"Basically the tree of cells are arranged in different angles so that as the sun moves some of them will always be receiving optimal sunlight when their normal is parallel with that of the incident light."
Except it's trivially impossible to generate more energy like this (as he claims)! If a set of solar panels is independent and non-interacting (e.g., they don't shade or heat each other), then their total power output is simply the sum of their individual power outputs. And so their total energy outputs (integrated power) is the sum of their individual energy outputs. (Necessary distinction because their power peaks may occur at different times).
P_tot = Σ P_i
∫ P_tot dt = ∫ (Σ P_i) dt = Σ (∫ P_i dt)
E_tot = Σ E_i
If there is an unqiue optimal orientation for a single static panel (which under realistic assumptions there is) -- optimal in terms of maximizing energy output, E -- then the optimal orientation of a array of independent solar panels is just the same orientation, iterated. Combine inferior panel orientations, and their total remains inferior.
E_i < E_opt (for i <- 1..N)
Σ E_i < N * E_opt
And all those panel angles in his "tree", facing non-South, up, down, towards a wall... they are individually inferior to the single 45° south-facing panel in his static array. So combined together they are still inferior.
Where did his experiment go wrong? I'd start with the shade issue. The whole setup is in intermittent shade (note the tree shadows); and one experiment is sitting near the ground, the other is mounted on a pole. Looks like different shade environments. His output graphs show that the entire flat array is sometimes in total shade, at a time when the pole-mounted "tree" is not:
Another fatal flaw (I had to look this up to check) is that he is measuring voltage, not power, and they are not linearly related in photovoltaics. Actually, the open-circuit voltage (what you get if you stick a voltmeter over a solar cell, when it's not hooked up to a load) is practically independent of irradiance:
Open-circuit voltage going from 200 W/m^2 to 1,000 W/m^2 only increases from 34 V to 38 V. Power output, with a real load, would go up by about 5 times.
So, he never really measured energy production, or anything that remotely approximates it.
Agreed with the experiment flaws, plus the article is way too shady to make anything even remotely reliable out of it. It reads awfully like "Child pwns scientists, saves world and his dog"
Still, what if you consider a fixed volume as the constraint in establishing the comparison? For a flat panel assembly, a given volume would constrain the surface quite obviously. If you consider a tree structure one might be able to pack some more panels in the same volume, and Fibonacci might give us some solution about how to maximize panel density in the volume while minimizing shading interference from other panels and supporting infrastructure.
Doesn't it make sense that an empirical test would not reach a theoretical max? For instance you've discounted shade effects as where he went wrong. Maybe its exactly the shade issue that makes his findings significant. Every urbanite will have to deal with it, and his array may do a better job in intermittent shade.
If mass-distributed, most folks would not be optimizing anything - install-and-forget - so its a real solution to create something that works in most environments.
The shade issue implied by the parent post is that the tree configuration was mounted in a generally less shady position, as the Fibonacci-configured panels were located significantly further from the ground than the classic configuration. It is possible that the test was set up in a manner that puts the classic configuration at a disadvantage.
To control for that effect, the two panel configurations should be mounted at an equal average distance from the ground, and their positions swapped in different test runs.
This was where I was going too. (However, I'm not sure the power output of a panel is simply the sum of the individual cells. I thought that in a series array, the least illuminated panel essentially limits the panel current. Not that this matters in this case, since he wasn't actually measuring power. The voltage certainly adds.)
The experimental issue I thought about is that the experiment is situated next to a large wall. While the flat panel is facing away from the wall, it looks like some of the panels on the "tree" would also receive diffuse light from the wall. So effectively, the tree seems to have a larger collecting area than the flat panel. This would also make the tree less sensitive to shading.
>If there is an unqiue optimal orientation for a single static panel (which under realistic assumptions there is)...
If you have some kind of “continuous function attains its max” argument here, I think you should elaborate. Because it’s interesting but not really “trivial”.
No, results specific to solar panel orientation. Math is not trivial at all, but the results are intuitive: roughly, point the panel in the direction which, in mean, is in the direction of the sun. So tilted towards the equator, at the same angle as your latitude.
"The default value is a tilt angle equal to the station's latitude. This normally maximizes annual energy production. Increasing the tilt angle favors energy production in the winter, and decreasing the tilt angle favors energy production in the summer."
(Couldn't find a clear proof of this (under any modelling assumptions), sorry.)
A unique optimum isn't really necessary here (I shouldn't have required it!)
He used the extreme value theorem. The theorem asserts that a continuous function on a compact set (for our purposes closed and bounded set) has a maximum (and a minimum). In this case the the closed and bounded set is the sphere of all possible orientations for solar panels. The theorem is mentioned in most introductory calculus courses, but the proof is definitely not 'trivial'.
That's not actually sufficient to show unique extrema; there an be a set of points on which a function achieves the same, maximal, value.
i.e. cos(x) over x <- [0,4π] -- multiple maxima at {0,2π,4π}.
But we don't really need unique extrema here. Oversight on my part -- there can be (although I believe there aren't) multiple optimal panel orientations. Then the optimal array is an array with each panel having any one of the optimal orientations.
Well, the Calc 1 version isn't good enough because there are several variables here. You have to consider a continuous function defined on the sphere -- to capture all possible orientations of the panel. In fact, it could be a closed subset of the sphere -- to take care of all possible shadows, obstructions, etc. This is nice, you don't see a lot of direct applications of pure existence theorems...
Except it's trivially impossible to generate more energy like this (as he claims)! If a set of solar panels is independent and non-interacting (e.g., they don't shade or heat each other), then their total power output is simply the sum of their individual power outputs. And so their total energy outputs (integrated power) is the sum of their individual energy outputs. (Necessary distinction because their power peaks may occur at different times).
P_tot = Σ P_i
∫ P_tot dt = ∫ (Σ P_i) dt = Σ (∫ P_i dt)
E_tot = Σ E_i
If there is an unqiue optimal orientation for a single static panel (which under realistic assumptions there is) -- optimal in terms of maximizing energy output, E -- then the optimal orientation of a array of independent solar panels is just the same orientation, iterated. Combine inferior panel orientations, and their total remains inferior.
E_i < E_opt (for i <- 1..N)
Σ E_i < N * E_opt
And all those panel angles in his "tree", facing non-South, up, down, towards a wall... they are individually inferior to the single 45° south-facing panel in his static array. So combined together they are still inferior.
Where did his experiment go wrong? I'd start with the shade issue. The whole setup is in intermittent shade (note the tree shadows); and one experiment is sitting near the ground, the other is mounted on a pole. Looks like different shade environments. His output graphs show that the entire flat array is sometimes in total shade, at a time when the pole-mounted "tree" is not:
http://www.amnh.org/nationalcenter/youngnaturalistawards/201...
Another fatal flaw (I had to look this up to check) is that he is measuring voltage, not power, and they are not linearly related in photovoltaics. Actually, the open-circuit voltage (what you get if you stick a voltmeter over a solar cell, when it's not hooked up to a load) is practically independent of irradiance:
http://i.imgur.com/SWGjV.png
This from the solar module datasheet here:
http://www.bpsolar.us/products/3-series-solar-panels-polycry...
Open-circuit voltage going from 200 W/m^2 to 1,000 W/m^2 only increases from 34 V to 38 V. Power output, with a real load, would go up by about 5 times.
So, he never really measured energy production, or anything that remotely approximates it.