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Page1Resources – Renewables - Solar

Based on - R.L. Nersesian. 2007. Sustainable Energy Chapter 9 of Energy for the 21st Centrury: A Comprehensive Guide to

Conventional and Alternative Resources.- U.C.V. Haley and D.A. Schuler. 2011. Government Policy and Firm Strategy in the Solar Photovoltaic

Industry. California Management Review, Vol. 54, No. 1: 17-38.

Outline Astrophysics: Source of solar power Area Requirements Harvesting Solar Power

Thermal Photovoltaic

Solar power economics and policies

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Page2Feasibility of Solar EnergyThen and Now

• Solar energy in the siege of Syracuse, Sicily ≈ 214-212 BC. Syracuse was a Hellenistic city and was attacked by Romans.

• According to a legend, during the siege, Archimedes-designed mirrors burned Roman ships. At the end, Rome conquered the city and 75+ year old Archimedes was killed by Roman soldiers.Si

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Page3Nuclear energy Electromagnetic Energy Solar Energy

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4G network0.6-2.5 Giga Hz

Helium fusion reactions take place in the sun’s core and mass (4 mega tons / second) is converted into energy.

Sun’s temperature at the center ≈15,000,000 Celsius and on the surface ≈ 5,700 Kelvin (=273+5,423 Celsius).

A heated up metal (substance that does not burn) becomes first red and then yellow. It emits light.

An ideal model for sun’s light emission is blackbody radiation. A blackbody does not reflect any light but can emit its own. Ablackbody does not appear black to eye.

Depending on its temperature T, a black body emits power P 𝜆𝜆,𝑇𝑇 at different wavelengths 𝜆𝜆.

X-axis is the wavelength of emission in 10−9

meters = nanometer =nm. moreheat

less heat

Y-axis is the power P 𝜆𝜆,𝑇𝑇in 104 watts per 𝑚𝑚2, nm

and sr (steradian).

Source: Figure 3.8 Carroll & Ostite. 2006. Introduction to

Modern Astrophysics.

The total power emitted is the area under P 𝜆𝜆,𝑇𝑇 Planck curve. Higher temperature objects emit more power but at lower wavelengths.

Units of power is watts per squaremeter per steradian; see appendix.

An approximate formula for P 𝜆𝜆,𝑇𝑇 is offered by Planck. less energy, safe higher energy, danger

Infrared gogglesdetect infrared waves

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Page4

Take the blackbody radiation Planck curve at 5777 K from the previous page. Light is neither a wave nor a particle; it is both. This is known as wave-particle duality in quantum physics. Planck equation: Light has energy ∝ its frequency = 1/wavelength. From nuclear energy, 1 eV=1.6*10-19 J. The radiation that falls on the earth outside atmosphere is more than the radiation on the surface.

– Atmosphere (ozone, oxygen, water, carbondioxide) absorbs particular wavelengths of emission. E.g., Ozone O3 absorbs short wavelength (high energy) emissions. Lack of ozone ⟹ High energy emissions ⟹ Skin cancer.

Instead of integral, approximate the integral by the area of the triangle above. The power that reaches the earth’s atmosphere is approximately the area of the triangle with base 1750-250 nm wavelength and height 1.75 𝑊𝑊/(𝑚𝑚2 ∗𝑛𝑛𝑚𝑚). The theoretical irradiance (area of the triangle) is 1312.5 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤/𝑚𝑚2.

– The empirical irradiance is 1361 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤/𝑚𝑚2.

Solar Power on Atmosphere and Surface

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Source: Figure 2 of JA Herron, J Kim, AA Upadhye, GW Huber & CT Maravelias. 2015. A general framework for the assessment of solar fuel technologies. Energy & Environmental Science, 8: 126-157.Remarks: 1) The unit of 𝑊𝑊/(𝑚𝑚2 ∗ 𝑠𝑠𝑠𝑠 ∗ 𝑛𝑛𝑚𝑚) on the right-hand vertical axis is

from the black body radiance theoretical model.2) 1 electron volt Empirical data are from Reference Solar Spectral

Irradiance: Air Mass 1.5, http://rredc.nrel.gov/solar/spectra/am1.5/, accessed June 2014. Units for these data are in 𝑊𝑊/(𝑚𝑚2 ∗ 𝑛𝑛𝑚𝑚).

3) Solar power reaches earth in several wavelengths (𝜆𝜆) not just one. The total power per area reaching earth is the area under the spectral irradiance curve Total Power per area of squaremetre = ∫0

∞𝑃𝑃 𝜆𝜆,𝑇𝑇 = 5777 𝑑𝑑𝜆𝜆

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Page5

Solar Power Accounting for Zenith Angle Solar power provides ~ 1360 W/m2 outside the earth’s atmosphere. This is also called AM0 irradiance. It provides less after passing through the atmosphere. When the sun is falling directly, air mass (AM)=1.

Half of 1360 W/m2 reaches solar panels. The rest is reflected back to space or absorbed by the atmosphere.

– Solar radiation bouncing from atmospheric particles move in various directions. This called diffuse radiation, which is higher on hazy or cloudy days.

When the zenith angle is wider, the light must travel longer in the atmosphere and drops more of its energy at the atmospheric particles.

AM1.5 irradiance is generally about 827 W/m2. Industry rounds AM1.5 up to 1000 W/m2. Earth wide average solar power per area ≈ 650 W/m2.

480AM =

1Cos 𝑍𝑍𝑍𝑍𝑛𝑛𝑍𝑍𝑤𝑤𝑍 𝐴𝐴𝑛𝑛𝐴𝐴𝐴𝐴𝑍𝑍

cos 48 = 0.67; AM= 1.5

cos 0 = 1; AM= 1

600

cos 60 = 0.5; AM= 2

900

cos 90 = 0; AM= ∞

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Page6

Solar Power Accounting for Nights & Clouds AM1.5’s irradiance 1 kW/m2 (=1,000 W/m2) is available during the day when the sun is visible.

1 9 17 1 9 17

1 kW/m2

Day 1 total 5.4 kWh/m2 Day 2 total 4.4 kWh/m2

Average daily 4.7 kWh/m2

Average annual 1715.5 kWh/m2

based on only day 1 and 2

Power/area

HoursNight Night Night

Pyranometer measures solar radiation

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Page7

T/X Questions1. Diffuse radiation outside the atmosphere is the largest.

X, there is no diffuse radiation outside the atmosphere.

2. The energy generated by the sun is due to fusion reaction. T

3. Only visible light generated by the sun reaches the earth.X, light reaches in different wavelengths including infrared and ultraviolet.

4. Lower frequency electromagnetic waves carry less energy. T

5. A hot substance emits power through electromagnetic waves at different wavelengths. T

6. A hotter substance emits a higher amount of power at some but not all wavelengths.X, it emits higher amount of power at all wavelengths.

7. Depletion of ozone in the atmosphere increases the solar power reaching the panels on earth’s surface. T

8. Air Mass (AM) in solar energy refers to the mass of air in 1 cubicmetre volume at various altitudes.X, AM is the 1/cos(zenith angle) and is used to measure the air mass the light travels through in the atmosphere.

9. Zenith angle at dawn is zero degrees. X, zenith angle at dawn is 90 degrees.

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Page8

Solar Power Area Requirement Overton, North Texas receives maximum irradiance of about 6 kW/m2 in August and minimum of about 2 kW/m2 in

December. Texas solar irradiance database is www.me.utexas.edu/~solarlab. Southwest US receives more solar energy about 6 kW/m2.

– Optimistically suppose sun is out 12 hours/day and 30% conversion efficiency, so we get 0.9 kW/m2

(=(6/2)*0.3) of radiation at every hour. How many m2 required to generate 350,000 kW (capacity of Mojave desert panels)?

» 350,000/0.9 = 388,888 m2 which can be made up by a 623 metre x 623 metre square. A (American) football field is 4,500 m2. About 90 football fields are needed.

– More likely scenario has sun out 8 hours/day and 20% conversion efficiency, so we get 0.4 kW/m2 of radiation at every hour. How many m2 required to generate 350,000 kW?

» 350,000/0.4 = 875,000 m2 which can be made up by a 935 metre x 935 metre square. About 200 football fields are needed.– Where to find such a large deserted land? In a desert!

Total US generation capacity was 1,000,000,000 kW in 2007, which requires 1,100,000,000 – 2,500,000,000 m2 or 250,000 – 560,000 football fields.

Source: http://sroeco.com/solar

Mojave desert 5000-6000 W/m2

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Page9Roofs for Solar Power at UTD and around In 2013, UTD installs 220 kilo watt solar panels on the new Parking Structure, see the photo below. With this, UTD was

able to participate in a Solar Program funded by Oncor and qualified to receive $203,722. UTD also qualified for another $98,371 in incentives during 2012 for efficient chiller installations and by constructing

buildings that are more efficient than the code requirement. This brought the total incentive in 2013 to $302,093.

Utility companies are required to invest in energy efficiencies in their service areas. – Oncor runs the “Take a Load off Texas” project, which funded $62 million in 2013.– CenterPoint invested $42 million in projects in 2013.

Source: Merit Report: Plugging into Energy Efficiency. 2014. Z. Cologlu, D. Flom, T. Junt, S. Patel and A. Pizaňa.

In Fall 2013, parking lotnorthwest of SOM building

ATT distribution center Lancaster, TX

In 2015, ATT installs 677 kilo watt ground mounted solar panels. 2,000 panels provide about 40% of building’s annual power need.

UTD’s Energy Revolving Fund reinvests these incentives in energy efficiency projects on campus. Savings from the projects refill the fund.

FedEx distribution center Hutchins, TX

In 2014, FedEx installs 1,400 kilo watt roof mounted solar panels. 4,622 panels provide about 20% of building’s annual power need.

~0.3 kW/panel

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Page10

Harvesting Solar Power Sun’s energy can be harvested as thermal solar power and as photovoltaic solar power.

NuclearPower

SolarPower

ThermalSolar Power

PhotovoltaicSolar Power Electricity

Turbine

Direct Use ofHot Water

Thermal solar power capacity of 374 GW end of 2013, http://www.iea-shc.org/solar-heat-worldwide Installed thermal solar power capacity: China 262 GW; Europe 44 GW; US and Canada 17 GW.

Dwarfed photovoltaic capacity of 139 GW end of 2013, p.17 of “Global Market Outlook for Photovoltaics 2014-2018” published by Solar Power Europe http://solarpowereurope.org.

Wind

Steam

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Page11

Thermal Solar Power Thermal Direct Use: Solar power heats water that later is

used for heating buildings or to serve hot water needs. Thermal → Steam: Indirect Use to generate Electricity

(Concentrated Solar Power): – Collector absorbs the solar heat and passes to the water as it

passes through it towards the storage tank. – Storage tank keeps the water hot and insulated. A

secondary power source (electric or gas) can be used.– Hot water is used to generate steam for electricity turbines.– Collectors: Parabolic troughs (convex mirrors), Fresnel

concentrator (flat mirrors), dish stirling (dish shaped mirror).– Heat transfer agent:

» Water, which can be hard to come by in a desert. » Synthetic oil heated upto 735 F by convex mirrors,

Mojave Desert, CA. » Molten salt heated upto 1050 F in a tower by Sunlight

tracking mirrors. When power is needed molten-salt flows through a heat exchanger and cools down. Salt’s temperature drops at least down to 550 F, it has to remain molten.

collector storage tank

Sources: www.dipol.com.tr/solar_energy.htmwww.global-greenhouse-warming.com/solar-parabolic-trough.html

Ivanpah Solar Thermal Power Field

Focal point

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Page12

Thermal → Wind: Solar Updraft Tower Wind is due to air pressure difference If hot air is collected and fed into a tower, it will generate wind inside the tower while rushing to the top of the tower. This wind can operate turbines at the bottom of the tower. Air at the bottom can be warmed up with thermal solar power mirrors.

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2010 Jinshawan Tower, Inner Mongolia, China200-kilowatt power generating unithttp://news.xinhuanet.com/english2010/china/2010-12/27/c_13666710.htm

Challenges:– Cost & stability of a tall tower– Resistance of tower against strong winds

Solutions proposed a tower that– Is constructed from lighter materials, e.g., cloth– Floats, inflated with light gas– Bends to minimize the effect of strong winds

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Page13Photovoltaic Solar Power:Excited Electron’s Excursion from Valence to Conduction Band

Photovoltaic effect: Movement of electrons by materials exposed to light. Each electron resides in its regular orbit (its valence band) around the nucleus. While residing in its valence band, an electron has a certain amount of energy. Energy of an electron increases upon receiving photovoltaic energy (through a light

photon). A rise in an electron’s energy,

• If small, warms up the material.• If large, moves an electron from its valence band to the conduction band.

Conduction band is further away from the nucleus and accommodates electrons that are ready to break away from their nucleus.

If an electron in the conduction band is not directed through a circuit to come out away from its nucleus, it can drop back to its valence band.

– In the process of falling back, the electron emits light: Fluorescent lamps excite gas electrons in the lamp with electricity. The electrons go up to their conduction band, fall back to their valence band and emit light.

Conductionband

Valenceband

Light increases the energy of an electron from valence band level

to conduction band level.

= Conduction band - Valence bandInSb: Indium Antimonide, crystal of Indium and Antimony.Narrow band-gap semiconductor used in infrared detectors.InAs: Indium Arsenide, crystal of Indium and Arsenide.Ge: GermaniumGaSb: Gallium Antimonide, crystal of Gallium and Antimony.Si: Silicone, crystal. InP: Indium Phosphide, crystal of Indium and Phosphorus. GaAs: Gallium Arsenide, crystal of Gallium and Arsenide.GaxAl1-xAs: Gallium Aluminum ArsenideWide band-gap semiconductor.

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Page14

p-layern-layer

Excitable Semiconductors to Harvest Photovoltaic Solar Power

Photovoltaic cells (PV) have two layers (as your pizza) n-layer is rich in electrons so it is negatively charged. p-layer is deprived of electron so it is positively charged.As cheese topping in your pizza does not go into the crust, electrons do

not naturally move to p-layer in an n-p two-layer semiconductor. Light brings electrons to a valance band of an n-layer. A circuit takes them to

the p-layer and finally returns to n-layer.

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Silicone has 14 electrons; 10 of them are stable in lower orbits; 4 (valence electrons) are in the external orbit.

Silicone crystal (c-Si) Fosfor with extra electron in c-Si

Boron with space for electron in c-Si

Energy level to excite electrons depend on the composition of n-layer. Multiple n-layers (as multiple topping pizza) can catch different wavelengths and increase efficiency. Toppings are: Al, Ga, As, Ge, Si and their n- and p-layers.

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Electron flow n-layer → p-layerCircuit flow p-layer → n-layer

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Page15Efficiently Capturing Solar Energy is Matching

Efficiently Capturing Solar Energy is matching the wavelengths of solar radiation with the wavelengths (energy) of valence band gaps

by using appropriate mix of semiconductors.

Sun

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Page16

Research: Photovoltaic Cells Types & Efficiencies

Source: NREL (National Renewable EnergyLaboratory) as of Aug 2015

Active research by using materials (Gallium, Arsenide) other than silicon and by using multiple layers (junctions). The highest solar-to-electricity conversion efficiency at the time of writing this document: 46%. This is higher than coal-to-electricity efficiency. Such high efficiency cells are not produced at industrial scale yet but are only for space applications.

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Page17

Industry: Photovoltaic Cells Types

Efficiency of converting solar energy to electric energy can range from 7-13%: Thin-layer PV cells deposited on plastic (glass or steel). 14%: Low-end polysilicon 15-18% : High-end polysilicon made up of cleaner more pure silicon

wafers. 30%+: Gallium Arsenide PV cells used in the space program.

Types of Photovoltaic Cells Thin film solar cells, limited market share, low cost and efficiency Crystalline silicon

Monocrystalline: Crystallization starting from a single cell Polycrystalline: Crystallization starting at multiple cells Multi-junction: Multiple p-n junctions in tandem

Silicon wafers used in PV cells are as thin as 200 microns (10-6 meters) and about 15-20 square centimeters.

A PV cell is the building block producing about 1-2 watts. It is assembled into modules and arrays with serial connections.

Source: http://science.nasa.gov

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Page18

Industry: Photovoltaic Cells Global Production

Global production of polysilicon is 69,000 tons in 2009. Chinese production is 20,000 tons. Half of this came from a single company: GCL Poly. Polysilicon plants should have more than 5,000 tons of annual capacity to be competitive. A 10,000-ton

plant can cost about $1 billion. GCL Poly managers believe that the price should be about $28/kg; others suggest higher prices. US has competitive edge on thin-film PV production. Chinese firms Yingli and Trina are closing the gap.

PV and Module Production in 2009 Haley and Schuler (2011).

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Page19PV Capacity, Supply and Demandfrom Haley and Schuler (2011)

PV Installed Capacity in 2010 PV Overcapacity

Germany, Spain and USA are the first three in terms of leading installed capacity. Governments offer incentives to consumers to adopt solar energy.

In a FiT (Feed-in-Rate) contract a government pays consumers to reduce the gap between solar and conventional energy for extended period of time (20 years in Germany).

Germany introduced FiT subsidies in 2000 and has been reducing them. Spain introduced FiT subsidies in 2007 at about €0.45/kWh and reduced it by 35% in 2008 and also by

30% in 2010. Consequently, polysilicon price collapsed from its heights of $450/kg.

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Page20Government Policies: Production and Consumption Assistance

No production and no consumption assistance – Solar is not cost-competitive without assistance.– Upstream (Production) companies: Consolidate or withdraw.– Midstream companies: Fierce competition can lead to acquisitions of midstream companies by

production companies.– Downstream companies: Find non-traditional customers such as municipal power companies, public

utilities, residential users. Remove restrictions on widespread use such as limitations on rooftop installations in residential areas.

Production assistance but no consumption assistance– Production grows, upstream companies benefit– Midstream companies want to sell in the international markets– China’s experience– Solyndra case in US: Inability to compete with the manufacturing cost caused failure to pay $535

million federally guaranteed loans. No production but consumption assistance

– All benefit from higher demand. – Upstream companies may want to stop import boom

– Use domestic content requirement as in Ontario’s to qualify for FiT. – Downstream companies should not be addicted to consumption assistance

– Expand demand: Net-zero affordable homes in Arizona, California, Colorado, Nevada– Spain’s experience

Both production and consumption assistance– Both may be unnecessary

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Page21Economics of Solar PowerConsumer’s Perspective

Scenario: The installation cost of panels at $8 per W and tax credit of 50% (consumption assistance).– The installation cost to generate 2000 W is $16,000. – Tax credit compensates for 50% of this cost so the consumer is left with $8,000. – The total energy generated by these panels: 100,000 kW – hour = 2 * 5.5 * 365 * 25

» 2 kWh per hour.» 5.5 hours per day: accounting for clouds, sun at the horizon twice during the day.» 365 days per year.» 25 years of lifetime for panels.

– If the retail price of electric is $0.08 per kW-hour, the saving over panel’s life time is $8,000.– Without 50% consumption assistance, consumers will not install panels.

The latest gains in efficiency is puling the cost down. Say $3 per W. Realizing this, the governments are reducing the tax credit. Say 30%.

Scenario: The installation cost of panels at $3 per W and tax credit of 30%.– The installation cost to generate 2000 W is $6,000. – Tax credit compensates for 30% of this cost so the consumer is left with $4,200. – The total energy generated by these panels: 100,000 kW – hour = 2 * 5.5 * 365 * 25– If the retail price of electric is $0.08 per kW-hour, the saving over panels life time is $8,000.– With or without 30% tax credit, consumers will install panels.

A 2,000 squarefeet house needs approximately 2,000 W of power.

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Page22

Roof-top Solar Panel Savings at UTD Go to http://pvwatts.nrel.gov and use UT-Dallas campus address 800 West Campbell Road, Richardson. Select the weather data for this location. The closest data location is TMY3 Dallas/Addison. Enter PV system information to obtain columns A and E below

Consider a 4 kW system with standard (polycrystalline silicon) module type. If modules are for residential use, they are most likely to be polycrystalline silicon.

A 4 kW system is likely to have 16 solar panels. With panels of the standard 1.6 x 1 m2, we need about 30 m2 of roof space.

Most residential panels are fixed (as opposed to one-axis or two-axis sun tracking) on the roof. If your panel is facing south directly, the azimuth angle is 180 degrees. The tilt angle is the slope of the roof for fixed panels, you can put 20 degrees for the tilt angle in Dallas.

You can keep the default system loss of 14% for taking into account shading, soiling, wiring, etc.

Averages over

A:kWh/(m2*day)

B: 30*AkWh/day

C:Days/month

D: Input, B*C kWh/month

E: Output kWh/month

F: % Efficiency

Jan 2.76 82.8 31 2,566.8 284 11.06

Apr 4.55 136.5 30 4,095.0 433 10.57

Jul 6.69 200.7 31 6,221.7 591 9.5

Oct 3.96 118.8 31 3,682.8 379 10.29

Annum 51,112.5 5,216

Finally, the saving with electricity cost of $0.1 per kWh is $521 per year.

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Page23

Roof-top Solar Panel Costs Generic solar panel costs

PV components Cost $/Watt

Module 0.48

Electrical 0.40

Inverter 0.06

Racking 0.10

Labor 0.10

Miscellaneous 0.02

Total 1.16

Cost of 4,000 Watt panel is $4,640 Savings $521/year Time to recover investment: About 9 years

Can we really save $521 without storage (battery)? Tesla is selling solar panel-battery bundles!

20 years lifetime for panels 10 years lifetime for battery

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Page24Consumer’s PerspectivePrice of Electricity

Time-of-day electric pricing advocates for charging higher during the day when the demandis higher than during the night when the demand is lower.

If the retail price of electricity is $0.16 per kW-hour during the day as opposed to $0.10 when the panel is generating electricity, the saving over panels life time is $16,000 as opposed to $10,000.

Without a consumption assistance but with time-of-day electric pricing, consumers will install panels.

Who offers time-of-day electric pricing in zip code 75080 in March 2012? Go to http://powertochoose.org, enter zip code, pick variable rate, companies include:

Veteran Energy, Pennywise Power, Southwest Power & Light, YEP, Mega Energy, Direct Energy, GexaEnergy, Reliant, Smart Prepaid Electric, Frontier Utilities, Bounce Energy, Just Energy, First Choice Power, Entrust Energy, Texpo Energy, Power House Energy, Stream Energy, Andeler Power, Nueces Electric, Payless Power, Ambit Energy, Texas Power.

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Page25Summary – Renewable - Solar

Astrophysics: Source of solar power Area Requirements Harvesting Solar Power

Thermal Photovoltaic

Solar power economics and policies

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Page26

Radiance Intensity in Watt/Steradian 1 steradian ≈ The solid angle creating the cone with bottom has area of 𝑠𝑠2

– A steradian of Ω covers Ω𝑠𝑠2 surface area on the sphere– Ω = 1 covers 1 squaremetre on a sphere with radius 1 metre– Area of 𝐴𝐴 is covered by steradian of 𝐴𝐴

𝑟𝑟2

Surface area of a sphere with radius 𝑠𝑠 is 4𝜋𝜋𝑠𝑠2 ≈ 12.57𝑠𝑠2

1

12.57≈ 8% of the surface area is spanned by 1 steradian

Think of two concentric spheres with a power source in the center The inner sphere has radius 𝑠𝑠; the outer has 2𝑠𝑠 Which sphere receives more power on its entire surface?

– They receive the same power as no energy is lost when emitted outward – Inner sphere has area 4𝜋𝜋𝑠𝑠2, the outer has 4(4𝜋𝜋𝑠𝑠2)– The power received per area by the inner sphere is 4 times that of the outer

sphere– The power received by 1 steradian in the inner sphere is the same as the

power received by 1 steradian in the outer All of the power is received by 12.57 steradians, no matter which sphere 1 steradian gets 8% of the power; 1/8 steradian gets 1% of the power Using steradian in measurement allows us to capture geometric variable

(the ratio of area on the surface to the square of radius)

1 Steradianboth in theinner & outer spheres

Radiance intensity is radiant flux (arrows) emitted per unit of solid angle. 1 candela (luminous intensity of 1 candle) = 1

683watt/steradian.

1 lumen = 1 candela over 1 steradian. 1 candle yields 4𝜋𝜋 lumens of light.

𝑠𝑠𝑠𝑠

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Page27

Irradiance in Watt/(Steradian*Squaremetre)

Larger area gets larger power Eg, surface area of 8𝑠𝑠2 gets 2 times the power of 4𝑠𝑠2

Both small and large blue circles get 1Ω

lumen Small blue circle gets 1

2r2Ωlumen/m2 and the larger blue circle gets 1

4r2Ωlumen/m2

The distance between the sun and earth is seasonal (max variation 1.7%) but predictable, as well as the surface of the earth ⇒ drop steradian

The solar power density reaching earth’s atmosphere is 1360 W/𝑚𝑚2

Distance between sun and earth 1.5*1011 metres

Earth radius 6*106 metres

Steradian = 𝜋𝜋 6∗1062

1.5∗1011 2 = 16𝜋𝜋 ∗ 10−10

5𝑠𝑠

𝑠𝑠2 4𝑠𝑠2

5𝑠𝑠

𝑠𝑠2

𝑠𝑠2

4𝑠𝑠2

4𝑠𝑠2

Power falling onboth small & large circles 1/25 lumen

Ω = 125

steradian Ω fixed

Sun emits 4*1030 W in all directions; 1.8*1017 W hits the earth