mechanisms create both soluble and insoluble particles. The soluble particles generally show an increase in their size with the relative humidity and ambient temperature. Insoluble particles tend not to be affected by this phenomenon, except in supersaturated conditions; their size being determined almost entirely by the process that created them (Monger, 1996). The vertical population distribution of large particles, while exhibiting some irregularities, shows an exponential decrease with altitude (Penndorf, 1954). This, combined with the fact that the shape and irregularities of the large particles vary, illustrates that the small angle scattering of the incident solar beam is heavily dependent on the local atmospheric conditions. The Lawrence Berkeley Laboratories (LBL), in the mid to late 1970s and early 1980s, compiled a large number of sunshapes in what is termed the reduced database (RDB). Eleven sites were chosen across the United States that would give a range of atmospheric characteristics from such factors as altitude, proximity to sources of large particulates, humidity and climates of coastal and desert regions (Table 3.1). These data and the methodology by which they were collected are described in more than 16 publications by Grether, Evans, Hunt and Wahlig (see reference list in Noring et al., 1991). The team from LBL documented a broad range of distributions, the spectral nature of scattering, trends exhibited in the 11 sites and the effects these have on a variety of solar concentrating systems. CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 28

Site Latitude Longitude Elevation # location (degrees) (degrees) (feet)

1 Albuquerque, NM 34◦ 57’ 44 ” 106◦ 30’ 32” 5589 STTF 2 Albuquerque, NM 35◦ 03’ 106◦ 40’ 5600 TETF 3 Argonne, IL 41◦ 43’ 87◦ 58’ 725 4 Atlanta, GA 33◦ 46’ 84◦ 24’ 990 5 Barstow, CA 34◦ 53’ 117◦ 00’ 2180 6 Boardman, OR 45◦ 42’ 32” 119◦ 52’ 54” 620 7 China Lake, CA 35◦ 39’ 117◦ 40’ 2700 8 Colstrip, MT 45◦ 48’ 28” 106◦ 31’ 09” 3060 9 Fort Hood, TX 31◦ 04’ 97◦ 24’ 800 Bunker 10 Fort Hood, TX 31◦ 03’ 97◦ 31’ 1030 TES 11 Edwards AFB, CA 34◦ 59’ 30” 117◦ 52’ 2300 Table 3.1: LBL reduced database summary of locations

Using the RDB, Rabl and Bendt (1982) created a standard solar scan which was a single sunshape proﬁle created by the averaging of an unstated number of LBL sunshapes. Their model was created to help analyse the average performance of solar concentrating systems and provided a solar proﬁle that better represented observed data. However, the model did not accommodate the short-term variations that are common in measured sunshapes.

Using a charged coupled device (CCD) camera, Schubnell et al. (1991); Schubnell (1992a,b); Steinfeld and Schubnell (1993); Neumann and Schubnell (1992), quantified the degree to which a variation in the sunshape influenced the overall optical efficiency of concentrators. By simulating observed solar data and a pillbox sunshape (where there is a constant intensity across the extent of the solar disc and zero elsewhere), a comparison was made between the performances of various concentrating devices using a ray-tracing simulation. The results illustrated variations of up to 20% in the optical performance from a change in the sunshape alone.

Neumann et al. (Neumann and Schubnell, 1992; Neumann and Groer, 1996; Neumann et al., 1998; Neumann and Witzke, 1999), continued this work, examining sunshape data collected at the DLR site in Cologne (Germany). Using smaller data sets of solar distributions collected using a similar CCD solar telescope to Schubnell, they illustrated the dependence of the solar distribution on the image created in the focal plane of various concentrators. They presented reports on the extent to which the relative amount of circumsolar radiation aﬀected a variety of optical systems, from both theoretical estimates and experimental observations. Expanding the studies by CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 29 the LBL, Neumann et al. (1998) illustrated some trends that were apparent in their data between global irradiance and the measured circumsolar ratio (CSR deﬁned in Section 1.5).

What was clear from this research was that the magnitude and distribution of the circumsolar radiation vary due to all influences mentioned, but the solar profile is most dependent on atmospheric conditions associated with geographic location, climate, season, time of day, and wavelength of the incident radiation (Noring et al., 1991). As previously stated, the amount of energy contained in the solar aureole affects the performance of solar energy converters. The exact degree to which optical systems are dependent on particular atmospheric condition is not accurately known, although it is clear that it is not trivial. Currently, to create an optimised optical design for a specific site, a reasonable statistical database of measured sunshapes must be used. This is not only time-consuming but also expensive to create, and there is little confidence that the resulting design will prove optimal in other locations. What is required is a sunshape model that adequately replicates the real solar distribution but is not location dependent.

Neumann et al. (2002) published a paper presenting such a study. Using a CCD camera they collected 2300 sunshapes from three locations: the German Aerospace Centre Cologne site, the Plataforma Solar de Almeria (Spain) and the CNRS Solar Furnace in Odeillo (France). Averaging these data for a speciﬁc CSR range, they provide six sunshape proﬁles that are typical of standard solar distributions.

Independently, by examining the LBL RDB, Buie and Monger (2001) published the preliminary work relating to this chapter, inferring the existence of a deﬁnitive sunshape. This present chapter extends the analysis, using both the RDB and the sunshapes created by Neumann et al. (2002). In doing so some parameters that are invariant under a geographic translation are illustrated, including perhaps altitude, allowing optical solar concentrating systems to be optimised for a broader range of conditions.

3.2 Sunshape data

Details of the two independent sunshape data sets used to create and test a simulated terrestrial solar distribution are explained within this section. CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 30

3.2.1 LBL reduced data base

The LBL collected approximately ten-station years of data from eleven different sites across the United States with site names: Albuquerque STTF & TETF, Argonne, Atlanta, Barstow, Boardman, China Lake, Colstrip, Ed- wards Air Force Base (AFB) and Fort Hood Bunker & TES (as given in Table 3.1). Each site was chosen to give a range of atmospheric characteristics. Four identical telescopes were used, purpose-built for measuring a radial profile of the sun to a maximum angular displacement of 56 mrad. The telescopes took a total of ten minutes to collect a full set of ten different profiles and ran autonomously for up to one week. Of these data, about one tenth was cleaned and compressed into a standard format (the RDB), and is available from a public ftp site (Noring et al., 1991), consisting of approximately 200 000 solar profiles.

Figure 3.1: Don Grether (left) and David Gumz stand with an LBL circumsolar telescope used to track the sun and measure the distribution of solar light scattered by the earth’s atmosphere. Tracking, control and data collection were done electronically. (Image courtesy of the LBL website). CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 31

The instruments used to collect the solar profiles were pinhole telescopes where the pinhole subtended an angular resolution of one twentieth of the solar disk, directing light onto a spectrally neutral pyroelectric (thermal) detector, calibrated for the range of solar wavelengths (Grether et al., 1975; Hunt et al., 1980; Noring et al., 1991). The intensity across the sun was digitised every 1.5 second of arc within 0.5 degrees of the solar centre and every 4.5 seconds of arc outside this region, giving 56 data points for each 2 1 pass of the camera, expressed in units of Wm− sr− . A set of measurements consisted of one scan at each of 10 filter positions including eight optical filters, as well as one open (or clear) and one opaque position. Normal incident flux from the active cavity radiometer (ACR) and two pyranometer readings, one tracking the sun and the other horizontal, were also recorded. Of these measurements only the unfiltered (clear) sunshape data, the ACR reading, both the pyranometer readings, a CSR whose origin was not described and data, camera, location and status flags were included in the RDB. To illustrate the data from the RDB, the first 20 scans from Albuquerque, New Mexico were plotted on a log-linear scale (Figure 3.2). As stated, each scan took one minutes to acquire and these unfiltered sunshapes are spaced 10 minutes apart, therefore Figure 3.2 contains 3 hours of measurements on one particular day. The initial data from Albuquerque represent typical profiles gathered by the LBL’s circumsolar telescope. The majority of the profiles illustrate trends typical of solar profiles, where the central part of the sun (zero angular displacement) contains the brightest portion of the solar image tending to lesser values as the angular displacement increases. Some of the profiles do not however posses these clear trends. It is assumed that those profiles have been interrupted with cloud moving across the solar image during the acquisition of the sunshape, or the solar telescope no longer was functioning within the tolerances designed to acquire accurate data. Clearly the data that doesn’t represent typical solar profiles needs to be removed before performing data analysis. The LBL circumsolar telescopes had an estimated uncertainty of 0.5% in the measured intensity readings. Using Equation 1.3, the LBL sunshape profiles were used to calculate the amount of energy contained within the field of view of their ACR. Figure 3.3 illustrates that the data from the line profiles are consistent with the ACR. The error associated with the relative displacement of the camera from the centre of the sun and each of the line profile positions were not supplied by Noring et al. (1991). CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 32

Figure 3.2: An example of raw data within the reduced data base compiled by the Lawrence Berkeley Laboratories

Figure 3.3: Correlation between the ACR reading of the direct beam insolation to that calculated from the sum of the solar proﬁles for the RDB. CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 33

The string of status flags contained in each of the data profiles indicated the quality of the RDB. A full list of these can be found in Noring et al. (1991). Attempts were made to ensure the RDB only contained good data, but a considerable number of the solar profiles still have crucial flags indicated as unknown. In addition, there is a requirement that the solar radiation comes from clear sky conditions only. We define this as solar insolation that is unimpeded by any atmosphere opacity over the entire direct beam and circumsolar regions. To ensure that both the poor data were removed and the sunshape data were from clear skies only, further processing was required. The stipulations on the data were:

• For a clear sky the data must fulﬁl,

d φ(θ) < 0 , (3.1) dθ

which implies that there can be no relative increase in the intensity of the solar profiles as the angular displacement increases. This requirement still leaves a statistically significant number of solar profiles within the RDB.

• Of the twenty-nine status flags indicating data quality within the RDB, only three were used to exclude data. These were: ‘Aperture in wrong position’, ‘Rain flap closed’ and ‘Sun not centred in scan’ corresponding to flags 3, 4 & 19. It was our belief that these three indicators, if triggered, would lead to an inadequate data quality and therefore not included in the data analysis.

Of the original 200 000 solar profiles that made up the RDB, 29 411 or just under 15% fitted the above criteria. Henceforth, when referring to the RDB we will only be referring to these solar profiles. The relative number of profiles in each of the 11 locations is represented in Figure 3.4.

The solar scans of the RDB were grouped according to both the location where the solar scans were acquired and their CSR. This allows the characteristics of individual sites and global characteristics illustrating commonality between locations to be examined. The CSR included in the RDB was not consistent with the CSRs calculated from the scan data and Equations 1.1- 1.3. As greater conﬁdence exists with the data in the solar scans, all the CSR data presented here are results from our calculations. The number of proﬁles in each of the CSR bins (grouped according to the integer value of the CSR CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 34

Figure 3.4: The relative number of ﬁltered sunshapes according their location.

Figure 3.5: The number of profiles represented in each of the CSR bins for the filtered RDB. CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 35 expressed as a percentage) for all of the 11 sites is given in Figure 3.5. As the number of profiles with CSRs greater than 0.8 are small or zero, these scans were also excluded. Finally, each solar profile was normalised against its central intensity reading (Figure 3.6).

Figure 3.6: Averaged profiles (filtered) of the RDB grouped according to their corresponding CSR. The curves represent profiles ranging from a CSR of 0.01 to 0.8 in steps of 0.01, leaving 80 individual averaged sunshape profiles.

3.2.2 DLR sunshape measurements

The DLR have designed and created a digital sunshape camera using an optical telescope together with a 12-bit digital resolution CCD camera (Fig- ure 3.7). Using this camera, 2300 solar profiles were acquired from three sites across Europe: the DLR Cologne site (Germany), the PSA (Spain) and the CNRS Solar Furnace in Odeillo (France). These profiles were grouped and averaged according to each sunshape’s CSR using a methodology similar to the one we applied to the RDB, however larger ranges of CSRs were accepted for each sunshape bin. A total of six profiles were published, comprising the averaged data of approximately 10% of the data pool (Neumann et al., 2002, recreated in Figure 3.8). CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 36

Figure 3.7: The system and equipment used by Neumann et al. (2002) within the DLR for collecting solar proﬁles.

Figure 3.8: Averaged DLR sunshape proﬁles published by Neumann et al. (2002) . CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 37

The optical system of the DLR’s telescope includes a band pass, neutral density filter. This is used in conjunction with the CCD camera to improve the spectral response of the CCD’s silicon wafer. For a detailed analysis of the spectral response of the optical system employed see Kaluza and Neumann (1998). The resulting uncertainty in the brightness measurements for each calibrated pixel, is quoted to be significantly below 10%, depending on the relative air mass and the assumption that the sunshape is spectrally neutral. This is different to the (practically) spectrally neutral telescope used by the LBL.

The solid angle subtended by each element in the CCD camera used by the DLR telescope is 0.065 millisteradians (msr). The equivalent ﬁgure for the LBL’s telescopes was 0.44 msr, which demonstrates the lower spatial resolution of the LBL data. This improvement by the DLR provides both a greater number of data points to be recorded across the transition between the solar disc and circumsolar region, and higher quality data at each imaging point. Also, the time required for the DLR to acquire an image with the CCD camera was virtually instantaneous, compared to scans from LBL lasting one minute. The shorter time span provides a higher probability of acquiring self- consistent proﬁles.

3.3 Creating a sunshape

In considering the recorded sunshapes, it is necessary to break the sunshape proﬁles up into two groups: the solar disc and the circumsolar region (Fig- ure 2.1). This is done to allow the proﬁles to be simulated by separate algorithms. These algorithms are discussed below.

3.3.1 Intensity proﬁle within the solar disc

Limb darkening and atmospheric attenuation are responsible for the energy distribution within the solar disk. Prior to 1982, this was modelled by what was called the Kuiper distribution (Kuiper, 1953), created from solar observations made prior to 1953. After the publication of the RDB it was argued that the Standard Solar Scan of Rabl and Bendt (1982) should be used as it was better able to describe the observed solar distributions.

Recently however, after the publication of the DLR sunshape models, Neu- mann et al. (2002) argued that the solar disc measurements conducted by CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 38

Figure 3.9: Solar limb data taken from both the RDB and the DLR. The profiles of the RDB are considerable lower than those of the DLR. the LBL do not adequately represent the more recently observed data, particularly near the solar limb, and should be replaced by the solar disc profiles gathered by the DLR (Figure 3.9). Neumann et al. (2002) stated that the squarer intensity of the profile in the DLR was primarily due to an averaging error, created by the broader acceptance angle of each of the data points within the LBL RDB. There is conjecture that the squarer shape of the DLR sunshapes (compared to those of the LBL) about the solar limb could be due to the spectral nature of both solar limb darkening and atmospheric scattering, as well as the smaller field of view of the DLR telescope. At longer wavelengths, the shoulder of the solar disc will be sharper than at shorter wavelengths (Grether et al., 1975; Cox, 1997). The CCD camera used by the DLR had a peak sensitivity at longer wavelengths (between 700 nm and 900 nm see Section 3.2.2) which could cause this feature. To quantify the difference between the solar disc profiles obtained from the two collection techniques, the amount of energy contained within the solar disc for the DLR data and the RDB were compared. Taking each sunshape to be the filtered normalised profiles with a central intensity reading of one CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 39 and using Equation 1.3, where the upper bound of the integral is 4.65 mrad the energy within the boundary was calculated. The data representing the energy contained within the solar disk was normalised against the largest value (DLR 5% profile) producing Figure 3.10. Both data sets illustrate a downward trend. The average amount of energy contained in the LBL data (line) is on average about 80% of the DLR data (plus signs) a result of the more rapid fall off of the LBL profiles. The spread in the amount of energy contained within the solar disc of the LBL data for sunshapes over all the CSR ranges is very large, varying by almost a factor of two.

Figure 3.10: The amount of energy contained within the solar disc for both the LBL & DLR normalised data.(NB only 5% of the RDB is displayed).

It is our belief that the effects of the atmospheric scattering, variations in the spectral component of limb darkening and a broader field of view of the measuring telescope could not account for such large variations in the measured energy by the LBL circumsolar telescopes when compared to the DLR measurement system. Considering the scatter in the amount of energy contained in the solar disc regions of the LBL RDB, there exists no mechanism, either in the formation of limb darkening or due to atmospheric attenuation, for there to be a considerable difference in the amount of energy contained within the solar disc for normalised clear sky sunshapes. For these reasons CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 40 we propose that the data collection process within the solar disc alone, for the LBL RDB is inconsistent and unreliable. We agree that the solar disc profiles of Neumann et al. (2002) more adequately represent the terrestrial solar disc profile. Examining the DLR data further, the variation in the amount of energy contained within the normalised solar profiles is 4% for the range of CSRs from 5% to 40%. Due to this small change in energy, we propose that one sunshape profile is sufficient to effectively model the energy distribution of the solar disc over all atmospheric conditions. Assuming that this 4% variation is more relevant to profiles with a smaller CSR than a larger CSR, an empirical fit to the DLR’s CSR 5% solar disc profile is generated and used to represent the terrestrial solar disc profile,

cos(0.326 θ) φ(θ) = , for θ 0 θ 4.65 mrad . (3.2) cos(0.308 θ) { ∈ $ | ≤ ≤ }

The fit of Equation 3.2 to the 5% DLR profile is shown in Figure 3.9. The correlation coefficient (explained in the next section) is 0.98.

3.3.2 Intensity proﬁle in the solar aureole

In examining average proﬁles of the circumsolar region, Noring et al. (1991) pointed out that there is a linear relationship between the solar intensity and the angular distribution of the aureole in log-log space. Figures 3.6 & 3.8 illustrate this linear relationship and also that the slope and intercept of this region of circumsolar radiation vary with each proﬁle’s CSR. This linear relationship between intensity and angular displacement in log-log space can be represented as a power function,

φ(θ) = eκθγ , (3.3) where γ is the gradient of the curve in log-log space and κ is the intercept of that curve at an angular displacement of zero. To test the validity of the power function of the circumsolar region a linear regression in log-log space of the RDB is undertaken. Setting,

xi = log(θi) , (3.4)

yi = log(φ(θi)) , (3.5) CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 41 we form the variance and covariance of the data,

1 S = (x x¯)2, (3.6) xx n i − 1 " S = (x x¯)(y + y¯), (3.7) xy n i − i 1 " S = (y y¯)2, (3.8) yy n i − " where,

a a¯ = i , (3.9) n #∆ = , (3.10) " "i=δ

and for brevity δ = log(θδ) and ∆ = log(θ∆). Using these relationships we can create the linear regression coeﬃcients,

S γ = xy , (3.11) Sxx y¯ γx¯ κ = − , (3.12) n and the square of the correlation coeﬃcient,

S2 r2 = xy . (3.13) SxxSyy

Conducting this statistical analysis on the RDB only, Figure 3.11 shows the average correlation coefficient for each of the profiles after being grouped according to their location and CSR. This validates Equation 3.3 as an appropriate representation of the circumsolar region with a typical correlation coefficient of 0.995 across all sites and CSRs. CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 42

Figure 3.11: r2 correlation between the normalised solar intensity levels and the angular displacement within the circumsolar region when represented in log-log space.

The average value of γ was calculated and plotted against each profile’s CSR bin (Figure 3.12). A strong correlation can be seen between all of the 11 sites, reinforced by the small variance in the data from the mean. This implies that the gradients of the circumsolar distributions when expressed in log-log space are the same, on average, irrespective of the 11 geographical locations. Some of the sites (for example Ft-Hood-Bunk, Ft-Hood-Tes & Colstrip), illustrate larger variances from the mean than expected for certain CSR values. This is due to the small number of profiles at these locations for the given CSR bin, but is insignificant in the overall statistical pool. A curve fit to these data is also illustrated in Figure 3.12, expressed by,

γ = 2.2 ln(0.52χ)χ0.43 0.1 , (3.14) − where χ is the CSR, and the resulting solar intensity proﬁle φ(θ) (Equa- tion 3.3) is expressed in milliradians.

A similar analysis can be carried out on the intercept in log-log space of the linear ﬁt in the circumsolar region of the solar proﬁles (Figure 3.13). The CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 43

Figure 3.12: Average gradient of the linear ﬁt of the data contained in the circumsolar region γ of the normalised solar proﬁles of the RDB when represented in log-log space.

Figure 3.13: Average intercept of the linear fit of the data contained in the circumsolar region κ of the normalised solar profiles of the RDB when represented in log-log space. CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 44 same trend can be seen with a small variance, and a fit to the data can be represented by,

0.3 κ = 0.9 ln(13.5χ)χ− , (3.15) again where χ is the CSR, and the resulting solar intensity proﬁle φ(θ) (Equa- tion 3.3) is expressed in milliradians.

It is also important to note that by combining Equations 1.3 & 3.3, the relative intensity within the circumsolar region can be approximated using the two parameters γ & κ;

θ∆ 6 κ γ+1 I = (10− ). 2πe θ dθ.dt , (3.16) θδ κ! 6 2πe γ+2 γ+2 = (10− ). (θ θ ).dt , (3.17) (γ + 2) ∆ − δ

6 where the scaling factor of 10− is to allow the working units to be milliradians.

3.4 Discussion

In the RDB, the intensity of the normalised direct beam varies by a factor of about two, averaging only 80% of the DLR figure. As stated earlier, there exists no mechanism either through solar limb darkening or atmospheric attenuation for this difference to be justified. It is the belief of the authors, in agreement with Neumann et al. (2002), that this difference is primarily a result of the broader instrumental acceptance angle associated with each data point in the RDB.

The small variation in the amount of energy contained in the normalised direct beam from the DLR sunshapes may be due to a combination of the spectral response of the DLR’s CCD camera and the narrow spectral band filter used. That is, the camera may not discern the greater scattering of the longer wavelengths due to the rounding of the solar disc profile. However, the variation in the total amount of energy represented by the profile will still remain relatively constant, and although the spectral response is important in CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 45 some respects, it is irrelevant to the ultimate flux distribution, which includes the sun and aureole for a good portion of applications. A single solar disc profile is sufficient then, to describe the sunshape of the solar disc at most locations and under most atmospheric conditions. With this knowledge, the RDB can be rescaled using the profiles of the solar disc provided by the DLR, improving the accuracy of the RDB. When considering the amount of energy contained in the circumsolar region, the corrected LBL data exhibits a strong correlation with the DLR data, for a given CSR. The reason this occurs is two-fold. Firstly, the direct beam is critical in determining the CSR. The measured flux for each step across the solar profile is an average of the intensity within the acceptance angle of each measurement point. Within the solar disk, and particularly at the solar limb, there is a rapid change in the relative intensity of the incident flux. The broader acceptance angle of LBL profile collection process, and the averaging that it causes, result in an underestimation of the solar energy within the solar disk, and therefore a misrepresentation of the CSR. Due to the relatively homogeneous nature of the solar aureole and the high signal-to-noise ratio of the LBL telescopes, these problems are not observed when measuring the greater proportion of the solar intensity within the circumsolar region. An overestimation of the incident radiation is experienced on the circumsolar side of the solar limb due to the aforementioned acceptance angle and explains the apparent contradiction between the behavior shown in Figure 3.3 and this argument. Recalculating the CSR using the method described allows a truer indication of the relationship between the amount of energy in the solar aureole and the CSR. Secondly, each geographic location has its own characteristic aerosol distribution. The relative size and composition of these particles determines the nature of the optical scattering. The physical process of scattering is common to all geographic locations. A location with a higher atmospheric turbidity will experience a solar energy distribution with a higher CSR, but will be in principle, based on the available sites and present data, similar to that at any other location where such a process occurs. A strong correlation within the statistical analysis confirms the validity of the power relationship between the normalised solar intensity and the angular displacement from within the solar aureole. The two variables κ and γ that define the distribution show a strong correlation among all 11 US sites. Combining this with a fit of the DLR’s 5% profile, the terrestrial solar flux distribution can be simulated using, CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 46

Figure 3.14: Correlation between the observed solar data from the DLR and their corresponding plot generated from Equation 3.18.

cos(0.326θ) where θ 0 θ 4.65 mrad , φ(θ) = cos(0.308θ) { ∈ $ | ≤ ≤ } (3.18) $ eκθγ where θ θ > 4.65 mrad , { ∈ $ | } where γ and κ are given by Equations 3.14 & 3.15 respectively.

This results in a normalised solar brightness distribution φ(θ) that depends on only one variable, the CSR, producing distributions that are generally independent of geographic location for the 11 measured US sites.

A strong correlation also exists when comparing the sunshape profiles collected by Neumann et al. (2002) for the three European sites to their corresponding profiles generated using Equation 3.18 (Figure 3.14). The large variance seen between the DLR 0% profile and the 0% profile generated using Equation 3.18 can be attributed to the fact that the DLR 0% profile has an actual CSR of 2.7% (Neumann et al., 2002), together with the fact that the recorded sunshape points measured by the DLR are pushing the bounds of the signal-to-noise ratio of the DLR’s CCD camera for low CSR values.

The gradient γ of the normalised energy distribution within the circumsolar region should tend to zero as the CSR tends to zero (seen in both Figures CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 47

Figure 3.15: The correlation between the required CSR (measured) and the generated CSR from Equation. 3.18. The dotted line illustrates where a one-to-one correlation would be.

Figure 3.16: Illustrating the competency of Equation 3.18 to its tolerance of 0.01 in the CSR. ± CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 48

3.12 & 3.14) implying that Equation 3.18 is a good representation of the ﬂux distribution for small CSR values.

Finally, by examining the correlation between a desired solar profile with a specific CSR and the corresponding profile generated from Equation 3.18 (Figures 3.15 & 3.16), the uncertainty in the generated profile’s CSR is no greater than 0.01. Three generated profiles representing sunshapes with ± CSR of 0.05 0.01 are plotted along with the DLR’s 5% profile. The close ± agreement demonstrates the competency of the algorithm for accurately simulating observed sunshape profiles.

There does exist an noticeable discrepancy between the profiles observed by Neumann et al. (2002) and those from the algorithm developed within this chapter in the range of 4.6 mrad - 4.9 mrad. The algorithm within this chapter does not allow for a smooth transition between the two distinct regions that may exist in nature. However, the majority of the smoothing seen in the DLR profiles in that region is due to an averaging that occurs when measuring flux regions of a rapid change. While the DLR camera has a greater approximation to observed data than the LBL telescope it too is vulnerable to the non-zero aperture measurement. Whether this smooth transition is present in the actual solar profile is irrelivant, as the amount of energy in that region is negligible to the total insolation for most applications.

3.5 Conclusion

When considering the terrestrial radial solar energy distribution, according to data from the 14 sites compared, there exists a strong geographic correlation when each of the profiles are represented according to their circumsolar ratios. This chapter presents an algorithm that defines the solar distribution for any circumsolar ratio that has a strong correlation with these observed data. The algorithm is invariant to changes in geographic location when representing averaged observed solar profiles, implying the existence of a generic solar model based upon the circumsolar ratio. The algorithm has value for improving the optical modelling of solar concentrating systems.

Using this model of the terrestrial solar beam, the next chapter goes on to investigate the actual size of the solar image for a range of CSRs. The broadening of the solar image from the reflection off a non-ideal mirrored surface is also investigated. The design limits for the surface normal deviation are also defined. CHAPTER 3. THE SPATIAL SOLAR ENERGY DISTRIBUTION 49

3.6 List of symbols

Symbol Definition θ Angular displacement from the solar vector θδ Angular limit of the solar disc θ∆ Angular limit of the circumsolar region δ log(θδ) ∆ log(θ∆) φ(θ) Radial solar energy profile Φ Total radiant solar flux (0 θ θ ) i ≤ ≤ ∆ Φcs Radiant circumsolar flux (θδ θ θ∆) χ Circumsolar ratio ≤ ≤ γ Gradient of the circumsolar region of the sunshape κ Intercept of the circumsolar region of the sunshape a¯ Mean of a Saa Variance of a Sab Covariance of a&b r2 Square of the correlation coefficient Table 3.2: List of symbols for Chapter 3 Chapter 4

The eﬀective size of the solar cone for solar concentrating systems

Using the sunshape model developed in the previous chapter, we define here a virtual solar cone, whose principle axis is aligned with the solar vector, having a radial angular displacement containing a pre-defined proportion of the terrestrial solar radiation. By simulating various sunshape profiles, the angular extent of the energy distribution is established giving the effective size of the solar cone for a range of atmospheric conditions.

By simulating the reflection of these solar distributions off a set of non-ideal mirrored surfaces, accounting for non-specular reflection and mirror shape errors, the combined effect of sunshape and mirror properties on the solar image is also obtained. Clear trends are presented that show the dependence of the effective size of the solar image on the accuracy of a mirrored surface for different sunshapes. We then identify the effective size of the solar image at the absorber plane that must be accommodated in the design and optimisation of solar concentrating systems. The majority of the material presented in this chapter was published in in Solar Energy (Buie et al., 2003a).

4.1 Literature review

The broadening of the solar image when sunlight travels from the near vac- uum of space to the Earth’s surface is due to interactions of the sunlight

50 CHAPTER 4. THE EFFECTIVE SIZE OF THE SOLAR CONE 51 with atmospheric particulates. As previously mentioned, the amount of energy in the circumsolar region is important for determining the overall ﬂux distribution at the focal region of imaging concentrators and, depending on the acceptance angle of a solar concentrating system, overestimation of the power output may occur if all the power is assumed to fall within the extent of the solar disc only.

This paper is concerned with the second of these issues: examining the amount of energy that is available in a given radial displacement about the solar vector that has been reflected off a mirrored surface in a generic solar concentrating system. To estimate this, an accurate model of the radial energy distribution of the sun must be established. Originally, work conducted by the Lawrence Berkeley Laboratories (LBL) (Grether et al., 1975; Noring et al., 1991) in the late 1970’s and early 1980’s measured the degree to which sunshapes varied from location to location. Work carried out by Rabl and Bendt (1982) in defining a sunshape using the RDB was followed by Schubnell et al. (1991); Schubnell (1992a,b); Steinfeld and Schubnell (1993);Neumann and Schubnell (1992); Neumann and Groer (1996); Neumann et al. (1998) and Neumann and Witzke (1999). All of these authors made contributions to the description of the radial energy distribution of the sun and the effect this distribution has on the ultimate flux distribution in the focal region of concentrating systems.

Neumann et al. (2002) described six sunshape profiles that are indicative of a typical range of atmospheric conditions. These profiles, combined with their statistical weights, represent a numerical database for calculating the influence of variable conditions of the sunlight scattering on solar concentrating systems. Buie et al. (2003c) extended the analysis of the LBL sunshape data, describing an empirical sunshape model that illustrates little variation over a range of geographic locations and presenting an algorithm that can be used to recreate sunshapes for simulating solar concentrators. In summary, work described in the literature to-date involves simulation of the radial energy distribution of the sun for clear skies showing good agreement with a vast amount of observed data.

Concentrating systems have the added complexity of reflecting sunlight off at least one mirrored surface. The reflection causes a broadening of the solar beam. The broadening is due to both the dispersion effects of non-specular surfaces, surface normal deviations from mirror shape errors and tracking errors (Bendt et al., 1979). Johnston (1995) showed that these sources of errors could be combined for a mirrored surface, and the broadening of the CHAPTER 4. THE EFFECTIVE SIZE OF THE SOLAR CONE 52 solar image could be simulated by considering a probability distribution of errors. Therefore, if sufficient statistical information is gathered characteris- ing mirror surfaces, the typical reflected solar distribution off that surface is predictable.

This chapter defines a solar cone whose principle axis is aligned with the solar vector and that has a radial angular displacement containing a predefined proportion of the terrestrial solar radiation. The effective size of the solar cone is then described for a range of solar conditions. The broadened distribution, or solar image caused by the reflection of the solar cone off a mirrored plane is then calculated. Clear trends are presented that show the dependence of the effective size of the solar image on the accuracy of a mirrored surface for different sunshapes. The effective size of the solar image at the absorber plane that must be accommodated in the design and optimisation of solar concentrating systems is then discussed.

4.2 The size of the solar cone

As illustrated in Chapters 2 & 3 the terrestrial sunshapes show little variation from location to location when they are described in terms of their CSR. If the CSR is well known then the spatial energy distribution φ(θ, χ) of the sun can be described by Equation 3.18. Using this distribution, the energy contained within both the solar and circumsolar region can be well approximated using Equation 1.3.

Using a combination of all of these equations, the proportion L of the incident radiation bounded by a speciﬁc radial displacement θL about the central solar vector can be estimated from,

θL φ(θ, χ)θ dθ L(χ) = 0 , (4.1) θ∆ %0 φ(θ, χ)θ dθ % for any given sunshape proﬁle.

The effective size of the solar cone was estimated for solar radiation with CSRs ranging from 0.01 to 0.8 (Figure 4.1). The six curves represent various percentages of the incident radiation of 70%, 80%, 85%, 90%, 95% and 98%. Obviously the upper asymptotic limit to the effective size of the solar cone is 43.6 mrad, encompassing 100% of the incident radiation, as defined in Section 1.5. CHAPTER 4. THE EFFECTIVE SIZE OF THE SOLAR CONE 53

Figure 4.1: The eﬀective size of a solar cone containing various percentages of the incident radiation. The angular displacement is deﬁned as the half angle of a solar cone that would encompass a certain percentage of solar insolation

By definition, a solar profile with a CSR of 0.3 has 70% of its energy contained within the solar disc. For each of the CSR sunshapes created using Equation 3.18 the effective edge of the solar disc was determined (Figure 4.2). The value of the angular displacement for all of the simulations should be 4.65 mrad, the accepted limit of the solar disc. This implies that the error associated with the prediction of the size of the solar disc is 1% for sunshapes ± with a CSR of less that 0.2 and 4% for the remainder of the sunshapes. ±

4.3 Reﬂection oﬀ a mirrored surface

If the terrestrial solar image is reflected off a perfect (planar) surface, the reflected image would be identical to the original. In real systems though, the reflection of an image off a surface causes a distortion in that image. Two principle effects cause this distortion in solar concentrators. Firstly, real surfaces interact with the reflected radiation, causing the expected specular reflection to form a dispersive cloud. Secondly, mirror shapes are not perfect, CHAPTER 4. THE EFFECTIVE SIZE OF THE SOLAR CONE 54

Figure 4.2: Demonstration of the accuracy of Equation 3.18 at predicting the edge of the solar disc for a range of atmospheric conditions. The angular displacement of the solar disc should be 4.65 mrad. that is to say, the variations in the surface normals from the ideal mirror shape are an additional inﬂuence on the reﬂected image.

While initially being addressed by Baum (1957), Johnston (1995) went on to show that by considering as a systemic mass the slope error (deﬁned as the angular deviation of the actual surface normal vectors from their ideal directions, measured in milliradians (Johnston, 1998)), the surface dispersion eﬀects and the tracking errors, the combined error could be treated by the probability distribution (P ),

dP θ θ2/2σ2 = e− , (4.2) dθ σ2 where σ is the standard deviation of the combined surface slope and dispersion error, and θ is the radial displacement of a reﬂected beam from the specular direction. Figure 4.3 is a recreation of the Johnston (1995) probability distribution with a standard deviation of 3.5 mrad. A real surface can then be characterised by the standard deviation of the probability distribution. As an example of diﬀerent surfaces, a high quality optical mirror CHAPTER 4. THE EFFECTIVE SIZE OF THE SOLAR CONE 55

Figure 4.3: Probability distribution of the slope error of a surface with a standard deviation of σ = 3.5 mrad. has a probability distribution with a standard deviation of about 0.2 mrad, whereas poorer quality solar reﬂectors could have a standard deviation as high as 8 mrad.

To investigate the reﬂection of the solar image oﬀ a non-ideal mirrored surface, the convolution of the solar image and the probability distribution from Equation 4.2 was conducted (Rabl, 1985, pages 202-205). The convolution involves redistributing the vectors representing the solar image according to a normalised two-dimensional representation of a radial Gaussian distribution. Initially a solar image consisting of one hundred thousand appropriately weighted vectors extending over the angular radius of the circumsolar region was created (discussed in full in Chapter 5). A normalised two dimensional representation of a radial Gaussian distribution was also created extending to an angular displacement of three standard deviations from each solar vector. The weight of each of the solar vectors was then redistributed according to the normalised Gaussian about the position of each vector, simulating a real surface with a standard slope error of σ.

It must be pointed out that the convolution of a radial error distribution and a solar image is only truly representative of a reflected beam at normal CHAPTER 4. THE EFFECTIVE SIZE OF THE SOLAR CONE 56 incidence. For non-normal reflection, being the majority of reflections, the error distribution would be asymmetric (or elliptical). Therefore this method is only an approximation of a real system.

Figure 4.4: The effective size of the solar cone after reflection off a surface with a standard deviation of surface errors of 3 mrad.

The energy distribution of various solar images was calculated including the effective size of the solar cone, after the reflection off assumed planar mirrors of differing optical quality. Figure 4.4 illustrates a typical result similar to Figure 4.1, only the original solar image was reflected off a surface with a standard deviation of errors of 3 mrad. The results of the simulations have been presented in two ways: firstly, the effective size of the solar cone was compared primarily to the optical quality of mirrors for particular solar conditions (Figures 4.5-4.8) and secondly, the size of the solar image was primarily compared to various solar conditions where the concentrator has certain predefined parameters (Figures 4.9-4.11). CHAPTER 4. THE EFFECTIVE SIZE OF THE SOLAR CONE 57

Figure 4.5: The angular displacement of the reflected solar cone for different mirror errors for a CSR of 0.05. The different curves represent different percentages of total energy collected.

Figure 4.6: The angular displacement of the reflected solar cone for different mirror errors for a CSR of 0.1. The different curves represent different percentages of total energy collected. CHAPTER 4. THE EFFECTIVE SIZE OF THE SOLAR CONE 58

Figure 4.7: The angular displacement of the reflected solar cone for different mirror errors for a CSR of 0.2. The different curves represent different percentages of total energy collected.

Figure 4.8: The angular displacement of the reflected solar cone for different mirror errors for a CSR of 0.3. The different curves represent different percentages of total energy collected. CHAPTER 4. THE EFFECTIVE SIZE OF THE SOLAR CONE 59

Figure 4.9: The angular displacement of the reﬂected solar cone as a function of the circumsolar ratio; incident radiation collection eﬃciency - 80%.

Figure 4.10: The angular displacement of the reﬂected solar cone as a function of the circumsolar ratio; incident radiation collection eﬃciency - 90%. CHAPTER 4. THE EFFECTIVE SIZE OF THE SOLAR CONE 60

Figure 4.11: The angular displacement of the reﬂected solar cone as a function of the circumsolar ratio; incident radiation collection eﬃciency - 95%.

4.4 Discussion of results

Figures 4.5 to 4.8 illustrate the dependence of the effective size of the reflected solar cone on the quality of the reflecting surface, for a generic solar concentrating system. As in Figures 4.1 and 4.4, a range of incident radiation proportions (L) were chosen so as to get a clear indication of the optical performance of concentrating systems. Of the four solar conditions ranging from a CSR of 0.05 to 0.3, the effective size of the reflected solar cone increases with the degradation of the optical quality of the mirrored surface (defined by the standard deviation of the combined surface errors). Also as the incident beam becomes broader (defined by a larger CSR), the effective size of the solar cone increases substantially, particularly for small surface errors.

An approximately linear relationship exists between the angular size of the reflected solar cone and the standard deviation of errors, as the standard deviation of errors increases. This is qualitatively different to the upper asymptotic limit in the effective size of the solar cone that exists in reflecting the solar image off an ideal surface. As the standard deviation of the surface error increases, the influence of the sunshape distribution diminishes, effectively reducing the sunshape distribution to a point source. Therefore, CHAPTER 4. THE EFFECTIVE SIZE OF THE SOLAR CONE 61 the resulting reflected energy distribution is effectively described by a radial Gaussian distribution where θ σ. L ∝ Figures 4.9 to 4.11 illustrate the effective size of the solar cone for various solar conditions, described by an increasing CSR from 0.01 to 0.8. The higher the quality of the reflecting surface, the smaller the size of the angular spread of the solar image; analogous to a smaller absorber aperture that would be required in a generic solar concentrator. For locations in the world where there is little atmospheric scattering, represented by a low CSR, the optical quality of the mirror surface becomes more critical. For environments where there is generally large atmospheric scattering (signified by average CSRs of between 0.2 and 0.3), very little improvement in the collection efficiency can be achieved by using mirrors with an optical quality better than between 4.5 or perhaps 3 mrad in the standard deviation of the surface errors. Ideally, the size of the absorber aperture in solar concentrating systems should be approximately the size of the reflected solar image in the focal plane, plus an allowance to account for the typical tracking errors of the reflector modules. If all the power of the reflected beam was thought to be within the confines of the direct beam radiation (4.65 mrad about the solar vector) then an overestimation of the collector efficiency would occur. For the design of solar concentrating systems at a particular site, the size of the reflected solar image can be defined by two parameters: the CSR and the standard deviation of the surface error of the mirrors. If enough sunshape information about a site is known then average, typical, or effective maximum CSRs may be predicted and used as a design parameter (see Rabl and Bendt (1982)). The design solar distribution chosen may be related to improvements in the seasonal or daily output characteristics. The size of the absorber can then be determined by accurate knowledge of the standard deviation of the surface errors using Figures 4.4 to 4.11. Al- ternatively, given an optimum absorber aperture and therefore acceptance angle, the quality of the reflective surfaces can be chosen to ensure that a certain percentage of the incident radiation strikes the absorber. The absorber aperture may be determined by thermal considerations leading to a trade-off between optical efficiency and the total conversion efficiency of the concentrating system (Steinfeld and Schubnell (1993)). For concentrating photovoltaic systems, the sunshape spectral dependence could also be incorporated with a similar method, as previously described by Schubnell (1992b). Whilst this chapter does not outline an analytical design procedure for in- corprating CSR considerations into solar concentrators, as for example given CHAPTER 4. THE EFFECTIVE SIZE OF THE SOLAR CONE 62 by Rabl and Bendt (1982), the general relationship between the circumsolar ratio, mirror accuracy, absorber acceptance angle and optical efficiency has been established in design curves covering a wide range of the important parameters.

4.5 Conclusion

The inﬂuence of a sunshape, described by the circumsolar ratio, and mirror tolerance, described by the standard deviation of the mirror errors incorporating non-specularity and non-idea mirror shapes, on the angular distribution in the focal region of a generic solar concentrating system has been investigated. The results are presented as a series of graphs covering the parameters over their typical ranges, allowing broad design considerations to be investigated for solar concentrating systems.

4.6 List of symbols

Symbol Deﬁnition χ Circumsolar ratio

Φcs Energy contained within the circumsolar region

Φi Total terrestrial solar beam θ Angular displacement from the solar vector θδ Radial displacement of the solar disk θ∆ Radial displacement of the circumsolar region θL Radial displacement deﬁning the size of the solar cone φ Radial solar energy proﬁle (sunshape) per steradian κ Sunshape variable γ Sunshape variable L Percentage of the terrestrial solar beam P Probability distribution of the combined surface error σ Standard deviation of the probability distribution Table 4.1: List of symbols for Chapter 4 Chapter 5

An algorithm for the optical simulation of solar concentrating systems

This chapter describes an infrastructure for simulating the optics of dynamic abstract solar concentrators for the optimisation and design of solar energy conversion systems. Using the equations that have been developed in the preceding chapters, functions are written within the class structure of the computer language c++, that simulate the terrestrial solar beam. The library consists of four parts, the PSA Algorithm (Blanco-Muriel et al., 2001) for calculating the solar vector, Spectral2 (Bird and Riordan, 1986) simulating the spectral energy distribution and two new computer codes by the author, that describe the spatial energy distribution of the solar image and the broadening that occurs when a solar image is reflected off a non-ideal mirrored surface. The framework of these algorithms are created using a vector library, written by Kenji Hiranabe, which is a C++ port of the Java (TM) 3D API 1.1 vecmath package java.vecmath. The algorithms have been created with the highest priority given to speed and facilitate both a Euclidean environment and vector and point transformations used in computer graphics. The combination of these algorithms will generate any optical system which allows: a direct comparisons between various concentrating systems, the degree to which optical systems are dependent on individual solar profiles, and reproducibility in results which is essential in scientific research. Portions of the chapter has been submitted as a technical note to Solar Energy (Buie et al., 2003b).

63 CHAPTER 5. A MODELLING FRAMEWORK 64

5.1 Introduction

In the journal Philosphy of Science, Rosenbluth and Wiener (1945) wrote that,

No substantial part of the universe is so simple that it can be grasped and controlled without abstraction. Abstraction consists in replacing the part of the universe under consideration by a model of similar but simpler structure. Models ... are thus a central necessity of scientiﬁc procedure.

Experimental research conducted to optimise solar power applications is expensive and increasing. It is this reason that computer simulations are be- coming a necessary tool in the ﬁeld of solar energy.

There are a large number of computer resources, many freely available, that have been created to model individual components and entire systems for solar power applications. Generally, these resources vary in their structure, programming language, and relevance to a speciﬁc task. This hinders their usability for broader applications. Furthermore, reproducibility, which is essential in scientiﬁc research is also hindered as the operating parameters and details of one or more of the pieces of computer code used in modelling, are not always publicly available.

The purpose of this chapter is to provide the tools to accurately model the optics of any solar power system. To achieve this, the optics were divided into a number of components: the most obvious of these is the geometry of the optical system and its orientation relative to the sun. Of equal importance is the spatial and spectral distribution of the terrestrial solar beam, due to the dynamic nature of the atmosphere, and finally, considering concentrating systems, the effect to the beam of reflecting it off a non-ideal mirrored surface.

This information is presented in a class written in the computer language c++. It contains an abstract representation of the terrestrial solar beam solar.h, including the systemic effect of reflecting that beam off a non-ideal mirrored surface. A vector library vecmath.h is also included, creating the basis of a Euclidean space, together with functions that can facilitate a ray- tracing algorithm. Within this framework any optical system can be created, and, assuming all of these components are modelled representing real systems, then a good understanding of the overall optical performance can be gained. CHAPTER 5. A MODELLING FRAMEWORK 65

It must be pointed out that the computer codes and algorithms presented in this work do not specifically simulate a particular solar energy conversion system, whether it be a dish system or a central receiver. Rather, the codes provide the infrastructure to facilitate the creation of any such system, in which the solar input parameters that clearly describe the reflected terrestrial beam are common. This process facilitates reproducibility, which is essential in scientific procedure.

This chapter is divided in two parts. The first provides a brief analysis of the vecmath class that facilitates high-resolution optical maps for simulating the optical components of a concentrating systems. The second describes the principle components required to create an abstract solar image reflected off a mirrored surface, and the functions that generate them.

5.2 A tool to facilitate modelling

Blanco-Muriel and Alarcon-Padilla (2000) summarised the seven main computer codes currently available to carry out the task of modelling solar energy conversions systems (DBS (Maynard and Gajanana, 1980), HELIOS (Vitti- toe and Biggs, 1981), HFLCAL (Kiera, 1986), SIMSOL (Amannsberger and Bittner, 1982), DELSOL2 (Dellin et al., 1981) and RC (Pitmann and Vant- Hull, 1989)). While all of these codes have been effective at predicting the performance of solar collectors or optimising heliostat fields, all were developed at least fifteen year prior to this work. All of the aforementioned computer programs are complicated to read, are written in a computer language that does not allow advanced memory management nor a class structure, and use memory and performance short cuts to be compatible with the computers available at the time (lacking in both performance and memory).

Each of the codes were designed for a speciﬁc task. For example: SIMSOL was designed for addressing the performance of parabolic troughs, HELIOS, to analyse heliostat performance in central tower receivers. Converting these codes to address the performance of ones’ individual concentrating system is time consuming and requires as much understanding of the code as the people who originally developed them.

Two other more recent computer codes have been proposed by the solar energy community. EnerTracer (Blanco-Muriel and Alarcon-Padilla, 2000) (written in c) and Fiat Lux by Monterreal (1999) (written in MATLAB). CHAPTER 5. A MODELLING FRAMEWORK 66

Both codes were written to overcome the issues mentioned above but unfortunately never became publicly available. There exists no generic tool to model simple or complicated structures. All that is required is a generic library that can be used to build up the infrastructure of a concentrating system’s optical components and tools that facilitate the projection of a solar distribution, oﬀ those optical components to an absorber region.

5.2.1 Vector package

Kenji Hiranabe ([email protected]) implemented a vector library taking into account all of the components required to recreate Euclidean space, along with vector and point transformations often utilised in computer graphics. The library vecmath is a c++ port, written to conform to the Java(TM) 3D API speciﬁcation of Sun Microsystems. The included classes are the followings: Vector2, Vector3, Vector4; Point2, Point3, Point4; TexCoord2; Color3; Color4; Quat4; AxisAngle4; Matrix3, Matrix4. These classes are optionally in the namespace of vecmath and facilitate both a ﬂoat and double template. Great lengths have been taken by Kenji Hiranabe to optimise the algorithms for speed, removing any referenc- ing and he has implemented an extreme inline nature to the code. These algorithms lay the foundation of the following solar class.

5.3 Simulating the terrestrial solar beam

Four components are required to create an abstract model of the terrestrial solar beam: the position of the sun in the sky, its spectral and spatial energy distribution and the broadening of the spatial energy distribution after its reflection off a non-ideal mirrored surface. Combined these components completely characterises the reflected beam.

5.3.1 Literature review

The solar position

Solar tracking systems require a precise knowledge of the local horizontal coordinates of the sun. One of the ﬁrst to provide a complete program CHAPTER 5. A MODELLING FRAMEWORK 67 for calculating this was Walraven (1978) with a correction Walraven (1979) based on simpliﬁed equations from the American Ephemeris and Nautical Almanac. Walraven’s papers prompted correspondence and publications by: Archer (1980); Wilkinson (1981); Zimmerman (1981); Muir (1983); Wilkinson (1983); Ilyas (1983); Pascoe (1984); Ilyas (1984a); Wilkinson (1984); Ilyas (1984b), in relation to both the accuracy and thoroughness of the algorithm and the implementation of the program’s FORTRAN computer code.

Addressing the concerns of these papers, Michalsky (1988) published a simpler algorithm to that of Walraven, based on the The Astronomical Almanac (1986). Included in Michalsky’s algorithm was a model for atmospheric refraction that better ﬁtted observed data than the previous work by Archer (1980). However, Michalsky failed to correct the equations for Southern Hemisphere calculations, later presented by Spencer (1989).

Blanco-Muriel et al. (2001) examined the accuracy of both the corrected Walraven and Michalsky algorithms, along with some more simpliﬁed models, and stated that the average deviation from the true solar vector (the unit vector describing the position of the sun in the sky from a given location) was 0.271 and 0.207 minutes of arc respectively. Blanco-Muriel et al. (2001) wrote that, Even though the Michalsky algorithm may be considered good enough for most solar tracking applications, its accuracy, computing eﬃciency, and ease of use can still be improved.

The algorithm they proposed, the PSA Algorithm, has advanced memory management, greater speed and robustness and an average deviation from the solar vector of 0.147 minutes of arc. The PSA Algorithm is presently used as an open loop tracking system for the EuroTrough direct steam line at the Plataforma Solar de Almeria (PSA). The algorithm is written in ANSII C++ and is currently the most accurate model to predict the position of the sun at local coordinates not including atmospheric refraction.

Solar spectral energy distribution

The atmosphere acts as an active absorber, re-emitter and small angle scat- terer of solar radiation. As each of these processes is wavelength speciﬁc, the resulting terrestrial solar spectrum is dependent on both the constituents of the atmospheric proﬁle and the optical path length of the radiation. As the performance of most solar power systems is spectrally dependent, in order to simulate the optical performance the spectral characteristics of the terrestrial radiation must be determined. CHAPTER 5. A MODELLING FRAMEWORK 68

The spectral composition of solar radiation is pertinent to photovoltaics as well as for solar thermal applications. The absorption characteristics of selective surfaces, transmission of light through absorber windows and reflection of sunlight off metal coated mirrors are common to all solar concentrators, and are all spectrally dependent. There are presently three accepted methods for generating the terrestrial solar spectrum (Myers et al., 2002). Line-by-line (LbL) models consider very fine wavelength bands, simulating the quantum absorption and scattering of solar radiation off the relevant atmospheric constituents. These LbL models correlate well with observed data at the expense of computational speed. The second group are simpler models, based on the parametrisations of transmittance and absorption functions, for the basic atmospheric constituents. While this method of simulation possesses much lower resolution than LbL models, it has a rapid computation time. The third method of generating the terrestrial spectral simulations are Band models. These are simplified LbL models created by grouping together both layers of the atmosphere and wavelength bands. Band modes are more accurate than the simple parametrisation models but do not have the large computation times of the LbL models. For the purpose of establishing a comprehensive computational tool, speed is of greater importance than a model with a very high agreement with observed data. Myers et al. (2002) compared two simple atmospheric transmission simulations: SMARTS2 (Gueymard, 2001) and Spectral2 (Bird and Riordan, 1986), against a more accurate band model simulation (MODTRAN Anderson et al., 1993). Myers et al. (2002) concluded that SMARTS2 was both more accurate and more adaptable than Spectral2, and was the best alternative for fast simulation of the terrestrial spectral insolation. SMARTS2 is a flexible program with the ability to accurately simulate the solar spectrum through a broad range of atmospheric conditions. However, the source code for this simulation is not publicly available and for the pur- poses of optimising the optical performance of solar concentrating systems, a standard clear sky atmosphere is sufficient for most simulations (in particular, for generic simulations). In correspondence with Myers, he stated that, “... Spectral2 can be considered reasonably accurate where sufficient input data is lacking for the more detailed SMARTS2 approach”. Either of these models would then be sufficient to recreate the spectral solar energy distribution for generic simulations and Spectral2’s source code is publicly available. CHAPTER 5. A MODELLING FRAMEWORK 69

Solar spatial energy distribution

The literature review for the solar spatial energy distribution is recreated from Chapter 3 but has been included for completeness of this chapter. The Lawrence Berkley Laboratories began investigating terrestrial sunshapes in the mid to late 1970s. They compiled a large number of radial solar profiles, representing the spatial energy distribution of the sun (Noring et al., 1991), into what is called the reduced database (RDB). By using the RDB, Rabl and Bendt (1982) created a standard solar scan representing the average of an unstated number of solar profiles. This model was adopted as the new profile of the energy distribution of the terrestrial beam. Using both the RDB data and independently observed data Schubnell et al. (1991); Schubnell (1992a,b); Steinfeld and Schubnell (1993); Neumann and Schubnell (1992); Neumann and Schubnell (1992); Neumann and Groer (1996); Neumann et al. (1998); Neumann and Witzke (1999); contributed to both quantifying the degree to which the amount of energy in the circumsolar region of the sky effected solar concentrators and illustrated trends that were present in sunshapes at various locations. Later, Buie and Monger (2001) and Neumann et al. (2002) independently inferred that the spatial energy distribution of the sun, if represented by its circumsolar ratio, illustrated, on average, an invariance to a change in location for all of the sites where data had been collected. The spatial energy distribution was then described by Buie et al. (2003c) and is currently the most accurate model to simulate the terrestrial solar spatial energy distribution.

Reﬂection of the solar image

The literature review for the effects of the reflection of the solar image off a mirrored surface is included from Chapter 4 for completeness of this chapter. If the terrestrial solar image is reflected off a perfect (planar) surface, the reflected image would be identical to the original. However in real systems, the reflection of an image off a surface causes a distortion in that image. Two principle effects cause this distortion in solar concentrators. Firstly, real surfaces interact with the reflected radiation, causing the expected specular reflection to form a dispersive cloud. Secondly, mirror shapes are not perfect, that is to say, the variations in the surface normals from the ideal mirror shape are an additional influence on the reflected image. CHAPTER 5. A MODELLING FRAMEWORK 70

Johnston (1995) showed that by considering as a systemic mass both the slope error (deﬁned as the angular deviation of the actual surface normal vectors from their ideal directions, measured in milliradians (Johnston, 1998)) and surface dispersion eﬀects, the combined error could be simulated by a Gaussian (normal) distribution.

Mirror alignment and tracking errors are futher significant sources of error in predicting the resultant flux distribution in solar concentrators. It is only when a statistically significant number of mirrors or a long time average is considered, that a distribution in tracking error can be modelled by a radial Gaussian distribution (Bendt et al., 1979). If either of these criteria are met, the standard deviation of the alignment and tracking error can be simply added to the standard deviation of the combined surface slope and dispersion error. Movement in the absorber from the ideal position could be treated in the same manner. These results were investigatged in Chapter 4.

5.3.2 Terrestrial solar algorithm

To recreate the most accurate model of the terrestrial solar beam, four components of code are incorporated in the system developed here: the PSA Algorithm (Blanco-Muriel et al., 2001) to determine the solar position; Spec- tral2 (Bird and Riordan, 1986) to determine the spectral energy distribution; a solar image incorporating an accurate model of the spatial energy distribution of the sun (Buie et al., 2003c) and the eﬀect a mirrored surface has on that distribution (Johnston, 1995).

It is not sufficient to simply use each appropriate piece of code in its original form. Spectral2 uses an ansii c implementation of the Michalsky (1988) algorithm to calculate the solar vector. Spectral2 also uses a large number of globally defined variables making the algorithm difficult to read and hard to integrate into pre-existing code. The PSA algorithm implements a structure function for its variables different to that of Spectral2. They were all developed independently and considerable work is required to assimilate them into a consistent body of functions.

The decision was made to rewrite each of the appropriate algorithms within one c++ class structure solar.h. The algorithm would use:

• A template class facilitating both a single precision and a double precision implementation of the algorithms. While little performance is CHAPTER 5. A MODELLING FRAMEWORK 71

gained running a 32-bit algorithm to a 64-bit algorithm in modern computers and compilers, some calculations do not require 15 signiﬁcant ﬁgures to gain an indication of performance.

• Inline functions where possible, to minimise computation time

• No linking as all the functions are within one header ﬁle solar.h.

• Public access to class variables.

• An optional namespace for all the algorithm within the solar class.

• The standard template library for the GNU g++ compilers.

• An optional input-output support for the member functions is provided.

• Each of the pieces of code can still be used in their original form.

Using these principles, the algorithm solar.h was written and can be seen in Appendix C.

5.3.3 Solar Class

The library solar.h is a global library initiating three separate libraries: Spa- tial.h creates an abstract representation of the solar image, Spectral.h a c++ implementation of Spectral2 and Sunpos.h, a c++ implementation of the PSA algorithm. The algorithms are controlled by a ﬁle called solar conf.h which can switch the namespace and io-support on or oﬀ, and declares the dependent variables.

Sunpos.h

Apart from the reallocation of variables, one small modiﬁcation was made to the code to include the calculation of the day number which is required to generate the spectral solar energy distribution. The default variables declared during the initiation of the solar class with regards to calculating the solar vector are for the location of Sydney, Australia (Longitude 151.2◦, Latitude

-33.867◦ ) on the 1st of January 2000, at 12:06 - 11 Universal Time. CHAPTER 5. A MODELLING FRAMEWORK 72

Spectral.h

The code SPECTRAL2 was written by Martin Rymes from the National Renewable Energy Laboratories (NREL) and is available via public FTP from NREL’s website (http://www.nrel.gov). SPECTRAL2 is a c version of the algorithm by Bird and Riordan (1986). Small modiﬁcations were made to Rymes’s code: the solar position algorithm SOLPOS, a c implementation of Michalsky (1988) algorithm, was replaced with the more accurate PSA Algorithm (Blanco-Muriel et al., 2001) and the variables were reallocated to ﬁt more appropriately within the c++ class structure. However, the integrity of the code still remains and is embedded within this solar class. The default variables are described in Table 5.1.

Variables Description Default Units units Output units (1,2,3) 1 tau500 Aerosol optical depth at 0.5 µm 0.27 watvap Precipitable water vapor 1.42 cm alpha Power on Angstrom turbidity 1.14 assym Aerosol assymetry factor 0.65 ozone Ozone amount 0.34 cm press Air pressure 1013.25 kPa tilt Tilt on mirror 0.0 degrees

Table 5.1: The default atmospheric parameters.

Spatial.h

Within the solar class the spatial solar energy distribution of the direct beam and circumsolar region is simulated by an array of appropriately weighted vectors. Each vector deﬁnes an angular displacement from the solar vector

(θi) in a series of limbs (Figure 5.1). The resulting vector represents its corresponding segment of an arc whose weight is determined by:

θb φ(θi) sinθdθ Φ = θa , (5.1) θi θcsr n 0 %φ(θ)sinθdθ % where n represents the number of radial limbs, θa and θb represent the angular displacement of the inner and outer regions of each vector respectively and

θcsr refers to the outer limit of the circumsolar region (with a default value of 43.6 mrad), leaving a solar image with a total weight of 1, extending to an CHAPTER 5. A MODELLING FRAMEWORK 73

Figure 5.1: Graphical representation of an abstract solar array, with 100 limbs of 49 points along each limb giving 4901 individual vectors. The weight of each vector is represented by the z-axis per steradian. This particular picture represent a sunshape with a CSR of 0.3 Function Variable Default Units putCSR circumsolar ratio 0.05 putSigma stddev mirror error 3.5 mrad putCirSolarLimit cirSolarLimit 43.6 mrad putArrayDim no. spurs, no. radial points 100, 50 integer

Table 5.2: The input functions to create the solar distribution angular displacement of 43.6 mrad from the solar vector. The default values for the dependent variables are given in Table 5.2 The standard deviation of the mirror error, refers to the quality of the optical surface of the mirrors. A given surface’s deviation from the idea case (as described in Chapter 4) can be modelled by a normal distribution. A real surface can then be characterised by the standard deviation of the probability distribution. As an example of different surfaces, a high quality optical mirror has a probability distribution with a standard deviation of about 0.2 mrad, whereas poorer quality solar reflectors could have a standard deviation as high as 8 mrad. Simulating the solar image reflected off a mirrored surface can be performed by the convolution of the solar image created in Equation 5.1 and a two di- CHAPTER 5. A MODELLING FRAMEWORK 74

Figure 5.2: Radial energy distribution of the solar image after being reflected off non-ideal mirrors with varying standard deviation of errors mensional representation of the radial Guassian distribution in Equation 4.2. Figure 5.2 illustrates some radial energy profiles formed after the convolution of the solar image and the radial Gaussian for a range of surface qualities (represented by the standard deviation of the surface normal and dispersion errors combined).

5.4 Examples of applications and results

Within this section a series of concentrating systems have been generated for the purpose of illustrating the versatility of the code and typical outputs that the code can generate. All of the images have been generated using MATLAB and the movie was created using OpenDX.

Fresnel dish system

A concentrator has been constructed at the University of Sydney for testing, characterizing, and optimising solar energy conversion materials and components. The optical and heat transfer characteristics of the experimental concentrator are of great importance for the characterisation and optimisation CHAPTER 5. A MODELLING FRAMEWORK 75 of the solar concentrator technology used in the large-scale central receiver systems and can also assist development in any mid-to-high concentrating system. The test facility is a 2-axis tracking parabolic Fresnel concentrator. A spe- cial consideration in this project has been the particular application of the concentrator as a test facility for solar energy conversion materials and devices. Rather than trying to maximize the total solar energy collected at the receiver, the design process has focussed on the requirements of a well- defined source of high-intensity illumination and sufficient flexibility in test parameters. The concentrator is comprised of 18 spherical mirrors, each with an aperture area of 0.26 m2, giving a total aperture area of 4.8 m2. The mirrors have a radius of curvature of about 4.8 m and a rim angle of 7◦ (as seen from the focal point). The mirrors are bonded structures comprising an anodised aluminium reflective surface mounted on a lightweight aluminium honeycomb structure, enclosed by aluminium sheets at the back and along the sides. The reflective material is Alanod 410G with an as-new reflectivity of about 83 % quoted by the manufacturer. A hard oxide layer ( approximately 2 µm) protects the softer Aluminium surface in an outdoor environment and reduces the reflectivity deteriorating. The mirrors are attached to a 3-metre parabolic dish frame, and the focal length of the system is 2.4 m. The placement of the mirrors is 6 on an inner ring, and 12 on an outer ring. An advantage of this set-up is that individual mirrors can be covered to provide incremental control of the solar flux up to a maximum value of about 1000 suns. Even higher concentrations are possible by employing small secondary mirrors near the primary focus of the concentrator. Uncertainties resulting from misalignment, dispersion and wavelength dependent reflection characteristics of the mirrors require that the focal area be characterised theoretically, both with respect to intensity and homogeneity. Also, the solar flux reflected from the concentrator will vary with the number of mirrors used and the time of day. The environment of the Fresnel dish system was built within the vector class. As the optical quality of the mirror surface is relatively poor we assumed that the standard deviation of surface errors to be approximately 6 mrad. The default solar variables were accepted only that the CSR was set to 0.1, more typical for a costal city such as Sydney (Authors opinion). Figure 5.3 illustrates the flux distribution in the imaging plane from just one of the outer mirrors of this Fresnel dish system. The spherical aberration created from CHAPTER 5. A MODELLING FRAMEWORK 76

Figure 5.3: Theoretical concentration ratio predicted from one of the outer mirrors, (surface error 6 mrad)

Figure 5.4: Theoretical concentration ratio predicted from the 18 mirrored array, (surface error 6 mrad) CHAPTER 5. A MODELLING FRAMEWORK 77 reflecting the solar image off the optical axis and onto a flat plate can clearly be seen. Also the spot size of the mirror can be accurately determined.

Figure 5.4 illustrates the flux distribution in the imaging plane generated by reflecting the solar image off all 18 mirrors using the same surface error distributions. Because of the symmetric position of the 18 mirrors the astigmatic image seen in Figure 5.3 is replaced by a symmetric image, with peak concentration over 1000 suns. A high flux homogenous field exists in the imaging plane with an area of approximately 0.16 m2 assuming a zero tracking error in the parabolic collector. No experimental evidence has been collected to confirm this distribution.

Using this environment, accurate ﬂux images can be created and controlled. The desired optical characteristics can be theoretically generated and then translated to the real system.

EuroTrough

EuroTrough, created by a European consortium is a parabolic trough collector design based on the successful LS-2, LS-3 (Luz) collectors in California (Lupfert et al. 2000). As with all linear systems, there are inherent optical losses due to the limited degrees of movement of the reﬂector. Using the solar and vector class, a parabolic trough collector was created whose dimensions match those of the EuroTrough.

The purpose of the model was to identify portions of the mirror module whose reflected solar radiation didn’t strike the absorber line during peak fluxes. The simulations were conducted at the location of Sydney, Australia at two different times of year, summer and winter solstice.

Figure 5.5 illustrates the southern end of one of the mirror modules of the EuroTrough (South is positive). The red region represents a portion of the mirror whose reﬂected sunlight never hit the absorber line (summer or winter), the orange represents the reﬂector area that does not hit the absorber line during a Sydney winter and the yellow represents the distance from the end of the mirror line that continuously strikes the absorber line.

The redundant aperture area relates to approximately 2.5 m2 and 15 m2 of aperture area respectively. This must be placed in context of the total aperture area of the modules being a redundant area of approximately 0.5% and 2.5% of the total aperture area respectively. CHAPTER 5. A MODELLING FRAMEWORK 78

Figure 5.5: End losses of a parabolic collector whose dimensions match those of the EuroTrough for the location of Sydney at both Summer and Winter solstice (Red misses the absorber always, yellow never misses).

Central receiver system

The flux distribution in a central receiver system is extremely important for both defining the optical efficiency and determining the homogeneity of the beam. For the purpose of this simulation, a central tower receiver with a horizontal flat plate collector was generated. The absorber plane was positioned 10 m above the rotating axis of the mirrors and is centred amongst a square mirror field. The mirror field consists of 225 closely spaced circular reflectors 2 m in diameter. Each mirror is placed on the vertices of a regular square grid 30 m by 30 m (15 by 15 array, 2 m apart). Each of the mirrors has a paraboloidal curvature with a focal length equal to the path length between its pivot point and the absorber plane. This leaves 706.9 m2 of mirror (aperture) area. Each mirror reflects the suns energy to the central point in the absorber plane. A square section of the absorber 1.2 m by 1.2 m is investigated centred about the central aiming position of each of the mirror modules. All of the shading and blocking of each of the mirrors was calculated. The default values for the solar class were accepted, except that instead of using a specific location a particular optical airmass was adopted. One CHAPTER 5. A MODELLING FRAMEWORK 79 simulation had the sun directly above the central tower or an airmass of one. The other considered an optical airmass of 1.5, with an azimuth angle of 0.392 radians.

For the purpose of this paper, the ﬂux distribution was generated either side of the focal plane. 11 slices were considered 5 above and below the focal plane in intervals of 10cm. Figure 5.6 illustrates all 11 slices for the AM1 simulation. The colormap has been rescales to give a greater representation of the lower ﬂux levels.

The curves exhibit a high degree of symmetry. The focal plane (represented by a vertical displacement of 0 m) has the smallest spatial energy distribution and subsequently the highest concentration ratio. The image in the focal plane is almost spherical similar to the flux distribution in Figure 5.4. As you move out of the focal plane the shape of the field becomes more apparent in the flux distribution. The spatial energy distribution increases with a subsequent reduction in the concentration ratio as expected.

Figure 5.7 is somewhat different to Figure 5.6. The obvious symmetry within the image no longer exists. The concentration ratio is considerably lower primarily due to the fact that there is less that two-thirds of the energy in the focal plane (created from the cosine effect of the aperture area and the shading and block of each of the mirror modules). The peak flux, moving out of the focal plane migrates from the central position in the image. Over time this point migrates around the central position.

This illustrates an interesting point of central receiver systems. As the solar image migrates from above the tower (AM1 to higher zenith angles) the flux distribution changes considerable out of the focal plane. This high flux region rotates about the central position during the course of the day. While the energy coming from each area of the field is similar, the flux is higher from those mirrors where the reflected image in more aligned with the optical axis of each reflector.

The uniform flux fields created in this section represents a solution for creating homogenous fields in the imaging plane essential to photovoltaic absorbers. A movie has been created to better represent how these areas of homogeneity have been created. You can view this in the accompanying CD under conclusions. The movie file is called Flux.mpeg. The movie is in two formats, one is a large file of 500MB, while the other is a little more user friendly at 20MB. CHAPTER 5. A MODELLING FRAMEWORK 80

Figure 5.6: Flux distribution as a function of z-position through the absorber plane (AM1). The colormap is the concentration ratio.

Figure 5.7: Flux distribution as a function of z-position through the absorber plane (AM1.5). The colormap is the concentration ratio. CHAPTER 5. A MODELLING FRAMEWORK 81

5.5 Conclusion

This chapter provides the tools to accurately model solar concentrating systems. A class library is provided (written in the computer language c++), that contains an abstract representation of the terrestrial solar beam, including the systemic effect of reflecting that beam off a non-ideal mirrored surface. The code presented is an extension of the building blocks provided by both Blanco-Muriel et al. (2001) defining the solar position and Bird and Riordan (1986) simulating the solar spectrum.

A vector class was also presented that creates the basis of Euclidean space together with functions that can facilitate a robust ray-tracing algorithm. Within this framework any optical system can be created and theoretically analysed. No graphical user interface is provided with this structure. The power of the code is that it is adaptable to any environment and trades the ease of use, inherent in such systems with adaptability, vital for eﬀective comparisons and mainframe simulations. Chapter 6

Simulation of the performance of photovoltaic cells

Having developed the components of the terrestrial solar beam, it is also usefull to have a tool to simulate the optics of thin and thick optical ﬁlms to characterise the transmittion of light through all of the components of an optical system. To develop this tool the simulations of silicon photovoltaic cells was conducted.

This chapter presents an investigation into the use of thin films of silicon oxide and silicon nitride as anti-reflection coatings to minimise the reflection losses of incident solar insolation on silicon photovoltaic cells. The total theoretically reflectance under two insolation cases is investigated. Firstly, with AM 1.5 direct beam radiation at normal incidence, as is usually simulated, and secondly, with the direct beam simulated over a full day.

Assuming that a minimum silicon oxide layer of 20 nm is deposited on the silicon wafer for surface passivation, variations in optimised ﬁlm thicknesses between both simulations were approximately 4%. The theoretical results also showed a strong correlation to experimental results for the normal incident case. This demonstrates that surfaces optimised for normally incident radiation are adequate for non-tracking photovoltaic cells under real conditions. The seasonal variations of an optimised surface showed variations in performance of less than 0.1%.

This chapter also reports that the excellent surface passivation obtained with a thin thermal oxide is maintained after depositing a silicon nitride layer using low pressure chemical vapour deposition. The results presented in this chapter have been published in Solar Energy Materials and Solar Cells

82 CHAPTER 6. THIN FILM SIMULATIONS 83

(Buie et al., 2003b). The sections: Comparison to experimental results, Surface passivation, and portions of the literature review, were contributed by Michelle McCann from the Australian National University.

6.1 Literature review

Plain, polished silicon reflects more than 30% of the incident sunlight for wavelengths corresponding to an energy greater than the band gap of silicon. Minimising reflection losses from the top surface of a solar cell is therefore crucial for high efficiency cell design. Optimal results are achieved by combining an anti-reflection coating (ARC) with surface texturing, whereby reflected photons have a second (or further) chance of entering cells.

Both single and multiple layer ARCs can be formed from a number of materials, including Ta2O5 (Strauss et al., 1999), TiO2, ZnS, SiO2, Si3N4 (Palik,

1985) and Al2O3 (Palik, 1991). In addition to having a refractive index that is beneficial for reflection control on a silicon wafer, these materials do not inhibit light transmission through to the silicon due to their low or zero absorption component within the visible and near infra-red wavelengths. Multi-layer stacks of two materials with a large difference in their refractive indices are most suitable for reducing reflection losses.

A silicon nitride (nitride) layer with a thin silicon oxide (oxide) under-layer for surface passivation is a relatively simple construction that can behave as a good ARC. In addition, silicon nitride has many properties that allow for increased processing ﬂexibility and hence the realisation of novel cell structures. For example, when deposited at a high temperature, nitride layers are hard and therefore scratch resistant, nitride is etched much more slowly than oxide in solutions containing hydrogen ﬂuoride and a deposited nitride may mask against naturally occurring pinholes in a grown oxide.

Reflection losses off photovoltaic cells are usually calculated by considering the direct beam radiation at normal incidence. For a non-tracking system however, most of the input energy will have a non-normal angle of incidence. A more thorough approach is therefore to design an anti-reflection coating that minimises reflection losses over a full day, or even year, cycle. Depending on the final application, an anti-reflection coating could also be designed based on a desired energy profile. For example, power output at the beginning of the day may be increased at the expense of power output in the middle of the day. CHAPTER 6. THIN FILM SIMULATIONS 84

For a high efficiency cell design, any coating on the surface must provide not only good anti-reflection properties, but also good surface passivation. Silicon nitride is typically deposited using plasma enhanced chemical vapour deposition (PECVD), which is well known to enhance surface passivation properties for bare silicon wafers or wafers with an oxide layer. This has been demonstrated with many cell designs including the high efficiency bifacial cells and the MIS-IL cells developed by Hezel et al. (Aberle, 2000). Oxides, forming gas annealed oxides and aluminium annealed (alnealed) oxides have also been shown to provide excellent surface passivation (Kerr et al., 2001). The silicon nitride in this work was formed by low pressure chemical vapour deposition (LPCVD) using dichlorosilane and ammonia gases. This results in the deposition of amorphous, approximately stoichiometric silicon nitride,

Si3N4, with a hydrogen content typically in the range 2–10 atomic% (Habraken et al., 1986; Stein et al., 1983; Stein and Wegener, 1977). The LPCVD method was chosen as it is a reliable, mature technique with high through- put rates that is well suited to batch mode deposition. Within this chapter, the theoretical treatment of thin film optics is followed by an experimental justification of the results. Finally, the surface passivation resulting from an oxide/LPCVD silicon nitride stack on silicon is determined experimentally illustrating its usefulness as an anti-reflection coating.

6.2 Theoretical thin ﬁlm simulations

Theoretical reﬂection losses are determined as a function of both silicon oxide and silicon nitride thicknesses. In the ﬁrst instance we consider normal- incident radiation and then extend this to the case of a full day simulation of the direct beam.

6.2.1 Thin ﬁlm theory

The refractive indices of silicon oxide, silicon nitride and silicon as a function of wavelength are well known. The reflection losses of an oxide/nitride stack on silicon can therefore be calculated by considering the optical impedance of each of the individual layers. Creating a stack of thin film layers l1...lj...lL between two infinite layers of pottant l0 and silicon ls, each layer j has a characteristic matrix defined by, Mj i sin δj cos δj Yj j = , (6.1) M & iYj sin δj cos δj ' CHAPTER 6. THIN FILM SIMULATIONS 85

where δj is the phase delay of radiation passing through layer j:

2πN d cos θ δ = j j j , (6.2) j λ and Yj is the optical admittance, given by:

* Y = o N , (6.3) j µ j ( o where *o and µo are the permittivity and permeability of free space respectively and dj is the thickness of each layer, lj. Each layer has a complex refractive index, nj, for a given wavelength, λ, and complex incident angle,

θ, such that Nj is deﬁned by:

nj cos θj for transverse electric polarisation N = nj (6.4) j  for transverse magnetic polarisation  cos θj  The transmittance of the electric ﬁeld Ej and magnetic ﬁeld Hj at each of the layer boundaries (Macleod, 2001; Powles, 1999), can be calculated using,

Ej 1 j Ej j+1 − | = j | . (6.5) Hj 1 j M Hj j+1 & − | ' & | '

The whole series l . . . l then has a characteristic matrix, , that is the 1 L M1,L product of all the layer characteristic matrices. The elements of the product matrix are m , m , m and m . is given by: 11 12 21 22 M1,L

L m m = = 11 12 . (6.6) M1,L Mj j=1 & m21 m22 ' , Deﬁning C as,

C = Y m + Y Y m m Y m (6.7) n o 11 o s 12 − 21 − s 22 Cd = Yom11 + YoYsm12 + m21 + Ysm22 (6.8)

C Y m + Y Y m m Y m C = n = o 11 o s 12 − 21 − s 22 , (6.9) C Y m + Y Y m + m + Y m - d - - o 11 o s 12 21 s 22 ------CHAPTER 6. THIN FILM SIMULATIONS 86 the theoretical reﬂectance of a whole series of thin ﬁlms is,

Rλ = CC∗, (6.10)

where a∗ is the conjugate of a. Similarly, the transmittance T through to the ls layer and the absorption A from the thin ﬁlm stack, can be determined by,

4Yoreal(Ys 1) Tλ = − (6.11) CdCd∗ A = 1 R(λ) T (λ) (6.12) λ − −

for a given polarisation, wavelength and incident angle (Macleod, 2001; Powles, 1999).

This algorithm to compute Equations 6.2.1-6.2.1 was written into a c++ class Stack.h and imbedded within the solar class. The algorithm shows a 100% correlation to TFCalc, a commercial thin film software package. Infi- nite layers of thin films can be created with complex refractive indexes. The absorption, reflection and transmission of the s (transverse electric polarisation) and p (transverse magnetic polarisation) components can be modelled. The class is also written within the solar namespace. A copy of the code is given in the Appendix and on the accompanied CD-ROM.

6.2.2 Normal incident simulation

For simulations we consider an oxide/nitride stack (Figure 6.1) where the solar energy is incident normal to the surface, and with the objective to minimise the reﬂection losses.

Due to the quantum nature of photovoltiac cells, photons with energy less than the bandgap of the semi-conductor (1.1 eV 1.13 µm for silicon), ≈ do not contribute to the photo-generated current and photons with energy greater than 1.1 eV only contribute 1.1 eV of potential. Therefore simply considering the total energy that is reflected is insufficient to maximise the available energy. The important quantity is the photon flux of the incident CHAPTER 6. THIN FILM SIMULATIONS 87

Figure 6.1: An oxide/nitride stack on silicon and encapsulated under pottant. ni are the refractive indices of each layer li of thickness di. solar spectrum that have energies greater than the bandgap of the semi- conductor. The solar weighted reﬂectance for a give wavelength (SW Rλ) is deﬁned as,

SW Rλ = RλΦpλ, (6.13)

where Φpλ is the photon ﬂux at a given wavelength. the total solar weighted reﬂectance (SW R), for the case of normal incident light, is given by:

λ2 R Φ dλ λ1 λ λ SW Rnorm = , (6.14) λ2 Φ dλ % λ1 λ % where λ1 = 0.295 µm, below which negligible solar radiation is incident on the Earth and λ2 is the bandgap of silicon. CHAPTER 6. THIN FILM SIMULATIONS 88

Refractive indices for both silicon and silicon nitride are taken from Green (1995) and the refractive indices of silicon oxide and pottant are assumed to be non-dispersive and equal to 1.46 and 1.4, respectively.

The photon flux was generated using the algorithms described in Chapter 5. All the default variables were accepted except the photon flux (input = 2, Table 5.1) and the optical path length was set to 1.5 . The generated energy flux for the extraterrestrial, diffuse, direct and global spectra are illustrated in Figure 6.2 and Figure 6.3.

From Equation 6.14, the theoretical SW Rnorm was calculated for a range of oxide and nitride thicknesses. Figure 6.4 illustrates the results as a high resolution contour plot where the solar weighted reflectance is as a function of both oxide and nitride thicknesses for an oxide/nitride stack under pottant. The maximum reflection shown in Figure 6.4 is 37.0%, which occurs with 96 nm of oxide underneath 116 nm of nitride. The minimum reflectance is 6.9%, which occurs with no oxide and 76 nm of nitride. For a 20 nm oxide between the silicon and the nitride, the minimum reflectance is 8.7%, which occurs with 54 nm of nitride. A thin oxide is necessary to achieve good surface passivation when LPCVD is used for nitride deposition.

6.2.3 Full day simulation of the direct beam radiation

The internationally accepted method for comparing the performance of silicon solar cells is to compare the output under airmass 1.5 direct (AM1.5D) radiation at normal incidence as used in the simulations in Section 6.2.2. This method does not necessarily indicate the performance of a silicon cell under real conditions. Non-tracking, single concentration cells experience radiation impinging on the surface from a full compliment of angles. Similarly, the spectral properties of the incident solar radiation would rarely match that of the AM1.5D spectrum.

In real systems the impinging radiation can be classed into two main groups: diffuse radiation (including Albedo radiation) and the direct beam. Diffuse radiation is principally isotropic in the atmosphere and usually contains only a small portion of the energy flux in comparison to that of the direct beam. The reflected component of the diffuse radiation and the Albedo radiation is heavily site specific and thus does not allow for any generic optimisation to be made of this component. So, although there exists an optimum anti- reflection coating for both the diffuse and Albedo radiation, it would produce CHAPTER 6. THIN FILM SIMULATIONS 89

Figure 6.2: The solar ﬂux for radiation for AM1.5D, including the extraterrestrial, global, diﬀuse and the direct beam as calculated using NREL’s Spec- tral2 (Bird, 1984) embedded within TSC.

Figure 6.3: Integral ﬂux of the solar insolation from 7am till 5pm. CHAPTER 6. THIN FILM SIMULATIONS 90

Figure 6.4: Theoretical values for total SW Rnorm for normal incidence sunlight as a function of oxide and nitride thickness for an oxide/nitride stack on silicon and under pottant. The colourmap indicates the SW Rnorm. a second order eﬀect on the ultimate performance of a photovoltaic receiver compared to that of the direct beam insolation. The results in this paper therefore only consider the case of direction beam radiation.

We are primarily interested in considering the light impinging on a ﬂat plate that is inclined to the declination of the sun, such that the plane of the sun’s path is perpendicular to the plane of the plate. This design would be the typical orientation of a non-tracking ﬂat plate photovoltaic collector. To correlate this with the direct beam insolation, the magnitude of the direct beam insolation is multiplied by the cosine of the solar angle and the photovoltaic’s surface normal, giving the actual radiation that would fall on one square meter of absorber at any time of the day.

Combining Equations 6.10 and 6.14, and investigating the performance of the cell over a full day, Equation 6.13 becomes,

λ2 R Φ dλdt day λ1 λ pλ SW Rday = . (6.15) λ2 Φ dλdt % day% λ1 pλ % % CHAPTER 6. THIN FILM SIMULATIONS 91

Again using the code from Chapter 5 the photon flux was for a full day of operation at summer solstice, gaining the total efficiency of the thin film for the dynamic system. The direct insolation values used as a function of time are presented in Figure 6.5.

Using the same oxide/nitride stack as was used for the normal incidence simulation, the SW Rday of the multi-layer was calculated using Equation 6.15. The result is shown in Figure 6.5. For the full day simulation the minimum reﬂectance is 7.2%, which again occurs with zero oxide thickness, but with a nitride thickness of 78 nm. Using a 20 nm thick oxide to provide surface passivation, the minimum reﬂection loss is 9.0%, which occurs with a 56 nm thick nitride layer.

Figure 6.5: Theoretical values for the SW Rday as a function of oxide and nitride thicknesses.

6.3 Comparison to experimental results

A range of films were grown to compare the theoretical values with real nitride/oxide films on silicon substrates. A spectrometer was used to measure reflectance as a function of wavelength. Figure 6.6 shows the measured and theoretical reflectance values for an oxide thickness of 17nm and a nitride CHAPTER 6. THIN FILM SIMULATIONS 92 thickness of 43nm. The theoretical SWR was calculated using the theory described earlier only replacing the infinite pottant layer with an air layer. The discrepancy between theoretical and measured values above 1000 nm is due to the finite thickness of the sample and reflection of long wavelength light from the rear of the wafer. There is a slight discrepancy between theoretical and measured reflectance curves for wavelengths below 1000 nm. Never the less, in the wavelength range 300-1000 nm, both theory and experiment show a total SWR of 9.6%.

Figure 6.6: Comparison between the experimental and theoretical reﬂection

6.4 Surface passivation properties

In addition to having good anti-reflection properties, the top surface of a silicon solar cell should also be well passivated. In order to determine the surface passivation provided by our anti-reflection coating, two float-zone, 100–400 Ωcm, p-type and (100) orientated wafers were prepared with a light phosphorous diffusion, thin thermal oxide (grown at 900◦C for 1 hour with a 30 minute nitrogen anneal) and a 30 minute forming gas (5% hydrogen in argon) anneal, at which point the surface was well passivated. The sheet resistance immediately after phosphorous diffusion was 210Ω/! and the oxide CHAPTER 6. THIN FILM SIMULATIONS 93 was 20 nm thick. Using low pressure chemical vapour deposition, with a deposition temperature of 750◦C, deposition pressure of 0.6 torr and a flow ratio of dichlorosilane:ammonia of 1:4, a 74 nm thick silicon nitride layer was deposited.

Effective lifetime measurements were made at a range of injection levels using the quasi-steady-state photoconductance (QSSPC) apparatus developed by Sinton. The data was analysed to extract both effective lifetimes and emitter saturation current using the equations of Sinton and Cuevas (1996) and Nagel et al. (1999). Effective lifetime measurements were also used to determine the implied Voc values Cuevas and Sinton (1997). Since a thermally grown oxide with a forming gas anneal provides good surface passivation, the purpose of this experiment was to determine the effect of nitride deposition on the wafer. Measurements were made immediately prior to and just after nitride deposition. The Joe, implied Voc and effective lifetime at an injection level of 1 1014 / cm3 are shown in Table 6.1. These results show that the high × effective lifetime and excellent surface passivation that were achieved with the thin oxide layer were maintained after the deposition of a thick nitride layer that would be suitable as an ARC.

Property Before nitride deposition After nitride deposition Wafer A Wafer B Wafer A Wafer B 2 14 14 14 14 J (A/cm per side) 1.5 10− 1.4 10− 1.5 10− 1.4 10− oe × × × × Implied Voc (mV) 695 691 698 697 Eﬀective lifetime (ms) 10 5 2.9 2.2

Table 6.1: Joe, implied Voc and eﬀective lifetime (measured at an injection level of 1014/cm3) of two samples before and after LPCVD of silicon nitride.

6.5 Implications for solar cell design

Table 6.2 shows the optimum nitride thickness for fixed oxide thicknesses of 0 and 20 nm and the total solar weighted reflectance for the case of normally incident light and a full day simulation. In both cases, the optimum nitride thickness was found to be slightly greater in the case of a full day simulation. Although there exists a change in performance of the two optimised surfaces, the film thicknesses are almost identical. Absolute variations of only one percent can be witnessed between sunlight incident at normal to the surface and sunlight representing a full day’s simulation (Figure 6.7). (SW Rdiff is the difference between Figures 6.4 & 6.5.) CHAPTER 6. THIN FILM SIMULATIONS 94

Figure 6.7: Theoretical values for the SW Rdiff as a function of oxide and nitride thicknesses.

Figure 6.8: The absolute percentage diﬀerence in the SWR predicted between simulations of a winter and summer solar spectrum for an oxide/nitride stack over a full day. CHAPTER 6. THIN FILM SIMULATIONS 95

This small variation in optimised film thicknesses for both of the simulations conducted is due to a range of factors. Primarily, similarities in performance can be attributed to the fact that the solar distribution contains a broad spectrum of wavelengths. A thin film stack of just two layers is not sufficient to create an ideal ARC for broad spectrum radiation (Macleod, 2001; Turner and Baumeister, 1966). Any decrease in performance due to the non-normal radiation is negated by the broad spectum radiation leaving similarities in the optical performance. Secondly, the fact that a greater proportion of the solar insolation occurs on or about normal to the surface, again reduces the difference between the normal incidence simulation and the full day simulation.

A further simulation was conducted to examine the performance of ARCs under different solar spectra resulting from a seasonal variation. A summer solstice day simulation was compared to that of a winter solstice simulation. Variations of less than 0.1% were observed from the theoretical simulations (Figure 6.8). The conclusion that can be drawn from these results is that a surface optimised for AM1.5 at normal incidence for an ARC of a few thin film layers is sufficient to address the overall performance a flat plate collector under real conditions.

Minimum reﬂ. loss With surface pass. Normal Day Sim. Normal Day Sim. Oxide thickness (nm) 0 0 20 20 Nitride thickness (nm) 76 78 54 56 SWR 6.9% 7.1% 8.7% 9.0%

Table 6.2: Results of the optimisation of the nitride/oxide thin ﬁlm stack for various simulations.

6.6 Conclusions

We have modelled reflection losses from a silicon wafer with a thin silicon oxide and an overlying silicon nitride anti-reflection coating, for the cases of normal incidence and full day direct beam radiation. The results show that an ARC optimised against the standard solar spectrum (AM1.5) at normal incidence is sufficient to address the overall performance of flat plate collectors. We have also illustrated that the two layer oxide/nitride stack is an effective ARC for single junction silicon solar cells. An oxide/nitride stack on silicon may be obtained using a thermally grown oxide and low CHAPTER 6. THIN FILM SIMULATIONS 96 pressure chemical vapour deposition of silicon nitride. We have shown that this results in excellent surface passivation, with Joe values before and after nitride deposition.

This chapter also constructed an algorithm to simulate the performance of thin films and included it within the solar class. The solar class now contains simulations of the spatial and spectral energy distribution, an algorithm to calculate the solar vector, the broadening of the distribution that occurs when a solar distribution is reflected off a non-ideal mirrored surface and an equation to simulate thin and thick optical films. Using these functions the next chapter goes on to illustrate some of the uses of these algorithm for the case of linear Fresnel concentrators. Chapter 7

Optical considerations in line focus Fresnel concentrators

Line focus Fresnel concentrators show great potential in reducing the capital cost of large scale solar concentrating systems. While not providing the most eﬃcient system for solar energy to electrical conversion their costs are consistently low enough to compete against more established forms of power production, though this still remains to be fully demonstrated.

This chapter investigates the optical performance of line focus Fresnel concentrators with regard to the collector field design. Array end-effects along with various layout parameters are examined to illustrate an optimum optical configuration. Using the algorithms developed in Chapter 5, the flux distribution in the absorber plane are also investigated and conclusions drawn. The results identified within this chapter were presented at the 11th SolarPACES International Symposium on Solar Thermal Concentrating Technologies held in Zuric¨ h (Buie et al., 2002)

97 CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 98

7.1 Literature review

As mentioned in Chapter 1, one of the greatest tasks that we have in creating viable solar energy collection systems is that of reducing the cost of such systems, so as to compete economically against established forms of power production. The solution to this problem does not necessarily lie in creating the most efficient system for solar to electrical conversion (say in terms of ground coverage or mirror area), but more so in the development of a system that has the lowest lifetime cost per megawatt-hour of electricity produced. A linear Fresnel reflector (LFR), has a single (or multiple) fixed linear absorber illuminated by a one-dimensionally tracked Fresnel mirror field. They have potential for greatly reducing the initial cost of establishing solar power production. LFRs may have lower system efficiency than other concentrating geometries, but their likely reduced cost may more than compensate, providing a solution for cost-effective solar energy collection on a large scale.

The concept of using a tracking reflector field to concentrate solar energy onto a single fixed absorber, thus removing the dependence on large mirrors for increasing the solar concentration, can be first attributed to Baum (1957) in the first publication of Solar Energy. The first person to apply this principle in a demonstration large-scale system though was Francia (1968) from the University of Genoa in Italy in 1968. Francia experimentally and theoretically addressed line focus and point focus systems stating that, on a large scale, LFRs were feasible for solar concentration ratios of approximately 100 suns, and comparable in cost to point focus systems.

It was not until 1979, under a contract from the United States Department of Energy, that a formalised treatment of LFRs took place. This treatment came at a time when there was growing interest in high concentration point focus systems under the US Central Receiver programme. Canio et al. (1979) from the FMC Corporation addressed the beneﬁts, design consideration and objectives in building large-scale linear systems compared with point focus designs. Canio et al. (1979) noted that system modularity, low tower cost, single axis tracking control, maximum usage of ground area, linear ﬁeld symmetry, and the adaptability of heliostats, receiver and tower to automatic factory production, transport and installation, were the main drivers in the reduction of cost. They argued:

“... the line focus concept is a feasible and competitive alternative for solar thermal generation of electric power.” CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 99

Canio et al. (1979) ﬁnal report was over 300 pages long and was written with the intention of building a trial 10 MW test plant followed by a full scale 100 MW solar thermal power facility. The proposed plant had a mirror ﬁeld on one side of a 1.69 km long linear cavity absorber mounted on a 61 m high tower. Unfortunately the construction bid for this design was unsuccessful. Instead, in 1982 the Solar One plant was constructed; a 10 MW two-axis tracking central receiver solar power plant which had eight years of successful operation and led to the very successful Solar Two (Vant-Hull, 1991).

The first large scale LFR was constructed in 1991 by the PAZ company at the Ben-Gurion Solar Electricity Technologies Test Center in Israel (Feuer- mann, 1993). This was a one-dimensional linear tracking Fresnel reflector field directing radiation onto an elevated secondary reflector, then to a linear tubular absorber. Unfortunately the system had serious optical problems resulting in very low solar to thermal efficiency. The shortcomings of the system were due to construction errors in the mirror field: the curvature of each of the mirrors was not within tolerance and there was mis-alignment of the mirror rows. No real performance data were published from the trial of this LFR but indications of the tolerances that are required to operate such plants were established.

Mills and Morrison (2000) took the next step in optimising the geometrical optics of LFRs and used a multiple tower configuration called the Compact Linear Fresnel Reflector, to increase the optical efficiency. Mills and Morrison theorised that individual mirror rows would have the option of directing reflected solar radiation to two linear receivers on separate tower lines. They stressed that the increased degree of freedom in mirror orientation allowed more closely packed mirror rows almost eliminating shading and blocking of consecutive mirrors.

More recently the Belgian company Solarmundo beginning in 2001 have been operating a 2500 m2 prototype linear Fresnel collector for steam generation in Liege, Belgium (Ha¨berle et al., 2001). Operating both in hybrid and pure solar modes, Solarmundo have been the ﬁrst to successfully demonstrate the feasibility of design and operation for linear Fresnel concentrators. The optical ﬁeld has been optimised against cost of production and more competitive prices for renewable energy have been suggested.

This chapter illustrates the performance of different field designs in single tower LFRs. Firstly, the lateral displacement of the solar radiation inherent in LFRs is quantified and an example is shown to minimise this problem. CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 100

The layout of the mirror ﬁeld is then optimised against mirror area to demonstrate the compromises in performance for various design strategies. Finally a discussion is presented on the optics of single tower LFRs with regard to ultimate performance and feasibility.

7.2 Collector ﬁeld end-eﬀects

In considering axis tracking systems, there exists a fixed path between the mechanical centre of the heliostats and the absorber over which the reflected solar radiation must always travel. That is to say, there are sufficient degrees of freedom in the mirror module to ensure a single focus point for a full day of operation. However, linear concentrators, which have only one degree of freedom in their motion, experience a displacement (an end-effect Send) between the reflected solar radiation and the imaging plane (Figure 7.1).

Figure 7.1: The end-eﬀect of a linear Fresnel concentrator

The end-effect must be addressed for two reasons: Firstly, it increases the path length between the mirror and the absorber, increasing the size of the solar image in the absorber plane. Secondly, it leaves one of the ends of the absorber without illumination for a portion of the day in a rectangular field layout. The first reason is important in determining the absorbers width in the course of optimising the performance of a solar array but is not addressed in this paper. The second can be alleviated by increasing the length of the CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 101

Analytical Computational Difference Distance from End Effect (m) End Effect (m) (m) Absorber (m) 13.24 13.25 0.010 2.8 13.92 13.93 0.010 5.51 15.07 15.08 0.011 8.3 16.61 16.62 0.012 11.0 18.44 18.45 0.014 13.8 20.47 20.49 0.015 16.6 22.66 22.68 0.017 19.3 24.96 24.98 0.019 22.1 27.35 27.37 0.021 24.9 29.80 29.82 0.023 27.6

Table 7.1: Comparison of the end-eﬀect of a line focus Fresnel system mirror ﬁeld with regard to the length of the absorber to ensure that it is long enough to illuminate the whole absorber during a year of operation.

The actual linear displacement for each mirror row is dependent on: the distance the mirror is from the tower d, the height of the tower h, and obviously the position of the sun in the sky. Analytically, the projection of the solar vector onto a single plane can approximate this. For a north-south field, if we make the approximation that the angle κ from Figure 7.1 is equal to the angle the reflected solar vector makes with the north-south plane, then the end-effect can be given by,

cos(θ)cos(φ) S = √h2 + d2 (7.1) end sin(θ) where θ and φ are the elevation angle and azimuth angle of the sun respectively. Alternatively the end-effect can be solved computationally using a computer model (discussed in Section 7.3.1). Table 7.1 illustrates the strong correlation between Equation 7.1 and the computational solution. The ray- trace simulation does nothing more than specularly reflect the solar vector about an individual mirrors surface normal and establishes the point at which the reflected beam strikes the imaging plane.

The compensation required for a north-south oriented field to ensure that the absorber is always illuminated for the full duration of a year is depicted in Figure 7.2. The specific simulation is of a plant positioned on the Tropic of Capricorn, the absorber plane is 13.5 m off the ground, and the lines represent the increase in length for both the northern and southern extremities of the mirror rows at various distances from the base of the absorber. If the solar CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 102

Figure 7.2: The end-effect compensation of a LFR concentrator represents a full year calculation and a summer trial for an equatorial north-south mirror field in the Southern Hemisphere. Eg. For a mirror row that is a distance of 10 m from the base of an absorber, the mirror row must extend 17 m to the northern end of the absorber and 5 m to the southern end. (The southern extension is due to the fact that the sun travels south of the east-west line in the winter months). array were 500 m long, sustaining 1.5 hectares of mirrors area, the extra mirror surface required to illuminate the absorber would account for less than 3% of the total mirror area of the field. LFRs with longer absorber lines will have even smaller reflector penalties.

7.3 Field design

7.3.1 Linear simulations

To carry out the simulations within this chapter, a single tower LFR array was created using the framework of the vecmath library developed in Chapter 5. Long rows of parabolic mirrors running exactly north-south or CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 103 east-west were modelled perfectly tracking the diurnal path of the sun onto the receiver. The mirrors had a finite width. An absorber was also created that was horizontal and flat and oriented parallel to the mirror modules. The absorber was wide enough to accept all of the radiation that was impinging onto that surface and considered initially as a black body and then using a selective surface. The solar insolation, position and distribution was generated using the solar.h libraries. The default settings were accepted, except that the solar array was positioned on the Tropic of Capricorn. The direct beam insola- 2 tion for the tropical location was 889 Wm− tracking normal and an average 2 of 704 Wm− during the course of the day. Each of the solar vectors was traced through the array, reflected off the mirrored surface. The interactions between adjacent mirrors including blocking and shading effects of solar radiation were considered. The simulations were to determine the performance of a solar array based on the mirror placement. The resulting energy in the absorber plane per m2 of mirror area was calculated along with the total amount of energy reaching the absorber plane for a finite solar array.

7.3.2 Field layout

In our simulations a single absorber design and a series of field designs are trialed. The first is a single absorber aligned north south with a parallel mirror field placed only on one side of the tower. Due to the symmetry of the field the output from either the west or east portions of the field, over the course of a day, gives similar results to its opposite half and including both only increases the running time of the simulation. The second absorber design trialed was an east west configuration. Here the array is placed such that the sun passes directly along the central line of the absorber during the course of the day to maintain the symmetric relationship between both the northern and southern portions of the field. For an equatorial mirror field this would only occur twice a year, but the simulation adequately illustrates the performance of the absorber configuration. The field layouts trialed are:

• 100% ground coverage - where mirror rows are placed next to each other so none of the ground could be seen if all the mirrors are placed horizontally. CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 104

• Compact ﬁeld design - each consecutive mirror away from the absorber is placed at the smallest distance away from the absorber with the condition that at no time can the reﬂected solar radiation be blocked by its adjacent mirrors.

• Energy optimised ﬁeld design - each mirror placed in the ﬁeld is optimised such that its placement with respect to each of its adjacent mirrors, the tower and the sun, produce the greatest possible energy output per unit mirror area for the entire array.

Two independent simulations were undertaken. Firstly, an arbitrary array is considered where the size of the mirrors in the field are calculated in terms of the total angular displacement the last mirror is from the central absorber position. This removes the performance of the field on a specific tower height and highlights the dependence of this performance on the tower height to field area ratio. (A field size of x degrees means that, from the downward viewpoint of the absorber plane, the outer mirror centre is positioned at x degrees from the vertical). The size of each of the mirrors in the array is not important for the ultimate performance of the field and is addressed in the discussion. The second simulation used a solar array with a 15 m high / 500 m long absorber that is completely illuminated for the full day of simulation for both north-south and east-west field simulations. This case was included to illustrate the performance of a more realistic solar array.

7.3.3 Spacing results

The results of the computational simulations of the mirror positions are presented in Figures 7.3 & 7.4 and are discussed in the following sections. The mirror ﬁeld is aligned north-south unless otherwise stated.

Width of the mirror modules

The average performance of each of the mirror modules in an array, based on the total ground area the mirror field would occupy (determined by the angle subtended between the outermost and innermost mirror modules as seen from the absorber) for each of the field design strategies mentioned previously is illustrated in Figure 7.3. Three simulations using a 100% ground covered field were conducted each using mirrors with different widths (0.5 m, 1m and 2m wide mirror modules). CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 105

Figure 7.3: The average power produced per m2 of mirror area required for different field strategies given an angular field width from the absorbing plane.

Clearly, the width of each of the mirror modules in the array does not greatly affect the ultimate performance of the solar array. For a smaller field, the larger mirrors have a greater efficiency than the smaller mirror modules. This anomaly is created because the effective ground area that the array is reflecting increases due to the real size of the mirrors for low solar incidence angles. As the size of the mirror field is increased this apparent increase in field size is reduced and the resulting performance approaches that of the smaller mirrors. Using these results, the width of the parabolic mirrors used in LFR arrays should be based around the limitations placed on: maintaining the image quality at the absorber, weight and modularity, ease of tracking and cost, and not on optical performance as no noticeable performance gains are exhibited for a specific mirror width in reasonably sized fields. In real arrays, the optical performance with smaller absorber widths can be degraded with the use of wide parabolic reflectors because astigmatism is introduced.

Best ﬁeld design

The total amount of power collected for a given field size and layout is illustrated in Figure 7.4. Using Figures 3&4 together, the greatest amount of CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 106 energy collected is from the 100% ground coverage as expected, but a good proportion of the mirror area is redundant due to blocking by adjacent mirrors. The compact field, where the mirrors are placed so as to eliminate the blocking of the reflected radiation, exhibits greater performance per mirror area and shows only a marginal drop in the total amount of energy reaching the imaging plane.

Figure 7.4: The average daily output from a linear Fresnel concentrator given various field strategies and field widths. The x-axis corresponds to the actual width of the mirror field either side of a single absorber. No thermal or reflection losses are considered, and the absorbing plane was positioned 15 m from the ground.

The optimal performance of the mirrors was achieved by a third mirror layout. Here extensive iterations were conducted to achieve optimum mirror positions in the field. The rapid decrease in performance with increasing field width for the more compact mirror fields is not seen in the optimal design. Understandably, there is a corresponding decrease in the total energy reaching the imaging plane but per mirror module, this decrease can be justified. Using the optimum field design 85% of the power can be collected using 80% of the reflector area of the compact mirror field. This increase in optical performance comes at the expense of increased ground area occupied CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 107 by the mirror field and in high-density areas, a compromise between the two designs could be made.

East-west ﬁeld

For the design of an east-west field, neither the mirror performance nor the amount of energy reaching the imaging plane could compete against its alternative north-south field. Secondly, the increased size of the mirror field required to illuminate the entire absorber after calculating the end-effects is noticeably bigger than for the north-south case.

7.4 Curved absorber simulation

An additional simulation was conducted where the ﬂux distribution in the imaging plane was considered. This was to demonstrate some of the optical considerations that also need to be taken into consideration when building and optimising similar arrays.

The abstract solar array consisting of 20 parabolic line focus mirrors with an aperture of 2 m each, centered about a single tower 13 m above the rotational axis of the mirrors was generated. The tower houses the absorber plane that runs parallel to the ground. Each mirror is free to direct its radiation to any point in the absorber plane. Adjacent mirrors are placed such that there is no gap between mirrors if both are facing directly upwards (compact ﬁeld design). The focal length of each of the mirrors is set to the minimum path length between the rotational axis of each mirror and the center of the absorber plane. The viewing half angle of the absorber is then approximately

57◦ to the outermost mirrors. The location and time of the simulation was defined as: Sydney, Australia for the 1st of January, 2000 at 13.00 - 11 UT. The direct normal radiation has a circumsolar ratio of 0.05. The discrete representation of the solar image was reduced to 10 radial profiles of 50 points in each limb (as opposed to the default value of 100 radial profiles in Chapter 5). The reflective surfaces are defined as having a reflectivity of unity and a distribution of surface errors described by a radial Gaussian with a standard deviation of 3.5 mrad.

For the first case a flat plate absorber one meter wide, acting as a perfect black body was placed in the imaging plane. The absorber is illuminated using an aiming strategy where each mirror’s reflected radiation is directed CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 108

Figure 7.5: Incident and absorbed flux profile on a flat plate thermal absorber with and without a selective surface, illuminated from a linear Fresnel concentrator. to achieve a smooth (flat) distribution. If you consider a portion of the absorber that would be illuminated by all of the mirror modules then a line profile is sufficient to evaluate the absorber plane (Figure 7.5 black line). (The aiming points across the absorber for each of the 20 mirrors in order are: -0.0, -0.05, -0.15, -0.25, -0.35, 0.0, 0.1, 0.2, 0.3, 0.4, -0.4, -0.3, -0.2, -0.1, -0.0, 0.35, 0.25, 0.15, 0.05, 0.0. The outer mirrors do not have a large range of aiming position due to their larger image size).

The optical efficiency of the system is 74%, (calculated from the amount of energy that falls on the the absorber plane, divided by the direct normal insolation multiplied by the mirror aperture area). The optical efficiency includes both the cosine effect of the Fresnel mirrors that aren’t normal to the direct beam and the blocking of the reflected beam created from the close packing of mirrors. It should be noted that this is not the ideal configuration for the optical design of a Fresnel mirror system (Buie et al., 2002), but represents a particular (typical) flux distribution. CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 109

7.4.1 Selective coating

Real systems do not have black body absorbers. For a thermal absorber a variety of materials are used to increase the conversion eﬃciency of collected solar radiation. For these simulations we chose to address the performance of various absorbers with a cermet (ceramic metal composite selective surface). The typical absorption characteristics as a function of angle can be simulated by the equation Zhang and Mills (1992):

α = 0.91776+0.11782θ 1.1108θ2+3.60018θ3 5.30095θ4+3.60829θ5(7.2) − − where theta is the angle of incidece(in radians) of a beam of radiation to the absorber surface normals.

Figure 7.6: Angular response of Zhang and Mills (1992) cermet

For the optical configuration of a flat plate receiver in a line focus concentrator, the greatest angle of incidece is approximately 57◦, therefore, for the majority of the incident radiation on the absorber has an absorption of 90% or more (Figure 7.6). After the selective surface properties of the cermet were included simulating a real absorber, 90.5% of the available radiation is absorbed by the flat plate (Figure 7.5). Note that no cover losses are assumed, and hence no secondary rays can be absorbed. In practice some sort of convective suppressive device is used, introducing a covering material such as glass or plastic. CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 110

7.4.2 Corrugated absorber

As an alternative design to a flat plate absorber, an absorber with macro-scale roughness is considered. A number of discrete corrugated shapes were created to investigate the effect of a such absorbers on the optical efficiency and flux distribution. Exposed pipes formed the lower portion of the profile while the gaps between were created by another semi-circle of a different diameter. Pipe diameters of between 0.01 m to 0.07 m in steps of 0.002 m were tested against equivalent spacings leaving a total of 961 individual absorber profiles (an example of two are shown in Figure 7.7)

Figure 7.7: Example of two corrugated absorbers. Absorber 1 has a pipe diameter of 0.07 m and spacing of 0.07 m, absorber 2 has a pipe diamter of 0.06 m and a spacing of 0.02 m

7.4.3 results

Again using the infrastructure developed in Chapter 5, the solar array was traced through the optical design and the flux distribution in the imaging plane was investigated. Each was reflected within the curved absorber and multiple bounces of each vector (where appropriate) were considered. Figure 7.8 shows the resulting flux distribution. Where the curved absorber is flat the flux distribution approaches that of the flat plate as expected. The average lower flux distribution can be attributed to the increase in surface area of the wavy absorber.

A range of pipe spacings were simulated and the optical performance of the absorber design determined (Figure 7.9). The reference to the number represented in Figure 7.9 is that a ﬂat plate with a selective surface absorbs 90.5% of the incident radiation. CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 111

Figure 7.8: Flux distribution on a corrugated absorber

Figure 7.9: Performance of a corregated absorber coated with a selective surface, represented by various pipe spacings. The colourbar represents the solar weighted absorption. The reference case is a ﬂat plate selective surface absorbing 90.5%. CHAPTER 7. LINE FOCUS FRESNEL CONCENTRATORS 112

Clearly, the a wavy designs increase the amount of radiation absorbed by the absorber, averaging over 94% of all incident radiation almost 4% above that of the ﬂat plate. Obviously it must be taken into account that the increasing size of the absorber will increase the radiating surface area and a trade oﬀ will need to be taken with the thermal design.

7.5 Conclusion

Within this chapter, a range of optical conditions for a linear Fresnel concentrator have been generated. Using the algorithm developed in Chapter 5, the end-effects associated with linear systems was presented. Similarly, the performance of an array based on the position of its mirror fields was gained. Three types of field spacing were trailled where the optimum was dependent on what was your performance criterion, energy or optical efficiency. Finally, a complex absorber line was considered. Chapter 8

Conclusion

This thesis describes a computational model of the terrestrial solar beam used in optical simulations of solar concentrating systems that has a strong correlation with observed data. The algorithms take into account: the small variations that exist in the spatial energy distribution created from atmospheric scattering; the systemic effect of reflecting that beam off a non-ideal mirrored module; the spectral energy distribution; and, a solar vector calculator. Accompanying these algorithms is code that simulates the transmission, absorption and reflection characteristics of thin and thick optical layers. Combined, these tools and overall framework can be applied to high detailed optical modelling of solar concentrating systems.

Initially a model was developed that simulates the terrestrial spatial energy distribution of the direct solar radiation. The model was based on the existence of a strong correlation when each of the sunshape proﬁles were represented according to their circumsolar ratios independent on location. The algorithm has a high correlation to all of the collected data and invariant to changes in geographic location when representing averaged observed solar proﬁles, implying the existence of a generic solar model based upon the circumsolar ratio.

The inﬂuence of variations in the solar energy distribution described by the individual circumsolar ratios, and the optical quality of the mirrored surfaces in concentrating systems, described by the standard deviation of the mirror errors incorporating non-specularity and non-idea mirror shapes, was quantiﬁed. The results are presented as a series of graphs covering the parameters over their typical ranges, allowing broad design considerations to be investigated for solar concentrating systems.

113 CHAPTER 8. CONCLUSION 114

A class library is then developed that contains an abstract representation of the terrestrial solar beam, including the systemic effect of reflecting that beam off a non-ideal mirrored surface. The code presented is an extension of the building blocks provided by both Blanco-Muriel et al. (2001) defining the solar position and Bird and Riordan (1986) simulating the solar spectrum and incorporates the sunshape model developed as part of this thesis. These algorithms represent an improvement over previous models due to its speed, advanced memory management, object oriented nature and a strong correlation with observed data.

The reflection losses from a silicon wafer with a thin silicon oxide and an overlying silicon nitride anti-reflection coating were then modelled, for the cases of normal incidence and full day direct beam radiation generated using the solar algorithm. The results showed that an anti-reflection coating, when optimised against the standard solar spectrum (AM1.5) at normal incidence is sufficient to address the overall performance of flat plate collectors. The algorithms to simulate both thin and thick optical layers were incorporated into the solar class to give a toolbox of functions that characterise the terrestrial solar beam, and the components that make up the optical system of a solar concentrator.

A range of optical conditions for a linear Fresnel concentrator were also generated. The end-effect of the linear array were quantified, along with the trade-offs of the mirror placement and the trade-offs of the absorber design. While not presenting a full example of all of the functions of the developed solar algorithm, it does demonstrate some interesting characteristics of linear Fresnel systems. Previously, simulations of this nature have not been required because the intensity distribution of the solar flux falling on a thermal absorber did not greatly affect the performance of a concentrating systems. Presently there are designs to place photovoltaic absorbers at the focal plane of large concentrating systems, where uniform flux distributions are essential to operation. Similarly, as the need for economically viable systems is paramount, optimising the optical performance of a system is an essential component of successful application of renewable energy.

These algorithms described in this thesis have the ability to investigate and optimise the solar flux fields again with a high correlation to observed data. Using a central tower receiver as an example, the movie flux.mpeg exam- ines the creation of homogenous fields for this solar concentrator. Series of high resolution flux maps are create about the imaging plane which are con- verted into a data volume containing all of the information with regards to CHAPTER 8. CONCLUSION 115 the incident solar flux. Using these data-sets, regions of homogeneity can be identified represented by isosurfaces of the solar concentration ratio. Sea- sonal, monthly and instantaneous flux distribution can be compared. More so, using these algorithms, optimised designs can be generated by combining these algorithms with more robust optimisation routines, which is the next step for this body of work. Bibliography

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File format of the Reduced Data Base

The Lawrence Berkeley Laboratories collated 200 000 radial energy ﬂux pro- ﬁles of the sun from 11 sites across the United States of America forming what was termed the Reduced Data Base (RDB).

1 Albuquerque STTF, NM 28971 2 Albuquerque TETF, NM 13851 3 Argonne, IL 9702 4 Atlanta, GA 38405 5 Barstow, CA 36632 6 Boardman, OR 4782 7 China Lake, CA 10683 8 Colstrip, MT 616 9 Fort Hood Bunker, TX 5150 10 Fort Hood TES, TX 8250 11 Edwards AFB, CA 27344

Table A.1: Lawrence Berkeley Laboratories Reduced Data Base

124 APPENDIX A. FILE FORMAT OF THE REDUCED DATA BASE 125

1 2 77/04/01 14:00 10 01 Time: 14:11 Alt: 49.02 Azi: 229.43 ESD: 0.9996 1 2 77/04/01 14:00 10 02 Flag Field: 00000 00000 00101 00000 00000 0000 1 2 77/04/01 14:00 10 03 Pyranometers. Trk:1241.8 1048.7 Hor: 922.5 794.1 1 2 77/04/01 14:00 10 04 Pyrheliometer: 928.6 (clear), filtered next line 1 2 77/04/01 14:00 10 05 31.8 62.4 -1.0 83.8 25.5 108.8 62.4 2.6 1 2 77/04/01 14:00 10 06 SolRad: 912.6 Circum: 20.5 C/(C+S): 0.0219867 1 2 77/04/01 14:00 10 07 Misc. ACR: 0.01724 NIP: 0.01965 CC: 6.794E+07 1 2 77/04/01 14:00 10 21 1.715E+07 1.706E+07 1.694E+07 1.674E+07 1.655E+07 1 2 77/04/01 14:00 10 22 1.616E+07 1.560E+07 1.478E+07 1.383E+07 1.133E+07 1 2 77/04/01 14:00 10 23 2.501E+06 1.464E+05 4.578E+04 3.117E+04 2.439E+04 1 2 77/04/01 14:00 10 24 2.095E+04 1.881E+04 1.713E+04 1.511E+04 1.389E+04 1 2 77/04/01 14:00 10 41 1.197E+04 9.781E+03 8.142E+03 6.822E+03 5.789E+03 1 2 77/04/01 14:00 10 42 4.988E+03 4.303E+03 3.726E+03 3.374E+03 3.092E+03 1 2 77/04/01 14:00 10 43 2.782E+03 2.480E+03 2.259E+03 2.073E+03 1.899E+03 1 2 77/04/01 14:00 10 44 1.781E+03 1.689E+03 1.574E+03 1.414E+03 1.287E+03 1 2 77/04/01 14:00 10 45 1.197E+03 1.138E+03 1.083E+03 1.056E+03 9.691E+02 1 2 77/04/01 14:00 10 46 8.898E+02 8.417E+02 7.986E+02 7.494E+02 7.086E+02 1 2 77/04/01 14:00 10 47 6.775E+02 6.483E+02 6.284E+02 6.020E+02 5.236E+02 1 2 77/04/01 14:00 10 48 4.627E+02 ======End of Brightness Data ======1 2 77/04/01 14:00 10 99****************************************************

Table A.2: A sample the data within the Reduced Data Base APPENDIX A. FILE FORMAT OF THE REDUCED DATA BASE 126

Flag Number Meaning 1 Steps out of tolerance 2 Filter wheel problem 3 Aperture in wrong position 4 Rain flap closed 5 Half speed scan 6 Quarter speed scan 7 Scan points outside tolerance 8 Solar guider off 9 Voltage reference comparison out of tolerance 10 Voltage reference comparison out of tolerance 11 Average noise outside tolerance 12 RMS noise outside tolerance 13 Pyheliometer diff. out of tolerance 14 Bad match, 1st aperture switch 15 Bad match, 2nd aperture switch 16 Scan failed 1st quality criterion 17 Scan failed 2nd quality criterion 18 Saturated scan (pathological) 19 Sun not centered in scan 20 Recouped scan 21 Solar radius fixed 22 Aperture drop fixed 23 Pyrheliometer reading fixed 24 Scan smoothed 25 Scan would not smooth on failed test 26 Record had parity error on Kennedy tape 27 Failed QTEST reality criterion 28 Failed QTEST effective radius criterion 29 By QTEST, radius can be fixed

Table A.3: Summary description of error and status ﬂags Appendix B

Justiﬁcation of the intercept of a line focus concentrator

It is necessary to address the angle of the surface normal of a linear tracking parabolic reflector so that the sunlight is reflected into the absorber plane. Let ˆı, ˆ and kˆ be the unit vectors in the east, north and upwards directions. Let T be the point which is the top of the tower and M a point on your reflector due east of the tower. Then the position vector of T relative to M is a linear combination of ˆı and kˆ. Let tˆ be a unit vector in this direction: say tˆ = aˆı + ckˆ. For any other point M’ on the reflector, this same vector tˆ gives the direction from M’ of some point on your collector. Let

sˆ = -ˆı + mˆ+ nkˆ (B.1) be a vector pointing in the direction of the sun. Project this onto the plane spanned by ˆı and kˆ (ie. remove the ˆcomponent) and let sˆ$ be the unit vector in this direction. So,

- n sˆ$ = ˆı + kˆ (B.2) χ χ where

χ = √-2 + n2 (B.3)

The reﬂector should be angled so that the normal to its surface points in the direction of the vector sˆ$ + tˆ. The unit normal is thus,

- + aχ n + cχ ˆ = ˆı + kˆ (B.4) ℵ χζ χζ

127 APPENDIX B. LINE INTERCEPT JUSTIFICATION 128 where

- n ζ = ( + a)2 + ( + c)2 (B.5) . χ χ

The formula for the vector vˆ giving the direction of the reﬂected light in terms of the vector sˆ and the normal is then, ℵ

vˆ = sˆ + 2(s.ˆ ˆ)ˆ (B.6) − ℵ ℵ -(- + aχ) n(n + cχ) = ( -ˆı mˆ nkˆ) + 2( + )ˆ (B.7) − − − χζ χζ ℵ

Since

a2 + c2 = 1 (B.8) and

-2 + n2 = χ2 (B.9) we have

n2 + -2 = χ2(a2 + c2) (B.10) and so

(χa + -)(χa -) = (n + χc)(n χc) (B.11) − − which gives

(χa + -)2 + (χc + n)2 = 2-(χa + -) + 2n(χc + -) (B.12)

Hence

1 vˆ = ( -ˆı mˆ nkˆ) + [(χa + -)2 + (χc + n)2]ˆ (B.13) − − − χζ ℵ

The coeﬃcient of ˆı in this is

1 (χa + -) - + [(χa + -)2 + (χb + n)2] (B.14) − χζ χζ APPENDIX B. LINE INTERCEPT JUSTIFICATION 129 and since

(χζ)2 = [(χa + -)2 + (χc + n)2] (B.15) this simpliﬁes to

- + (χa + -) = χa (B.16) −

Similarly the coeﬃcient of kˆ in vˆ is

1 (χc + n) n + [(χa + -)2 + (χc + n)2] = n + (χc + n) (B.17) − χζ χζ − = χc (B.18)

Thus

vˆ = χaˆı mˆ+ χckˆ = χtˆ mˆ. (B.19) − −

The plane through M and through the north-south line through the top of the tower (ie. the line of collectors) contains the vector tˆ and the vector ˆ, and so contains all linear combinations of these two. So the line through M in the direction of the vector χtˆ mˆ does indeed meet the collectors. − Appendix C

Computer code

C.1 C++ libraries

C.1.1 solar.h

#ifndef SOLAR_H #define SOLAR_H

/* Copyright (C), 2003 * Damien Buie, * * This program is free software. Permission to use, copy, * modify, distribute and sell this software * and its documentation for any purpose is hereby granted without fee, * provided that the above copyright notice appear in all copies and * that this permission notice appear in supporting documentation. * Damien Buie or the Solar Energy Group at the University of Sydney, * makes no representations about the suitability of this software for any * purpose. It is provided "AS IS" with NO WARRANTY. */

#include

#ifdef INCLUDE_VECMATH # include # include #endif

#include #include #include

SOLAR_BEGIN_NS typedef Sunpos Sunposd; typedef Sunpos Sunposf; typedef Spatial Spatiald; typedef Spatial Spatialf;

130 APPENDIX C. COMPUTER CODE 131

typedef Spectral Spectrald; typedef Spectral Spectralf; typedef Stack Stackd; typedef Stack Stackf;

#ifdef VECMATH_H # define xnormal VM_VECMATH_NS::Vector4(1.0,0.0,0.0,0.0) # define ynormal VM_VECMATH_NS::Vector4(0.0,1.0,0.0,0.0) # define znormal VM_VECMATH_NS::Vector4(0.0,0.0,1.0,0.0) # define xnormalf VM_VECMATH_NS::Vector4(1.0,0.0,0.0,0.0) # define ynormalf VM_VECMATH_NS::Vector4(0.0,1.0,0.0,0.0) # define znormalf VM_VECMATH_NS::Vector4(0.0,0.0,1.0,0.0) #endif

SOLAR_END_NS

#endif /* SOLAR_H */ solarconf.h

#ifndef SOLAR_CONFIGURE #define SOLAR_CONFIGURE

/*******************************************/ /************* USER EDIT *******************/ /*******************************************/

/* Include to allow * ’std::cout << vec;’ * ’std::cout << spatial;’ */ #define SOLAR_INCLUDE_IO

/* * Uses the namespace option. Not available in g++ version * prior to gCC 1.1.2 * */ #define SOLAR_INCLUDE_NAMESPACE

/* * The algorithm Sunpos and Spatial can be used (limited) APPENDIX C. COMPUTER CODE 132

* without using the vecmath library. However the Spatial * library uses the vecmath library as its basis and will * not be included with these functions * */ #define INCLUDE_VECMATH

/*******************************************/ /********** END OF USER EDIT ***************/ /*******************************************/

#ifdef INCLUDE_VECMATH # ifdef SOLAR_INCLUDE_IO # define VM_INCLUDE_IO # endif # ifdef SOLAR_INCLUDE_NAMESPACE # define VM_INCLUDE_NAMESPACE # endif #endif

#ifdef SOLAR_INCLUDE_NAMESPACE # define SOLAR_NS solar # define SOLAR_BEGIN_NS namespace solar { # define SOLAR_END_NS } #else # define SOLAR_NS # define SOLAR_BEGIN_NS # define SOLAR_END_NS #endif

/* * This program using the gCC Standard Template Library * and does not facilite using the older libraries */

/* * A type default is declared */ #define TYPE_DEFAULT double template struct solarUtil { // prefer to cmath for portability static T abs(T t) { return t > 0 ? t : -t; } static T sin(T x) { return T(std::sin(x)); } static T cos(T x) { return T(std::cos(x)); } static T atan2(T y, T x) { return T(std::atan2(y, x)); } static T acos(T x) { return T(std::acos(x)); } static T sqrt(T x) { return T(std::sqrt(x)); } static T pow(T x, T y) { return T(std::pow(x, y)); } static T log(T x) {return T( std::log(x) ); } static T exp(T x) {return T( std::exp(x) ); } };

#endif // SOLAR_CONFIGURE APPENDIX C. COMPUTER CODE 133

Spatial.h

#ifndef SPATIAL_H #define SPATIAL_H

#include #include

SOLAR_BEGIN_NS template class Spatial{ public: /* The constructor builds a solar distribution with 100 radial limbs with 50 points * along each limb about the znormal vector. The spatial extent of the solar * distribution goes out to 43.65 mrad. The distribution has a cirsumsolar * ratio of 0.05, and the whole distribion has been simulated after the reflection * off a mirrored surface with a stand deviation of surface errors of 3.5 mrad. */ Spatial( T b = 0.04365, T c = 0.05, T d = 0.0035, int e = 50, int f = 100): cirSolarLimit(b), cirSolarRatio(c), standDevSurfErrors(d), noRadialPoints(e), noSpurs(f) {};

/* Destructor */ ~Spatial(){ delete []rotatedSolarArray; delete []normalSolarArray; delete []weight; delete []theta; delete []thetaLim; delete []phi; }

/* Input the standard deviation of surface errors */ void putSigma(const T sigma){ standDevSurfErrors = sigma; }

/* Input the limit of the spatial energy distribution */ void putCirSolarLimit(const T csl){ cirSolarLimit = csl; }

/* Input the circumsolar ratio */ APPENDIX C. COMPUTER CODE 134

void putCSR(const T csr){ cirSolarRatio = csr; }

/* Define the number of point that make up the solar array */ void putArrayDim(const int nrp, const int ns){ noRadialPoints = nrp; noSpurs = ns; sizeSolarArray = nrp * ns; }

/* Generate the solar energy distribution */ void generate(void){ allocateDimensions(); allocateProfile(); generateSolarArray(); if (standDevSurfErrors > 0.0) broadenDist(); allocateWeight(); }

/* Generate the solar energy distribution * calculating the spatial distribution at each * of these angular displacement from the solar * vector and boundary regions */ void generate(const T externTheta[], const T externThetaLim[], const int n){ noRadialPoints = n; theta = new T[noRadialPoints]; thetaLim = new T[noRadialPoints];

for ( int i = 0; i < noRadialPoints; i++){ theta[i] = externTheta[i]; thetaLim[i] = externThetaLim[i]; }

allocateProfile(); generateSolarArray(); if (standDevSurfErrors > 0.0) broadenDist(); allocateWeight(); }

/* Generate the solar energy distribution * using this spatial energy profile */ void generate(const T externTheta[], const T externThetaLim[], const T externPhi[], const int n){

noRadialPoints = n; theta = new T[noRadialPoints]; thetaLim = new T[noRadialPoints]; phi = new T[noRadialPoints];

for ( int i = 0; i < noRadialPoints; i++){ theta[i] = externTheta[i]; thetaLim[i] = externThetaLim[i]; externPhi[i] = externPhi[i]; }

generateSolarArray(); if (standDevSurfErrors > 0.0) broadenDist(); allocateWeight(); APPENDIX C. COMPUTER CODE 135

}

/* Rotate the solar array to be about the solar vector */ void rotate(VM_VECMATH_NS::Vector4 solarVector){

if (solarVector.length() != 1.0 ) solarVector.normalize();

VM_VECMATH_NS::Vector4 axis; axis = solarVector + VM_VECMATH_NS::Vector4(0,0,1,0); axis.normalize();

T cosAngle = axis.dot(solarVector);

axis.cross( axis , solarVector ); axis.w = cosAngle;

VM_VECMATH_NS::Matrix4 mat; mat.set(axis);

for ( int i = 0; i < sizeSolarArray; i++) rotatedSolarArray[i] = mat * normalSolarArray[i];

}

/* Return the size of the solar array */ void size( int *i){ *i = sizeSolarArray; }

/* Return the size of the solar array */ int size(void){ return sizeSolarArray; }

/* Return the solar array about the znormal vector */ void getNormalArray(VM_VECMATH_NS::Vector4 array[], T we[]){ for ( int i = 0; i < sizeSolarArray; i++){ array[i] = normalSolarArray[i]; we[i] = weight[i]; } }

/* Return the solar array about the solar vector */ void get(VM_VECMATH_NS::Vector4 array[], T we[]){ for ( int i = 0; i < sizeSolarArray; i++){ array[i] = rotatedSolarArray[i]; we[i] = weight[i]; } }

private:

T calcIntensity(const double tempAngle){ /** Sunshape distribution defined by Buie et al. * "Sunshape distribution for terrestrial solar simulations" * Solar Energy 74(2):111-120 */

// Solar Disk limit APPENDIX C. COMPUTER CODE 136

const T solarDiskLimit = 0.00465; // Output variable T intensity; // Algorithm is for milliradians const T tempAngleMrad = tempAngle * 1000.0; // cirsumsolar region variables const T kappa = 0.9 * std::log( 13.5 * cirSolarRatio ) * solarUtil::pow( cirSolarRatio , -0.3 ); const T gamma = 2.2 * std::log( 0.52 * cirSolarRatio ) * solarUtil::pow( cirSolarRatio , 0.43 ) - 0.1; // Calculating the intensity if ( tempAngle <= solarDiskLimit ) intensity = std::cos( tempAngleMrad * 0.326 ) / std::cos( 0.308 * tempAngleMrad); else intensity = std::exp( kappa ) * solarUtil::pow( tempAngleMrad , gamma ); // Returning the value of the normalised intensity per steradian return intensity; }

void allocateDimensions(void); void allocateProfile(void); void generateSolarArray(void); void allocateWeight(void); void broadenDist(void); void gaussianWeight(const VM_VECMATH_NS::Vector4*,T*, const int); void gaussianDist(VM_VECMATH_NS::Vector4 *, int*,const T); void convolution(const VM_VECMATH_NS::Vector4*,const T*,const int ); public:

/* Angular size of the virtual solar array * cirumsolar angular limit (radians) * default value of 0.0436 rad */ T cirSolarLimit;

/* Cirumsolar ratio, defined as the ratio * of the amount of energy contained within the * solar disk, to the total amount of energy * contained within the solar disk and circumsolar * region (out to cirSolarLimit). * default value of 0.05 */ T cirSolarRatio;

/* The value of the error associated with regards * to non-specula reflection and surface normal * deviations. This number can also include the * tracking error only when there are a statistically * significant number of mirrors. * The number represents the standard deviation * of surface errors (Radians). * Default value of 0.0035 rad */ T standDevSurfErrors;

/* The virual solar array consists of a number of * spurs propagating about the solar vector (noSpurs), * with a noRadialPoints in each spur. Leaving the * total number of points in the solar vector as * sizeSolarArray. */ int noRadialPoints; int noSpurs; APPENDIX C. COMPUTER CODE 137

int sizeSolarArray;

/* The solar array is contained as a series of vectors * centered about the znormal array and generated using * create. The array can also be rotated to be about * the solar vector */ VM_VECMATH_NS::Vector4 *normalSolarArray; VM_VECMATH_NS::Vector4 *rotatedSolarArray;

/* Each of the vectors in the solar array has a * corresponding weight defined by the spatial * energy distribution of the sun (phi). * Descretely this distribution is defined at each * point theta some radial displacement from the * solar vector. Each of the regions are defined * by the solar displacment thetaLim*/ T *weight; T *phi; T *theta; T *thetaLim;

}; // end class Spatial template void Spatial::allocateDimensions(void) { const T ang = cirSolarLimit / (noRadialPoints-1);

/** Angular displacement of each vector */ theta = new T[noRadialPoints]; for ( int i = 0; i

/** Angular displacement of each vector boundary */ thetaLim = new T[noRadialPoints];

for ( int i = 0; i < noRadialPoints - 1; i++) thetaLim[i] = ( theta[i] + theta[i+1] ) / 2.0;

thetaLim[noRadialPoints - 1] = 2 * theta[noRadialPoints - 1] - thetaLim[noRadialPoints - 2]; } template void Spatial::allocateProfile(void) { phi = new T[noRadialPoints]; for ( int i = 0; i < noRadialPoints; i++) phi[i] = calcIntensity(theta[i]); } template void Spatial::generateSolarArray(void) { /* Define the maximum size of the solar array */ sizeSolarArray = noRadialPoints * noSpurs;

/* Dynamically allocate two arrays */ rotatedSolarArray = new VM_VECMATH_NS::Vector4[sizeSolarArray]; normalSolarArray = new VM_VECMATH_NS::Vector4[sizeSolarArray];

/* Initiate auxiliary variables */ APPENDIX C. COMPUTER CODE 138

const T ang = 6.28318530717956 / noSpurs; int counter = 0;

/* Generate the solar array */ normalSolarArray[counter++].set(0.0,0.0,1.0,0);

for( int j = 1; j < noRadialPoints; j++){ const T sinang = std::sin(theta[j]); const T cosang = std::cos(theta[j]); for( int i = 0; i < noSpurs; i++) normalSolarArray[counter++].set( sinang* std::cos(ang * i), sinang * std::sin(ang * i), cosang,0); }

/* Redefine the size of the solar array */ sizeSolarArray = counter;

} template void Spatial::allocateWeight(void) { weight = new T[sizeSolarArray]; T *noSter = new T[noRadialPoints];

const T TWOPI = 6.28318530717956;

for( int i=0; i

int counter = 0; T normalisingConstant = 0;

weight[counter++] = noSter[0] * phi[0]; normalisingConstant = noSter[0] * phi[0];

for ( int j = 1; j < noRadialPoints; j++){ for ( int k = 0; k < noSpurs; k++){ weight[counter++] = noSter[j] * phi[j] / noSpurs; normalisingConstant += noSter[j] * phi[j] / noSpurs; } }

for ( int i = 0; i < sizeSolarArray; i++) weight[i] /= normalisingConstant;

delete []noSter; } template void Spatial::broadenDist(){

int size = noRadialPoints;

VM_VECMATH_NS::Vector4 *gaussian = new VM_VECMATH_NS::Vector4[size*size]; gaussianDist(gaussian,&size, 3.0* standDevSurfErrors);

T *gaussWeight = new T[size]; gaussianWeight(gaussian,gaussWeight,size); APPENDIX C. COMPUTER CODE 139

convolution(gaussian,gaussWeight,size);

delete []gaussian; delete []gaussWeight; } template void Spatial::gaussianDist(VM_VECMATH_NS::Vector4 gaussian[], int *n,const T extent){

int counter = 0; const T unit = 2 * extent / (*n-1);

for( int i=0; i < *n; i++) { for( int j = 0; j < *n; j++) { gaussian[counter].set(-extent + i*unit, -extent + j * unit , 1,0); gaussian[counter++].normalize(); if (gaussian[counter-1].dot(VM_VECMATH_NS::Vector4(0,0,1,0)) < std::cos(extent)) counter--; } }

*n = counter; } template void Spatial::gaussianWeight(const VM_VECMATH_NS::Vector4 gaussian[],T gaussWeight[], const int n){

const T precalc = 2.0 * standDevSurfErrors * standDevSurfErrors; // precalculate

for( int i = 0; i < n; i++) { const T angle = std::acos(gaussian[i].dot(VM_VECMATH_NS::Vector4(0.0,0.0,1.0,0.0))); gaussWeight[i] = angle * exp( -angle * angle / precalc ); } } template void Spatial::convolution(const VM_VECMATH_NS::Vector4 array[],const T weight[],const int n){

T *tempPhi = new T[noRadialPoints]; T *energy = new T[noRadialPoints]; T *noSter = new T[noRadialPoints];

const T TWOPI = 6.28318530717956;

for( int i=0; i < noRadialPoints; i++ ) {

tempPhi[i] = 0.0; noSter[i] = TWOPI * ( 1.0 - std::cos( thetaLim[i] ) ); if ( i != 0 ) noSter[i] -= noSter[i-1]; energy[i] = noSter[i]*phi[i];

}

for( int i=0; i

VM_VECMATH_NS::Matrix4 rotMat; rotMat.rotY(theta[i]); APPENDIX C. COMPUTER CODE 140

for ( int j=0; j tempVec = rotMat * array[j]; const T angle = std::acos(tempVec.dot(VM_VECMATH_NS::Vector4(0,0,1,0))); int k = 0; while ( angle > thetaLim[k] ) k++; tempPhi[k]+= energy[i] * weight[j]; } }

for( int i=0; i

delete []tempPhi; delete []energy; delete []noSter; }

SOLAR_END_NS

#ifdef SOLAR_INCLUDE_IO template std::ostream& operator<<(std::ostream& o, const SOLAR_NS::Spatial& s1) { return o <<"Circumsolar Limit > "< "< "< "< "< "<

#endif

#endif // SUNPOS_H APPENDIX C. COMPUTER CODE 141

Sunpos.h

#ifndef SUNPOS_H #define SUNPOS_H

#include #include

/* The algorithm Sunpos can be used without calculating * the solar position as a vector (that is without using * the vecmath library). Simply uncomment the following * line or edit solar_conf.h */ //#undef INCLUDE_VECMATH

#ifdef INCLUDE_VECMATH #include #endif

#ifdef SOLAR_INCLUDE_IO #include #endif

SOLAR_BEGIN_NS template class Sunpos { public:

/** The default constructor is for the locations of Sydney, Australia, * at 1:06 pm 1/1/2000 ESDT (Eastern Standard Daylightsavings Time) */ Sunpos(T a = 151.2,T b = -33.867,T c = 11.0,int d = 1,int e = 1, int f = 2000,T g = 13.0,T h = 6.0 ,T i = 0.0 ): longitude(a),latitude(b),localTime(c), day(d), month(e), year(f), hours(g - c), minutes(h), seconds(i) {}

/** Containing the dependent variable in an array */ Sunpos( const T a[] ): longitude(a[0]), latitude(a[1]), localTime(a[2]), day(int(a[3])), month(int(a[4])), year(int(a[5])), hours(a[6] - a[2]), minutes(a[7]), seconds(a[8]) {}

#ifdef VECMATH_H APPENDIX C. COMPUTER CODE 142

/** Directly imput the solar vector on initialtion */ Sunpos( VM_VECMATH_NS::Vector4& vec, const int dn = 1): longitude(151.2),latitude(-33.867),localTime(11.0), day(1), month(1), year(2000), hours(13.0 - 11.0), minutes(6.0), seconds(0.0), dayNumber(dn), solarVector(vec) {}

/** Input the solar vector directly */ void putSolarVector(const VM_VECMATH_NS::Vector4& vec){ solarVector = vec; calculateDayOfTheYear(); }

/** return the terrestrial solar vector */ void get(VM_VECMATH_NS::Vector4& vec){ vec = solarVector; }

#endif

/** Input the Zenith and Azimuth values directly (radians) */ void putZenithAzimuth(const T zen, const T azi){ zenithAngle = zen; azimuthAngle = azi; sphericalToVector(); calculateDayOfTheYear(); }

/** Input longatude latitude and localtime. For information on * local time zones and regions see: * http://aa.usno.navy.mil/faq/docs/worldtzones.html */ void putLongLatLoc(const T lon, const T lat, const T loc){ longitude = lon; latitude = lat; localTime = loc; }

/** Input the day, month and year */ void putDMY(const int d, const int m, const int y){ day =d; month =m; year =y; }

/** Input the local time (floating point number of hours) */ void putHour(const T h){ hours = h - localTime; minutes = 0.0; seconds = 0.0; }

/** Input the local time (hours, minutes, seconds) */ void putHMS(const T h,const T m,const T s){ hours = h - localTime; minutes = m; seconds = s; }

/** return the azimuth, zenith and daynum */ void getAzimuth(T *azi) {*azi = azimuthAngle;} void getZenith(T *zen) {*zen = zenithAngle;} void getDayNum(int *dn) {*dn = dayNumber;} APPENDIX C. COMPUTER CODE 143

/** Calculates the solar vector in both spherical and vector coordinates * written by Blanco Muriel et. al. from the PSA. The azimuth and zenith * angles are now stored as radians */ void generate(void){ calcSolarVector(); } private:

/* The original PSA algorithm */ void calcSolarVector(void);

/** Calculate both the difference in days between the current Julian Day * and JD 2451545.0, which is noon 1 January 2000 Universal Time * written by Blanco et al. and the day of year added by Buie et al. */ void calculateDayoftheYear(void){

T dJulianDate; long int liAux1; long int liAux2; decimalHours = hours + (minutes + seconds / 60.0 ) / 60.0; liAux1 =(month-14)/12; liAux2=( 1461 * ( year + 4800 + liAux1 )) / 4 + (367 * ( month - 2 - 12 * liAux1 ) ) / 12 - ( 3 * (( year + 4900 + liAux1 ) / 100 ) ) / 4 + day -32075; dJulianDate = T(liAux2) - 0.5 + decimalHours / 24.0; elapsedJulianDays = dJulianDate - 2451545.0;

dayNumber = liAux2 - (( 1461 * ( year + 4799 )) / 4 + 336 - (3 * (( year + 4899 ) / 100 )) / 4 - 32075 ); }

#ifdef VECMATH_H /**Converts spherical azimuth, zenith solar coordinates to a local vector */ void sphericalToVector(void){ solarVector.set( std::sin( azimuthAngle ) * std::sin( zenithAngle ), std::cos( azimuthAngle ) * std::sin( zenithAngle ), std::cos( zenithAngle ), 0.0 ); } #endif public: /* Both longitude and Latitude are * represented as floating point * numbers. Local time can be a floating * point number. */ T longitude; T latitude; T localTime;

/* Day of the month, month of the year * (0-12) and year (1995 - 2005) */ int day; APPENDIX C. COMPUTER CODE 144

int month; int year;

/** Local time at the give location * hours minutes and seconds can be * given as floating point number */ T hours; T minutes; T seconds;

/** decimalHours and elapsedJulianDays * are both required to determine the * day of the year */ T decimalHours; T elapsedJulianDays; int dayNumber;

/** Both the zenith angle and the * azimuth angle are now represented * in radians */ T zenithAngle; T azimuthAngle;

#ifdef VECMATH_H /** Solar vector represented as a three vector */ VM_VECMATH_NS::Vector4 solarVector;

#endif

}; // End of class sunpos template void Sunpos::calcSolarVector(void){ /* * The PSA algorithm :: This Code was initially written by * Blanco Muriel et. al. from the PSA. The original * ansii c code can be downloaded from http://www.psa.es/sdg/sunpos.htm * The code is called the PSA algorithm for solar positioning * B. Muriel et al.,"Computing the solar vector " * Solar Energy, 2001. */

// Defining const variables const T dEarthMeanRadius = 6371.01; // In km const T dAstronomicalUnit = 149597890; // In km const T twopi = 6.28318530717956; const T rad = 0.01745329252;

// Main variables T dEclipticLongitude; T dEclipticObliquity; T dRightAscension; T dDeclination;

// Auxiliary variables APPENDIX C. COMPUTER CODE 145

T dY; T dX;

// Calculate difference in days between the current Julian Day // and JD 2451545.0, which is noon 1 January 2000 Universal Time

calculateDayoftheYear();

// Calculate ecliptic coordinates (ecliptic longitude and obliquity of the // ecliptic in radians but without limiting the angle to be less than 2*Pi // (i.e., the result may be greater than 2*Pi) { T dMeanLongitude; T dMeanAnomaly; T dOmega; dOmega=2.1429-0.0010394594*elapsedJulianDays; dMeanLongitude = 4.8950630 + 0.017202791698 * elapsedJulianDays ; // Radians dMeanAnomaly = 6.2400600 + 0.0172019699 * elapsedJulianDays; dEclipticLongitude = dMeanLongitude + 0.03341607 * std::sin( dMeanAnomaly ) + 0.00034894 * std::sin( 2 * dMeanAnomaly ) - 0.0001134 - 0.0000203 * std::sin(dOmega); dEclipticObliquity = 0.4090928 - 6.2140e-9 * elapsedJulianDays + 0.0000396 * std::cos( dOmega ); }

// Calculate celestial coordinates ( right ascension and declination ) in radians // but without limiting the angle to be less than 2*Pi (i.e., the result may be // greater than 2*Pi) { T dSin_EclipticLongitude; dSin_EclipticLongitude= std::sin( dEclipticLongitude ); dY = std::cos( dEclipticObliquity ) * dSin_EclipticLongitude; dX = std::cos( dEclipticLongitude ); dRightAscension = std::atan2( dY,dX ); if( dRightAscension < 0.0 ) dRightAscension = dRightAscension + twopi; dDeclination = std::asin( std::sin( dEclipticObliquity ) * dSin_EclipticLongitude ); }

// Calculate local coordinates ( azimuth and zenith angle ) in degrees { T dGreenwichMeanSiderealTime; T dLocalMeanSiderealTime; T latitude_InRadians; T dHourAngle; T dCos_Latitude; T dSin_Latitude; T dCos_HourAngle; T dParallax; dGreenwichMeanSiderealTime = 6.6974243242 + 0.0657098283 * elapsedJulianDays + decimalHours; dLocalMeanSiderealTime = ( dGreenwichMeanSiderealTime * 15 + longitude ) * rad; dHourAngle = dLocalMeanSiderealTime - dRightAscension; latitude_InRadians = latitude * rad; dCos_Latitude = std::cos( latitude_InRadians ); dSin_Latitude = std::sin( latitude_InRadians ); dCos_HourAngle= std::cos( dHourAngle ); zenithAngle = std::acos( dCos_Latitude * dCos_HourAngle * std::cos( dDeclination ) + std::sin( dDeclination ) * dSin_Latitude ) ; dY = - std::sin( dHourAngle ); dX = std::tan( dDeclination ) * dCos_Latitude - dSin_Latitude * dCos_HourAngle ; azimuthAngle = std::atan2( dY, dX ); if ( azimuthAngle < 0.0 ) azimuthAngle = azimuthAngle + twopi; APPENDIX C. COMPUTER CODE 146

// Parallax Correction dParallax = ( dEarthMeanRadius / dAstronomicalUnit ) * std::sin( zenithAngle ); zenithAngle = zenithAngle + dParallax; }

#ifdef VECMATH_H // Convert to vector coordinates sphericalToVector(); #endif }

SOLAR_END_NS

#ifdef SOLAR_INCLUDE_IO template std::ostream& operator<<(std::ostream& o, const SOLAR_NS::Sunpos& s1) { return o <<"Longitude > "< "< "< "< "< "< "< "< "<

#endif

#endif // SUNPOS_H APPENDIX C. COMPUTER CODE 147

Spectral.h

#ifndef SPECTRAL_H #define SPECTRAL_H

#include //#include #include

/* The algorithm Spectral can be used without using * the vecmath library. Simply uncomment the following * line or edit solar_conf.h */ //#undef INCLUDE_VECMATH

#ifdef INCLUDE_VECMATH #include #endif

#ifdef SOLAR_INCLUDE_IO #include #endif

SOLAR_BEGIN_NS template class Spectral{ public:

/* Constructor */ Spectral(int a = 1, T b = 0.27, T c = 1.42, T d= 1.14, T e = 0.65, T f = 0.34, T g = 1013.25, T h = 0.0, T i = 0.0 ): outputUnits(a), tau500(b), watvap(c), alpha(d), assym(e), ozone(f), airpressure(g), tilt(h), aspect(i) {};

/* Input parameters specifically */ void putUnits (int x){ outputUnits = x; }; void putAirPressure (T x) { airPressure = x; }; void putTau500 (T x) { tau500 = x; }; void putWaterVapour (T x) { watVap = x; }; void putAlpha (T x) { alpha = x; }; APPENDIX C. COMPUTER CODE 148

void putAssym (T x) { assym = x; }; void putOzone (T x) { ozone = x; }; void putTilt (T x) { tilt = x; };

// Return Function const int size(void) {return 122;}; // Size of spectral arrays void size( int *ret) { *ret = 122;}; // Size of spectral arrays

void getDifSpectrum (T ret[]) {alloc(ret, diffuseSpectrum);}; // Diffuse spectrum void getDirSpectrum (T ret[]) {alloc(ret, directSpectrum);}; // Direct spectrum void getExtTSpectrum (T ret[]) {alloc(ret, extraTSpectrum);}; // Extraterrestrial spectrum void getGloSpectrum (T ret[]) {alloc(ret, globalSpectrum);}; // globalRadiation_ spectrum void getWavelength (T ret[]) {alloc(ret, wavelength);}; // X-value of spectrum

void alloc(T ret[], T p[]) { for ( int i=0; i<122;i++) ret[i] = p[i];}

/** Returns the Diffuse radiation W/m^2 * Returns the Direct radiation W/m^2 * Returns the Global radiation W/m^2 */ void getDifRadiation(T *ret) { *ret = diffuseRadiation;}; void getDirRadiation(T *ret) { *ret = directRadiation;}; void getGloRadiation(T *ret) { *ret = globalRadiation;};

/* generate the solar spectrum + insolation values using the zenith, azimuth * and day number as input arguments */ void generate(const T ze, const T az, const int dn, const int onlyif = 1){ zenithAngle = ze; azimuthAngle = az; dayNumber = dn; calcAirmass(); calcSpectrum(); if ( onlyif ) calcInsolation(); }

#ifdef SUNPOS_H /* generate the solar spectrum + insolation values using the * variables contained with the Sunpos::sunpos */ void generate(const Sunpos& sunpos, const int onlyif = 1){ zenithAngle = sunpos.zenithAngle; azimuthAngle = sunpos.azimuthAngle; dayNumber = sunpos.dayNumber; calcAirmass(); calcSpectrum(); if ( onlyif ) calcInsolation(); } #endif

#ifdef VECMATH_H /* Generates the solar spectrum + insolation for a given solar vector and daynumber */ void generate( VM_VECMATH_NS::Vector4 solarVector, const int DN = 1, const int onlyif = 1){ if (solarVector.length() != 1) solarVector.normalize();

zenithAngle = std::acos(solarVector.z);

if (zenithAngle == 0) azimuthAngle = 0.0; else azimuthAngle = std::acos(solarVector.y / sin(zenithAngle)); APPENDIX C. COMPUTER CODE 149

dayNumber = DN;

calcAirmass(); calcSpectrum(); if ( onlyif ) calcInsolation(); } #endif

/** Kasten, F. and Young, A. 1989. Revised optical air mass * tables and approximation formula. Applied Optics 28 (22), * pp. 4735-4738 */ void calcAirmass( void ) {

const T RAD = 0.01745329252;

if ( zenithAngle / RAD > 93.0 ) { airmass = -1.0; airmassPC = -1.0; } else { airmass = 1.0 / ( std::cos(zenithAngle) + 0.50572 * solarUtil::pow( (96.07995 - zenithAngle / RAD) , -1.6364) ); airmassPC = airmass * airpressure / 1013.25; } }

/** * Calculate the solar insolation values * using the trapazoidal method. This is * probably the most appropriate * method as the wavelengths were chosen at * absorption lines. The equation includes * a correction for the corresponding units. */ void calcInsolation(void);

/** * Spectral2 C++ implementation using the day number and position * as imput arguments. */ void calcSpectrum( void ); public:

int outputUnits; /* Spectral2 output units */

T tau500; /* Aerosol optical depth at 0.5 microns, base e */ T watvap; /* Precipitable water vapor (cm) */ T alpha ; /* Power on Angstrom turbidity */ T assym ; /* Aerosol assymetry factor */ T ozone ; /* Atmospheric ozone (cm) */ T airpressure; T tilt; T aspect; T airmass; T airmassPC; /* optical airmass pressure corrected */

T zenithAngle; T azimuthAngle; APPENDIX C. COMPUTER CODE 150

int dayNumber;

/* Spectral arrays */ T extraTSpectrum[122]; T diffuseSpectrum[122]; T directSpectrum[122]; T globalSpectrum[122]; T wavelength[122];

/* Local insolations values (W/m^2) */ T diffuseRadiation; T directRadiation; T globalRadiation;

}; // End of class Spectral2; template void Spectral::calcInsolation(void){ /** * Calculate the solar insolation values * using the trapazoidal method. This is * probably the most appropriate * method as the wavelengths were chosen at * absorption lines. The equation includes * a correction for the corresponding units. */

diffuseRadiation = directRadiation = globalRadiation = 0.0;

for(int i=0; i<121; i++) {

T tStep = ( wavelength[i+1] - wavelength[i] ); T tDif = ( diffuseSpectrum[i] + diffuseSpectrum[i+1] ); T tDir = ( directSpectrum[i] + directSpectrum[i+1] ); T tGlo = ( globalSpectrum[i] + globalSpectrum[i+1] ); T var;

switch (outputUnits){ case 1: diffuseRadiation += tDif / 2.0 * tStep; directRadiation += tDir / 2.0 * tStep; globalRadiation += tGlo / 2.0 * tStep; break;

case 2: var = tStep / ( wavelength[i+1] + wavelength[i] ) * 2.9979244e8 * 6.6261762e-24; diffuseRadiation += tDif * var; directRadiation += tDir * var; globalRadiation += tGlo * var; break;

case 3: var = ( wavelength[i+1] + wavelength[i] ) / 4.0 * tStep * 1.6021891e-15; diffuseRadiation -= tDif * var; directRadiation -= tDir * var; globalRadiation -= tGlo * var; break; } APPENDIX C. COMPUTER CODE 151

} } template void Spectral::calcSpectrum(void) { /* Spectral.h :: Developed from the anscii c code * S_spectral2 (National Renewable Energy Laboratories’s * simple spectral model), initially written by Martin Rymes * from the National Renewable Energy Laboratory (NREL) on * 21 April 1998. Spectral2 is based on the SERI (now NREL) * technical report SERI/TR-215-2436, "Simple Solar Spectral * Model for Direct and Diffuse Irradiance on Horizontal and * Tilted Planes at the Earth’s Surface for Cloudless * Atmospheres", by R. Bird & C. Riordan */

/* wavelegths points */ const float tempWavelength[122] = { 0.3, 0.305, 0.31, 0.315, 0.32, 0.325, 0.33, 0.335, 0.34, 0.345, 0.35, 0.36, 0.37, 0.38, 0.39, 0.4, 0.41, 0.42, 0.43, 0.44, 0.45, 0.46, 0.47, 0.48, 0.49, 0.5, 0.51, 0.52, 0.53, 0.54, 0.55, 0.57, 0.593, 0.61, 0.63, 0.656, 0.6676, 0.69, 0.71, 0.718, 0.7244, 0.74, 0.7525, 0.7575, 0.7625, 0.7675, 0.78, 0.8, 0.816, 0.8237, 0.8315, 0.84, 0.86, 0.88, 0.905, 0.915, 0.925, 0.93, 0.937, 0.948, 0.965, 0.98, 0.9935, 1.04, 1.07, 1.1, 1.12, 1.13, 1.145, 1.161, 1.17, 1.2, 1.24, 1.27, 1.29, 1.32, 1.35, 1.395, 1.4425, 1.4625, 1.477, 1.497, 1.52, 1.539, 1.558, 1.578, 1.592, 1.61, 1.63, 1.646, 1.678, 1.74, 1.8, 1.86, 1.92, 1.96, 1.985, 2.005, 2.035, 2.065, 2.1, 2.148, 2.198, 2.27, 2.36, 2.45, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 4.0}; /* extraterrestrial spectrum (W/sq m/micron) */ const float horizExtraSpectrum[122] = { 535.9, 558.3, 622.0, 692.7, 715.1, 832.9, 961.9, 931.9, 900.6, 911.3, 975.5, 975.9, 1119.9, 1103.8, 1033.8, 1479.1, 1701.3, 1740.4, 1587.2, 1837.0, 2005.0, 2043.0, 1987.0, 2027.0, 1896.0, 1909.0, 1927.0, 1831.0, 1891.0, 1898.0, 1892.0, 1840.0, 1768.0, 1728.0, 1658.0, 1524.0, 1531.0, 1420.0, 1399.0, 1374.0, 1373.0, 1298.0, 1269.0, 1245.0, 1223.0, 1205.0, 1183.0, 1148.0, 1091.0, 1062.0, 1038.0, 1022.0, 998.7, 947.2, 893.2, 868.2, 829.7, 830.3, 814.0, 786.9, 768.3, 767.0, 757.6, 688.1, 640.7, 606.2, 585.9, 570.2, 564.1, 544.2, 533.4, 501.6, 477.5, 442.7, 440.0, 416.8, 391.4, 358.9, 327.5, 317.5, 307.3, 300.4, 292.8, 275.5, 272.1, 259.3, 246.9, 244.0, 243.5, 234.8, 220.5, 190.8, 171.1, 144.5, 135.7, 123.0, 123.8, 113.0, 108.5, 97.5, 92.4, 82.4, 74.6, 68.3, 63.8, 49.5, 48.5, 38.6, 36.6, 32.0, 28.1, 24.8, 22.1, 19.6, 17.5, 15.7, 14.1, 12.7, 11.5, 10.4, 9.5, 8.6}; /* Water vapor absorption coefficient */ const float waterVapourAdj[122] = { 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.075, 0.0, 0.0, 0.0, 0.0, 0.016, 0.0125, 1.8, 2.5, 0.061, 0.0008, 0.0001, 0.00001, 0.00001, 0.0006, 0.036, 1.6, 2.5, 0.5, 0.155, 0.00001, 0.0026, 7.0, 5.0, 5.0, 27.0, 55.0, 45.0, 4.0, 1.48, 0.1, 0.00001, 0.001, 3.2, 115.0, 70.0, 75.0, 10.0, 5.0, 2.0, 0.002, 0.002, 0.1, 4.0, 200.0, 1000.0, 185.0, 80.0, 80.0, 12.0, 0.16, 0.002, 0.0005, 0.0001, 0.00001, 0.0001, 0.001, 0.01, 0.036, 1.1, 130.0, 1000.0, 500.0, 100.0, 4.0, 2.9, 1.0, 0.4, 0.22, 0.25, 0.33, 0.5, 4.0, 80.0, 310.0, 15000.0, 22000.0, 8000.0, 650.0, 240.0, 230.0, 100.0, 120.0, 19.5, 3.6, 3.1, 2.5, 1.4, 0.17, 0.0045 }; /* Ozone absorption coefficient */ const float ozoneAdj[122] = { 10.0, 4.8, 2.7, 1.35, 0.8, 0.38, 0.16, 0.075, 0.04, 0.019, 0.007, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.003, 0.006, 0.009, 0.01400, 0.021, 0.03, 0.04, 0.048, 0.063, 0.075, 0.085, 0.12, 0.119, 0.12, 0.09, 0.065, 0.051, 0.028, 0.018, 0.015, 0.012, 0.01, 0.008, 0.007, 0.006, 0.005, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, APPENDIX C. COMPUTER CODE 152

0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 }; /* Uniformly mixed gas "absorption coefficient" */ const float uniMixedGasAdj[122] = { 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.15, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 4.0, 0.35, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.05, 0.3, 0.02, 0.0002, 0.00011, 0.00001, 0.05, 0.011, 0.005, 0.0006, 0.0, 0.005, 0.13, 0.04, 0.06, 0.13, 0.001, 0.0014, 0.0001, 0.00001, 0.00001, 0.0001, 0.001, 4.3, 0.2, 21.0, 0.13, 1.0, 0.08, 0.001, 0.00038, 0.001, 0.0005, 0.00015, 0.00014, 0.00066, 100.0, 150.0, 0.13, 0.0095, 0.001, 0.8, 1.9, 1.3, 0.075, 0.01, 0.00195, 0.004, 0.29, 0.025 };

const T c = 2.9979244e14; // Used to calculate photon flux const T cons = 5.0340365e14; // Used to calculate photon flux const T evolt = 1.6021891e-19; // Joules per electron-volt const T h = 6.6261762e-34; // Used to calculate photon flux const T OMEG = 0.945; // Single scattering albedo, 0.4 microns const T OMEGP = 0.095; // Wavelength variation factor const T e = h * c / evolt; // energy in electron volts const T PI = 3.1415926535; const T RAD = 0.01745329252;

/* Cosine of the angle between the sun and a tipped flat surface, useful for calculating solar energy on tilted surfaces. */ T ci = std::cos(zenithAngle) * std::cos ( tilt ) + std::sin ( zenithAngle ) * std::sin ( tilt ) * ( std::cos ( azimuthAngle ) * std::cos ( aspect ) * + std::sin( azimuthAngle ) * std::sin( aspect ));

/* If the tilt angle is greater than PI assuming tracking */ if (fabs(tilt) > PI){ tilt = zenithAngle; ci = 1.0; }

/* Precalculate the cosines */ T ct = std::cos( tilt ); T cz = std::cos( zenithAngle );

/* Ground Reflection have been hardwired - * these are for solar energy calculations*/ T wv[6] = {0.3, 0.7, 0.8, 1.3, 2.5, 4.0}; T rf[6] = {0.2, 0.2, 0.2, 0.2, 0.2, 0.2};

/* Equation 3-14 */ T alg = std::log( 1.0 - assym );

/* Equation 3-12 */ T afs = alg*( 1.459 + alg * ( 0.1595 + alg * 0.4129));

/* Equation 3-13 */ T bfs = alg*( 0.0783 + alg * ( -0.3824 - alg * 0.5874));

/* Equation 3-15 */ T fsp = 1.0 - 0.5 * std::exp( ( afs + bfs / 1.8 ) / 1.8); APPENDIX C. COMPUTER CODE 153

/* Equation 3-11 */ T fs = 1.0 - 0.5 * std::exp ( ( afs + bfs * cz ) * cz);

/* Ozone mass */ T ozoneMass = 1.003454 / std::sqrt( cz * cz + 0.006908 );

/* Current wavelength range */ int nr = 1;

/** ERV calculation, Day angle * Iqbal, M. (1983) An Introduction to * Solar Radiation. Academic Press, NY., page 3 */ T dayang = RAD * 360.0 * ( dayNumber - 1.0 ) / 365.0;

/** Earth radius Vector * solar constant = solar energy * Spencer, J. W. 1971. Fourier series representation of the * position of the sun. Search 2 (5), page 172 */ T erv = 1.000110 + 0.034221 * std::cos( dayang ) + 0.001280 * std::sin( dayang ) + 0.000719 * std::cos( dayang * 2 ) + 0.000077 * std::sin( dayang * 2 );

for(int i=0; i<122; ++i) { /* MAIN LOOP: step through the wavelengths */

/* Input variables */ T lambda = tempWavelength[i]; T H0 = horizExtraSpectrum[i] * erv; T WatVap = waterVapourAdj[i] * watvap; T Ozone = ozoneAdj[i] * ozone; T UniMixAdj = uniMixedGasAdj[i];

/* Equation 3-16 */ T omegl = OMEG * std::exp(-OMEGP * solarUtil::pow(std::log(lambda/0.4), 2));

/* Equation 2-7 */ T c1 = tau500 * solarUtil::pow(lambda * 2.0, -alpha);

/* Advance to the next wavelength range for ground reflectivity */ if (lambda>wv[nr]) ++nr;

/* Equation 2-4 */ T Tr = std::exp( -airmassPC / ( solarUtil::pow( lambda, 4 ) * ( 115.6406 - 1.3366 / solarUtil::pow( lambda, 2 ))) );

/* Equation 2-9 */ T To = std::exp( -Ozone * ozoneMass );

/* Equation 2-8 */ T Tw = std::exp( -0.2385 * WatVap * airmass / solarUtil::pow (( 1.0 + 20.07 * WatVap * airmass ), 0.45 ));

/* Equation 2-11 */ T Tu = std::exp( -1.41 * UniMixAdj * airmassPC / solarUtil::pow(( 1.0 + 118.3 * UniMixAdj * airmassPC), 0.45 ) );

/* Equation 3-9 */ T Tas = std::exp( -omegl * c1 * airmass); APPENDIX C. COMPUTER CODE 154

/* Equation 3-10 */ T Taa = std::exp(( omegl - 1.0 ) * c1 * airmass );

/* Equation 2-6, sort of */ T Ta = std::exp( -c1 * airmass );

/* Equation 2-4; primed */ T Trp = std::exp( -1.8 / ( solarUtil::pow( lambda, 4 ) * ( 115.6406 - 1.3366 / solarUtil::pow( lambda, 2 ))) );

/* Equation 2-8; primed */ T Twp = std::exp( -0.4293 * WatVap / solarUtil::pow( 1.0 + 36.126 * WatVap, 0.45) );

/*Equation2-11;primedairmassM = 1.8 */ T Tup = std::exp( -2.538 * UniMixAdj / solarUtil::pow( 1.0+212.94 * UniMixAdj, 0.45) );

/* Equation 3-9; primed airmass M = 1.8 (Section 3.1) */ T Tasp = std::exp( -omegl * c1 * 1.8 );

/* Equation 3-10; primed airmass M = 1.8 (Section 3.1) */ T Taap = std::exp( ( omegl - 1.0 ) * c1 * 1.8 );

/* Direct energy */ T c2 = H0 * To * Tw * Tu;

/* Equation 2-1 */ T dir = c2 * Tr * Ta;

/* Diffuse energy */ c2 *= cz * Taa; T c3 = (rf[nr]-rf[nr-1])/(wv[nr]-wv[nr-1]);

/* Equation 3-17; c4 = Cs */ T c4 = 1.0;

if ( lambda <= 0.45 ) c4 = solarUtil::pow((lambda+0.55),1.8);

/* Equation 3-8 */ T rhoa = Tup*Twp*Taap*( 0.5*(1.0-Trp) + (1.0-fsp)*Trp*(1.0-Tasp) );

/* Interpolated ground reflectivity */ T rho = c3 * ( lambda - wv[nr-1] ) + rf[nr-1];

/* Equation 3-5 */ T dray = c2*(1.0-solarUtil::pow(Tr,0.95))/2.0;

/* Equation 3-6 */ T daer = c2* solarUtil::pow(Tr,1.5)*(1.0-Tas)*fs;

/* Equation 3-7 */ T drgd = (dir*cz+dray+daer)*rho*rhoa/(1.0-rho*rhoa);

/* Equation 3-1 */ T dif = (dray+daer+drgd)*c4;

/* globalRadiation (total) energy */ T dtot = dir*cz+dif; APPENDIX C. COMPUTER CODE 155

/* Tilt energy, if applicable */ if ( tilt > 1.0e-4 ) {

/* Equation 3-18 without the first (direct radiation - beam) term */ c1 = dtot*rho*(1.0-ct)/2.0; c2 = dir/H0; c3 = dif*c2*ci/cz; c4 = dif*(1.0-c2)*(1.0+ct)/2.0; dif = c1+c3+c4; dtot = dir*ci+dif; }

/* Adjust the output according to the units requested */ switch (outputUnits) { case 1: c1 = 1; wavelength[i] = lambda; break; case 2: wavelength[i] = lambda; c1 = lambda * cons; break; case 3: wavelength[i] = e/lambda; c1 = lambda * lambda * lambda *cons / e; break; }

globalSpectrum[i] = dtot * c1; directSpectrum[i] = dir * c1; diffuseSpectrum[i] = dif * c1; extraTSpectrum[i] = H0 * c1; } }

SOLAR_END_NS

#ifdef SOLAR_INCLUDE_IO template std::ostream& operator<<(std::ostream& o, const SOLAR_NS::Spectral& s1) { return o <<"Aerosol optical depth at 0.5 microns > "< "< "< "< "< "< "< "< "< "< "< "< "< "<

<<"Direct Insolation > "< "<

#endif

#endif // SPECTRAL_H APPENDIX C. COMPUTER CODE 157

Stack.h

#ifndef STACK_H #define STACK_H

#include

#ifdef INCLUDE_VECMATH #include #endif

#include #include

#define S 1 #define P 0

SOLAR_BEGIN_NS template class Stack{ public: Stack(const unsigned int a): noLayers(a) { n =new std::complex[noLayers]; theta = new std::complex[noLayers]; thickness =new T[noLayers]; }

~Stack(void){ delete []n; delete []theta; delete []thickness; }

void layer(const unsigned int a, const T b, std::complex c){ thickness[a] = b; n[a] = c; }

void layer(const unsigned int a, const T b, T c, T d = 0){ thickness[a] = b; n[a] = std::complex(c,d); } APPENDIX C. COMPUTER CODE 158

void getReflect(T *a){ *a = alloc(reflect); }

void getTransmit(T *a){ *a = alloc(transmit); }

void getAbsorb(T *a){ *a = alloc(absorb); }

T alloc(std::complex* comp){ std::complex tempa;

tempa = std::sqrt( comp[S] * conj( comp[S] ) ); tempa += std::sqrt( comp[P] * conj( comp[P] ) );

return (tempa.real()/2); }

void calc(const T wavelength, const T angle){ std::complex cangle(angle,0);

determineATR(wavelength,cangle);

}

#ifdef INCLUDE_VECMATH

void calc(const T wavelength, const VM_VECMATH_NS::Vector4 solarVector, const VM_VECMATH_NS::Vector4 surfaceNormal){

T angle = acos( solarVector.dot(surfaceNormal)); std::complex cangle(angle,0);

determineATR(wavelength,cangle);

}

#endif private:

void determineATR(const T , std::complex ); public: unsigned int noLayers;

std::complex* theta; std::complex* n; T* thickness;

std::complex reflect[2]; std::complex transmit[2]; std::complex absorb[2]; }; template APPENDIX C. COMPUTER CODE 159

void Stack::determineATR(const T wavelength , std::complex b){ /* Stack::determineATR calculates the reflection, transmittion and * absorption of a thin film stack into a substrate for a given * incident beam of radiation with a specific wavelength, and * complex angle on incidence onto the initial layer. */

theta[0] = b;

const std::complex im_i( 0, 1 ); const T pi = 3.14159265358979; const T mu0 = 4 * pi * 1E-7; const T eps0 = 8.85418782E-12;

/* asin wasn’t present in the library */ for ( unsigned int i = 1; i < noLayers; i++ ){ std::complex sinangle = n[i-1] * sin(theta[i-1]) / n[i]; theta[i] = conj(im_i) * std::log( im_i * sinangle + std::sqrt( std::complex(1,0) - sinangle * sinangle ) ); }

for (unsigned int sORp = 0; sORp < 2; sORp++){

std::complex delta[noLayers]; std::complex Y[noLayers];

for (unsigned int i = 0; i < noLayers; i++){ if (sORp) Y[i] = ( std::sqrt( eps0 / mu0 ) * std::cos( theta[i] ) ) * n[i]; else Y[i] = ( std::sqrt( eps0 / mu0 ) / std::cos( theta[i] ) ) * n[i];

delta[i]=( 2 * pi * thickness[i] * std::cos( theta[i] ) / wavelength * n[i] ); }

std::complex U[4];

std::complex M[4]; M[0] = std::complex(1,0); M[3] = std::complex(1,0);

for ( unsigned int j = 1; j < noLayers-1;j++){ for ( unsigned int k = 0; k < 4; k++) U[k] = M[k];

std::complex Temp[4];

Temp[0] = std::cos( delta[j] ); Temp[1] = im_i * std::sin( delta[j] ) / Y[j]; Temp[2] = im_i * std::sin( delta[j] ) * Y[j]; Temp[3] = std::cos(delta[j]);

M[0]=U[0]*Temp[0]+U[1]*Temp[2]; M[1]=U[0]*Temp[1]+U[1]*Temp[3]; M[2]=U[2]*Temp[0]+U[3]*Temp[2]; M[3]=U[2]*Temp[1]+U[3]*Temp[3];

//std::complex determinent=M[0]*M[3]-M[1]*M[2]; }

std::complex B = M[0]+M[1]*Y[noLayers-1]; std::complex C = M[2]+M[3]*Y[noLayers-1]; APPENDIX C. COMPUTER CODE 160

std::complex D = Y[0] * B + C;

std::complex temp = ( Y[0] * B - C ) / D; reflect[sORp] = temp * conj( temp );

transmit[sORp]= T(4.0) * Y[0] * Y[noLayers-1].real() / (D * conj( D ) );

temp = B * conj( C ) - Y[noLayers-1]; absorb[sORp]= T(4.0) * Y[0] * temp.real() / ( D * conj( D ) ); }

}

SOLAR_END_NS

#endif APPENDIX C. COMPUTER CODE 161

C.2 Fortran code

C.2.1 LBL

PROGRAM readLBL

implicit none integer i,p,limit(11) integer site,scope,time(2),date(3),solartime(2),realflag(2) integer flag(29) double precision alt,azi, esd , pyran(4) , pyr_clear , pyr_filt(8) double precision soldata(3), misc(3),datascan(56) double precision actdist, solardist,pi, rpd

pi = 3.14159265358979323846 rpd = pi / 180.

limit(1) = 45110520 limit(2) = 21607560 limit(3) = 15135120 limit(4) = 59911800 limit(5) = 57145920 limit(6) = 7459920 limit(7) = 16665480 limit(8) = 960960 limit(9) = 8034000 limit(10) = 4676880 limit(11) = 33120360

open(90,FILE=’zzcomb.txt’,STATUS=’OLD’)

do p=1,11 do i=1,limit(p)/1560

110 format(I2,I2,1x,I2,1x,I2.2,1x,I2.2,1x,I2,1H:,I2.2,1x,I1,I1, x 11X,I2,1x,I2.2,7x,f5.2,7x,f7.2,7x,f6.4) read(90,110) site, scope, date,solartime,realflag,time,alt,azi, esd 120 format(43x,5(5I1,1x),4I1) read(90,120) flag 130 format(45X,f6.1,1x,f6.1,6x,f6.1,1x,f6.1) read(90,130) pyran 140 format(42x,f7.1,28X) read(90,140) pyr_clear 150 format(29X,8f6.1) read(90,150) pyr_filt 160 format(34X,f7.1,9x,f6.1,11x,f10.7) read(90,160) soldata 170 format(40x,f8.5,6x,f8.5,5x,1pE10.3) read(90,170) misc 180 format(11(27x,5(1PE10.3),/),27X,1PE10.3,/) read(90,180) datascan

enddo enddo end