Optimal Panel Elevation Angle For a Single Day

[drcampbell]

... I'm a mathematician ... it may take me a little while.

I feel a little better now. I am an engineer, not a mathematician; contemplated how to craft an analytical optimum solution with the vector calculus I learned in the late 1970s, and didn't get very far.

 

[jaggedben]

Haven't these variables already been worked out, and available for every geographical location?

Yes, but to my knowledge it's all hidden in proprietary software and not exactly public.

 

[GoldDigger]

The calculations at PVWatts take into account average weather too.

 

[drcampbell]

The angles are published in Anderson's book.
(he didn't go into the temperature effects)

 

[wwhitney]

Sure, this is all simply 3-D geometry, I just haven't found a good write up of it, and so I've been working it out on my own.

A few useful things I've convinced myself of: the sun's path through the sky on any given day is a portion of a circle on the sky hemisphere. [This is for a model that ignores the earth's orbital movement during the day.] That circle is a great circle (meaning the center of the hemisphere is the center of the circle) precisely on the equinoxes.

That also means that on the equinoxes, you do want to point your solar panel at the elevation of the sun at solar noon. That's because all the sun vectors you are averaging lie in a single plane, so their average (or integral) does as well. And since the average will be due south, it must be due south at the sun's noon elevation. My earlier argument breaks down on the equinox because it is not the case that lowering the panel elevation slightly helps near sunset or sunrise; a slight deviation from solar noon elevation is a second order effect for the entire day on the equinox.

But on other days when sun's path is not part of a great circle, then the argument still applies. I'm still working out the details.

Cheers, Wayne

 

[wwhitney]

But on other days when sun's path is not part of a great circle, then the argument still applies. I'm still working out the details.

OK, it's only in the fall/winter than the argument applies. I think in the spring/summer, the optimal panel angle is actually above the elevation of the sun at solar noon. In the limiting case of summer at the north pole, the optimal angle is straight up, as the sun rotates around the horizon without ever setting.

Cheers, Wayne

 

[GoldDigger]

...
... In the limiting case of summer at the north pole, the optimal angle is straight up, as the sun rotates around the horizon without ever setting.

Cheers, Wayne

IMHO, no.
Straight up has the sun at a glancing angle the whole day. Vertical, facing in the right direction, gets full perpendicular insolation at least part of the day. I have not actually done the math though.
At the Equator, on the other hand, horizontal is the best over the entire year.

 

[drcampbell]

Rowe's Rule: If you don't know, don't babble.

 

[wwhitney]

Straight up has the sun at a glancing angle the whole day. Vertical, facing in the right direction, gets full perpendicular insolation at least part of the day. I have not actually done the math though.

OK, you are right depending on the elevation angle of the sun as it transits the horizons. I was not properly handling the case when the sun is visible in the sky but the angle to the solar panel normal is more than 90 degrees.

Anyway, just comparing the cases of elevation angle 0 and 90 degrees, when the sun is transiting the horizon at a constant elevation angle of a, and the day length is 2pi, the integrated dot products are 2pi * sin a and 2 cos a, respectively. So 90 degrees wins when tan a < 1/pi, or a < 17.7 degrees.

Thanks for the correction.

Cheers, Wayne

 

[kwired]

Could it depend on the intended use of the panel?

If sole purpose is to directly supply power to the grid then you want maximum performance at all times and probably want to consider something that moves automatically to attain max effect, or maybe a curved panel or array that at least can get max performance in certain portions of the setup throughout the day.

If you are more concerned about alternate source to reduce your demand from utility, you may consider what direction the sun will be at peak demand periods.

 

[jaggedben]

Could it depend on the intended use of the panel?

If sole purpose is to directly supply power to the grid then you want maximum performance at all times and probably want to consider something that moves automatically to attain max effect, or maybe a curved panel or array that at least can get max performance in certain portions of the setup throughout the day.

If you are more concerned about alternate source to reduce your demand from utility, you may consider what direction the sun will be at peak demand periods.

He stated the problem as simply maximizing production for a given day , for a 'portable', i.e. offgrid system.

 

[kwired]

He stated the problem as simply maximizing production for a given day.

OK. Wouldn't sun be at most optimal point in the sky at solar noon, but then you have elevation that changes throughout the year.

 

[GoldDigger]

I suspect that this is a one day thing, making elevaion adjustment useful if it will be installed at different parts of the country, but azimuth tracking (manual every hour?) can make a big difference in both time window and integrated power output.

 

[Hv&Lv]

Most utility-scale fixed-tilt solar photovoltaic systems are tilted 20 degrees-30 degrees - U.S. Energy Information Administration (EIA)

www.eia.gov

 

[Hv&Lv]

More a math/physics question, but say you're setting up a portable, adjustable ground mount PV system before dawn for a single day event. So you just want to maximize your solar production over the course of that day, which will be clear all day. Pointing your array due south is clearly optimal, but what is the optimal elevation angle?

A lot of discussions seem to assume that the sun's elevation angle at solar noon is the correct answer, but that can't be exactly true. A little lower has to be better, but is it on the order of 0.5 degrees, or 5 degrees? This must be a solved problem, so I was hoping someone could point me to the solution and save me from looking up the formula for the sun's angle over the course of the day and doing the calculus.

Cheers, Wayne

Here’s some more light reading..

Part 3: Calculating Solar Angles | ITACA

 

[GeorgeB]

While not an answer to the OP question, http://www.solarpaneltilt.com/ has lots of good info on this.

 

[BandGap1.1eV]

Are you writing a research paper? If so, please let us know how useless it is to reposition our array ever hour/minute/second of every day to maximize production.

At the end of the day, dollars and kWh are the only things that matter. How many kWh did you make, how many dollars did it make you, and how many dollars did you spend to make them. If an optimized system is your goal, that nut has already been cracked. All Earth Renewables makes a robust dual axis tracker that uses GPS positioning to track the path of the sun over the course of the day. From there it becomes a simple math problem; does the increase production economically justify the increased cost?

I think Zomeworks (sp?) had a passive dual axis tracker that uses differential heating of a working fluid to move a tracker of the course of a day. Neat idea, never installed or worked on one though to know if they actually work.

 

[LarryFine]

I'll repeat my previous suggestion, since there will be no installation: just move the panel a few times.

 

[electrofelon]

While not an answer to the OP question, http://www.solarpaneltilt.com/ has lots of good info on this.

Yes I was going to provide that reference. That site does not give a formula for a daily value, the best they do is adjusting 4 times per year. Anyway the optimum fixed annual tilt is lower that many people think (about 4-7 degrees less than latitude for much of the US IIRC) this is because the sun is east and west a lot more than it is south. I suspect that value range is in the ballpark of being a out the same as Wayne's question about a single day.

 

[electrofelon]

That previous post wasn't worded very well, but what I meant was one could approximate a given day ideal angle by looking at which quarter that day falls in (and where in that quarter), and using the formulas on the solar panel tilt website. For my latitude of 43 degrees, I get summer: 15 degrees, spring and fall: 40 degrees, winter: 62 degrees. The ideal fixed annual tilt is 36 degrees (which is cool since it is constructible with compass and straight edge).

Although as a math guy, I would like to see the exact formula.