1 - First-principles calculation of solar radiation

I started by figuring out a way to calculate the amount of solar radiation at a particular school. The input parameters are: 

1. School’s latitude as that would determine the sun’s position in the sky and 

2. The angle at which the solar panels are placed

Ideally, the solar panels would track the sun keeping the panels at 90 degrees to the sun for maximum efficiency (like a sunflower). However, that would require motors and controllers which would add cost and need energy and maintenance to operate. I observed that most school, home and commercial installations were fixed in place. So, I decided to keep it simple as well and assume that panels would be fixed, facing south at an angle of 10-15 degrees. 

After studying websites on photovoltaics (technical terms for solar power), I found PVeducation.org to explain the physics behind solar power the best. 

\( \alpha \) is the angle of the sun at a particular time in the day. It is computed as:

$$ \alpha = 90 - \phi + \delta $$

where \( \phi \) is the latitude and \( \delta \) is the declination and computed as:

$$ \delta = 23.45^{\circ} \sin \left[ \frac{360}{365} (d - 81) \right] $$

where \( d \) is the day of the year from 0 to 365. The \( 81 \) accounts for counting from the Spring equinox (around March 20) when the Earth's axis is tilted neither toward nor away from the sun.

\( \beta \) is the tilt angle of the solar panels from horizontal position, and set at 10 degrees.

\( S_{incident} \) is the amount of solar radiation measured perpendicular to the sun. It is measured at \( 1367 W / m^2 \) (per source). 

\( S_{module} \) is the radiation incident on a tilted solar panel array, and can be computed as: 

$$ S_{module} = S_{incident} sin (\alpha + \beta) $$

As the position of the sun varies during the day (from sunrise to sunset) as well as during each day of the year (highest in the summer and lowest in the winter), I divided the year into 1-hour chunks and computed the position of the sun using the following equations:

\( H \) is the hour angle at sunrise and sunset and can be computed as:

$$ H = \arccos \left[\frac{ \sin(-0.83^{\circ}) -\sin \phi \sin \delta } {\cos \phi \cos \delta }\right] $$

As earth’s angular velocity is \( 15^\circ / hour \) (as it makes a complete revolution of \( 360 ^ \circ \) in \( 24 \) hours), we can convert from hour angle degrees to time as follows:

$$ sunrise = 12 - H / 15 $$

$$ sunset = 12 + H / 15 $$

For my elementary school, Springer Elementary, located at \( (37.3713, -122.0952) \) latitude and longitude, the above equations yield the value of:

• Total Incident Solar Radiation = \( 2988.47 Wh / m^2 \) per year

• Total Solar Radiation on 10 degree tilted Solar array = \( 2634.74 Wh / m^2 \) per year

3 - Calculating annual solar radiation at a school 

I used Javascript and the Node.js environment to write a computer program to calculate the amount of solar radiation for each school in the 124,000+ school database.

Click on the image below to check out the code:

5 - Global Solar Atlas 

Incorporating the effects of atmosphere and geography meant moving beyond first-principles modeling. After a fair bit of research, I concluded that Global Solar Atlas (https://globalsolaratlas.info/) was the best dataset. It is built by Solargis and funded by the World Bank as a resource for all countries to “support the scale-up of solar power”. 

Photovoltaic power potential for the World: 

Photovoltaic power potential for the US: 

The site allows the user to enter a specific location. For my elementary school, it shows:

Some interesting things to point out: 

• Really striking to see how much more light we get in the summer vs the winter. Solar elevation gets to 75 degrees in the summer whereas it doesn’t even get to 30 degrees in the winter. 

• The optimal angle of solar panels is 32 degrees

• The Santa Cruz mountains are in the SW direction. It’s cool to see the program take them into account.

7 - Calculating the solar potential of 1MWp solar array

The Global Solar Atlas site allows users to model different systems. Using the assumption of 1MWp solar array installed at 10 degrees yields the following result:

• 1555 MWh / year of solar power output 

• 2065 KWh / m^2 per year of Global radiation (direct + diffuse) on tilted surface

Next, I automated this calculation for all schools in the US. Here, I got help from my dad who helped me discover that there is an undocumented REST API. He also helped me turn my one-school-at-a-time program into an asynchronous program so that schools could be processed in batches of 100 at a time. 

At this point, I have a data-set that looks like: 

Name,Address,City,State,Zip,Latitude,Longitude,PVOUT,GTI

MONTESSORI SCHOOL OF PALM SPRINGS,3692 E CHIA RD,PALM SPRINGS,CA, 92262,33.841823561,-116.506504363,1694437.9777940926,2349.268632633834

RAINBOW SCHOOL,11860 ROJAS DR,EL PASO,TX, 79936,31.72930976,-106.30660676,1711896.0395236588,2348.172209496066