So I am very happy to see some houses, where I live, get solar panels. The solar panels rarely cover all the roof since the roofs are typically trapezoidal or triangular in shape.
So assume the following :
Manufacturing costs of solar panels is not effected by the size/shape of panels
Transportation/Installation costs of solar panels is not effected by the size/shape of panels
Just like hexagonal leather pieces are the best geometry to make a spherical soccer ball, is there is a particular geometry that would give the best coverage of the median US house roofs ?
Please keep the analysis to a pure geometry problem.
ISTM, that if the shape doesn’t change the cost, then the best thing to do would be to have either the big squares like we already have for the field, and a variety of other shapes to fill in the boarders. Whether it’s panels that are trapezoidal with one size being square and the other side at an angle that lets it follow the roof line, or even smaller squares to fill the gaps along the edges.
Come to think of it, even just having them all made of squares would get you a much better fill, like finding the area under a curve, the smaller the panels, the closer to the edge you can get.
As I’m writing this, I’m thinking that it wouldn’t surprise me if over the next few decades, the panel sizes became standardized (if they’re not already) and people kept that in mind while building houses. Even if you don’t fit put them on while building the house, it would be easier to retrofit if the roof was designed in such a way that you could get edge to edge coverage.
You could get better coverage, regardless of shape, by making the panels smaller. But that would presumably carry its own decreases in practicality, or they’d be doing it already.
And while some roofs are trapezoidal (with various angles), some are still rectangular, and some are other, more complicated shapes. And not all roofs would be equally well-suited for solar panels, due to their orientation and environment. A proper answer to this question would require some sort of statistical analysis of typical roof shapes and sizes, probably cross-tabbed by latitude and climate.
I take this to mean that it costs a fixed amount to manufacture X square feet of panel, no matter how many panels are manufactured to cover that area. Given this (extraordinarily unrealistic) assumption, the answer is very simple: make them as small as possible. The shape doesn’t really matter as long as it tessellates the plane. Panels that are 1-inch squares will completely cover any roof, with only a tiny perimeter less than one inch wide uncovered.
Alternatively , if you follow the constraints of the OP, you make each facet of the roof a single panel since manufacturing costs won’t matter. I think squares would be the best bet overall for reasons mentioned above. Keep in mind you will only be covering the areas of the pitch that face southerly as covering the entire thing wouldn’t be worthwhile. This problem is the whole idea driving Solar Roof tiles, BTW. Tesla’s version is still not here but supposedly other companies are trying to make it economically reasonable.
Even with that clarification, I think you’re still up against roof shape/pitch and southern exposure to come up with any meaningful answer. The pedantic answer is “1, custom fit to fill the available space that has the proper orientation,” but that’s not really going to get you anywhere. Knowing the square footage of the median US home doesn’t help because you won’t know how that relates even to a hypothetical median roof structure, because you won’t know how many floors it’s spread across. A ranch-style house of 2600 sq. ft. has a hell of a lot more roof than a four-story townhouse of 2600 sq. ft.
N.b.: the linked stat, I realize in retrospect, is for new homes not all homes. Alas, I have to go offline now.
I think that, even if we’re assuming that all shapes are equally easy to make for any given area, we still want to assume a minimal number of different shapes, because the panel factory will have to re-tool in some way for each shape, which is usually expensive, so you want each of your models to be produced in as high a quantity as possible.
The people who are proposing tiny panels are forgetting one thing: we live in the “real world.”
That means, each panel must have connections to all the other panels. The more panels, the more connections, which means more points of failure, and more holes in the roof to leak. And, more expense to install.
That’s one of the reasons solar shingles have never caught on.
I agree the OP’s conditions are crazily unrealistic – both of them are clearly not true. And the OP is neglecting one of the more important components: solar panels produce DC current, and so each panel needs a DC-to-AC converter built in. It’s far more expensive than the glass & silicone that makes up most of the panel, so bigger panels are cheaper.
But I would say that you could adequately cover most roofs with about 4 sizes:
4’ x 8’ rectangular
2’ x 6’ rectangular
4’ x 4’ right triangle (4’ x 4’ square divided in half diagonally)
2’ x 2’ right triangle (2’ x 2’ square divided in half diagonally)
These shapes could be patched together to cover most roof geometries. You would start with the biggest sizes, and then place smaller ones around them to cover the rest of the roof. (Assuming US measurements here; the rest of the world would use metric equivalents.)
There’s a slippery slope here. Yes, that set of 4 panels would do a good job. But I could do an even better job with a set of 6 smaller panels. Somewhere (in the real world) there’s a point of diminishing returns where you have to stop making them smaller.
The OP’s attempted clarification makes the classic mistake of asking to optimize several things at once:
It’s a well-defined question to ask for the minimum number of panels that exactly cover a given set of roof shapes. It’s also a well-defined question to ask for the maximum area of roof that can be covered by a given set of panels. It’s not well-defined to ask for a minimum set of panels that will cover the maximum area of roof. For example you may find that you can cover 99% of roof area with 15 panels, but require 47 panels to cover 100%. In that case most people would say the 15 panel set is the best answer. But then what if reducing it to 14 panels drops the coverage to 85%? Is that set of 14 panels that covers 85% “better” than the set of 15 panels that covers 99%? You have to weigh a trade-off between number of panels and coverage and the OP hasn’t given us any guidance for how to do that.
Yes, you need a DC to AC converter, but I don’t think there’s any reason why that must be built into the thing that goes onto the roof. More likely, you’d have the things on the roof producing DC, all of that DC going into the house, and then being converted to AC in a single box in a closet somewhere.
In some places where the roof may get un-uniform shade or some panels may not get light while others do, they do put in individual inverters to improve net power production.
An analogy is like the lowest strength (highest internal resistance) battery in a series circuit decides the maximum current. So a string of solar panels gets limited by the production of the “shadiest” panel.
It’s not common though and very expensive and has shorter life.
Because to be efficient, the inverter has to be closely matched to the output current from the solar panels. So you’d need lots of different versions of this closet inverter to match various numbers of panels on each roof (or an adjustable one (more expensive, less efficient)). Plus during the day, as the sun moves and some of the panels fall into shadow, an efficient inverter would have to adjust for the reduced production – even more complexity. Also, as inverters get bigger to handle more current, they get more expensive. It’s much cheaper & easier to have a finely-tuned inverter built into each panel. That also makes inter-connecting them simpler, so less chance that your average construction worker could mess it up.
Also, modern inverters also pump the voltage up to standard household level. That reduces the size of the wires needed to carry it. And copper is expensive.
There are now people who, in addition to running their house on solar, also charge a large capacity battery for use when the sun goes down. The easiest way to do this is to have all power from the panels go to the battery and then run the house off that. In which case, you want DC going straight to the battery and then convert the power as it’s used. Whether they do it this way, I don’t know.
ETA: After reading T-Bonham’s post, it looks like they don’t.
When I had my system installed, all of the companies (plus the general consensus of the rabble on internet forums) I met with recommended going with micro inverters (the one inverter per panel solution), being that that solution was the wave of the future. I understand that there have been some innovations with string inverters that have allowed them to stay relevant, but I’m surprised to hear that micro inverters are not common. The three installs on my street (from three different companies) are all so configured.
This is much more in alignment with my actual experience having solar installed and the input I got from all directions.
Aside: Does it affect the solution to the proplem posed if we have to consider local fire and building codes? For example, I could not install solar panels to fill the space the way I had hoped for due to the requirements for clear space between the panels and my ridge line, and the panels to any other roof edge. If I understand correctly, clear walkable access in case of fire or whatever is required almost anywhere.