Fibonacci Solar Array Update

from The Capacity Factor

'Solar-panel "trees" really are inferior (or: "In which hopelessly inept journalists reduce me to having to debunk a school science project")

Some poor 13-year-old kid is all over the news as having made a "solar breakthrough". The news is to blame. All the usual suspects -- popular environment blogs, tech magazines -- blindly parrot the words of this very misinformed (not to blame him, he's an unguided 13 year old) kid.

Usual suspects:

[Popular Science] 13-Year-Old Designs Super-Efficient Solar Array Based on the Fibonacci Sequence

[Slashdot] 13-Year-Old Uses Fibonacci Sequence For Solar Power Breakthrough

[The Atlantic Wire] 13-Year-Old Looks at Trees, Makes Solar Power Breakthrough

[UberGizmo] 13 year old uses Fibonacci sequence to improve solar efficiency

[Inhabitat] 13-Year-Old Makes Solar Power Breakthrough by Harnessing the Fibonacci Sequence

[TreeHugger] 13-Year-Old Makes A Solar Breakthrough With Fibonacci Sequence

This is his writeup:

[AMNH] The Secret of the Fibonacci Sequence in Trees

Essential summary:

- He's comparing arrangements of solar panels to maximize total electricity output
- One is a conventional, flat, 45°-tilted (roughly latitude), south-facing array
- The other is an oddly-arranged "tree" with panels facing all directions: up, down, towards a wall... (some Fibonacci mysticism involved here)
- (Writeup has photos of both setups, and graphs of data)
- Both experiments have equal numbers and types of solar cells
- He measures the (open circuit) output voltage over the cells connected in series
- He thinks the "tree" is superior (generates more electricity) than the optimal flat array

This is, I'm sad to say, clear nonsense. I'll take this in two parts: one, why his experiment is, unfortunately, completely broken (sorry again). Two, why the imagined result is impossible nonsense.

## Broken experiment

Most importantly, by mistake he did not measure power outputs from the solar cells. Instead he measured voltage, without a load attached ("open circuit"). They are barely related -- in solar cells, voltage is actually almost a constant, independent of power.

The actual power delivered by a solar cell is not linearly related to the open-circuit voltage; actually, as a semiconductor, it has a horribly nonlinear relationship. Here's the current-voltage (I-V) curve:

[National Instruments] Photovoltaic Cell I-V Characterization Theory

V_{OC} denotes the "open circuit" voltage: when there is no load attached, and no current flows (I = 0). Power goes as V*I; a real solar system will maximize efficiency, by working at the point on the I-V curve which maximies power (P_{MAX}).

The kid is measuring V_{OC}. As it happens, this is practically *independent* of power output! Here's how the I-V curve changes with incident solar power ("irradiance", areal density of radiation in [W/m^{2}]). As this solar module datasheet shows, V_{OC} is almost a constant, regardless of incident light!

[BP Solar] 3 series solar panels Polycrystalline (Data sheet)

In this module, V_{OC} stays close to 35 V, over a 5-fold range in irradiance. Whether the incident light is bright or dim, the open-circuit voltage is the same.

Of course, P_{MAX} (not shown) goes up roughly linearly with solar input. You must expect this: when you have 5x more solar power input, you have ~5x more electric output. But in Power = Voltage * Current, it is current, not voltage, which increases. (For those who are familar wiht physics, this is reasonable because: twice the brightness doesn't mean twice as much energy per photon, but twice the number of photons. The voltage reflects the energy of individual excited electrons: an electron with 1 eV energy can travel against a 1 V potential difference. The maximum current depends on the number of such excited electrons. More light means more excited electrons, but each with the same energy.)

End result: measuring the solar cells' V_{OC} over time, and adding them up, is garbage data, and has nothing to do with energy production.

## Unreasonable theory

As for why the result is impossible. I'm not sure I understand the confusion by which people think there *could* be some advantage, to orienting panels at sub-optimal angles. That somehow combining sub-optimal panels, together, makes them generate more energy in the net. Here's my argument, in case it helps clear up misconceptions.

Take an collection of solar panels (indexed by 'i'). Their power output is the some of their individual outputs. So, their total energy output (power integrated over time) is also the sum of their total energy outputs.

P_{tot} = Σ P_{i}

∫ P_{tot} dt = ∫ (Σ P_{i}) dt = Σ (∫ P_{i} dt)

E_{tot} = Σ E_{i}

Suppose some orientation of a solar panel, 'OPT' (maybe south-facing, latitude-tilted), is superior to some others, index them by 'i' (say, facing north, down, up, towards a wall...) Then adding together 'N' such panels, in any order, is strictly worse than a uniform array where all panels are at their individually-optimal angles:

E_{i} < E_{OPT} (for i <- 1..N)

Σ E_{i} < N * E_{OPT}

So: if the individual angles in the "tree" are worse then the 45°-tilted south-facing panels in the flat array (they obviously are), so is their combination.

(Implicit assumption: that the panels are non-interacting, e.g. they do not obstruct (shade) each other, or heat each other, etc. The panels in the "tree" do actually shade themselves, which makes them strictly worse and does not change this result).

How did this confused science project became international news?'

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PLZ LEEVE A MEZZAGE KTHNXBAI