On Photovoltaic Generators and Scattering Cross Sections

Subtitle: Dimensional Analysis again.

Our photovoltaic generator has about 5 kW rated ‘peak’ power – 18 panels with 265W each.

PV Generator: 10 modules oriented south-east
South-east oriented part of our generator – 10 panels. The remaining 8 are oriented south-west.

Peak output power is obtained under so-called standard testing condition – 1 kWp (kilo Watt peak) is equivalent to:

  • a panel temperature of 25°C (as efficiency depends on temperature)
  • an incident angle of sunlight relative to zenith of about 48°C – equivalent to an air mass of 1,5. This determines the spectrum of the electromagnetic radiation.
  • an irradiance of solar energy of 1kW per square meter.

The last condition can be rephrased as: We get 1 kW output per kW/minput. 1 kWp is thus defined as:

1 kWp = 1 kW / (1 kW/m2)

Canceling kW, you end up with 1 kWp being equivalent to an area of 1 m2.

Why is this a useful unit?

Solar radiation generates electron-hole pairs in solar cells, operated as photodiodes in reverse bias. Only if the incoming photon has exactly the right energy, solar energy is used efficiently. If the photon is not energetic enough – too ‘red’ – it is lost and converted to heat. If the photon is too blue  – too ‘ultraviolet’ – it generates electrical charges, but the greater part of its energy is wasted as the probability of two photons hitting at the same time is rare. Thus commercial solar panels have an efficiency of less than 20% today. (This does not yet say anything about economics as the total incoming energy is ‘free’.)

The less efficient solar panels are, the more of them you need to obtain a certain target output power. A perfect generator would deliver 1 kW output with a size of 1 m2 at standard test conditions. The kWp rating is equivalent to the area of an ideal generator that would generate the same output power, and it helps with evaluating if your rooftop area is large enough.

Our 4,77 kW generator uses 18 panels, about 1,61 m2 each – so 29 m2 in total. Panels’ efficiency  is then about 4,77 / 29 = 16,4% – a number you can also find in the datasheet.

There is no rated power comparable to that for solar thermal collectors, so I wonder why the unit has been defined in this way. Speculating wildly: Physicists working on solar cells usually have a background in solid state physics, and the design of the kWp rating is equivalent to a familiar concept: Scattering cross section.

An atom can be modeled as a little oscillator, driven by the incident electromagnetic energy. It re-radiates absorbed energy in all directions. Although this can be fully understood only in quantum mechanical terms, simple classical models are successful in explaining some macroscopic parameters, like the index of refraction. The scattering strength of an atom is expressed as:

[ Power scattered ] / [ Incident power of the beam / m2 ]

… the same sort of ratio as discussed above! Power cancels out and the result is an area, imagined as a ‘cross-section’. The atom acts as if it was an opaque disk of a certain area that ‘cuts out’ a respective part of the incident beam and re-radiates it.

The same concept is used for describing interactions between all kinds of particles (not only photons) – the scattering cross section determines the probability that an interaction will occur. Particles’ scattering strengths are represented by disk-shaped areas. The probability of a scattering event going to happen is equal to the ratio of the sum of all scatterers’ areas and the total area (Illustration).

9 Comments Add yours

  1. Peter Mander says:

    Interesting post, enjoyed reading it. I guess you could use the kWp unit to express efficiency in an analogous way to the (a-b)/a form used by Joule (1846) in connexion with the electric motor and by Rankine (1853) and others in connexion with the heat engine.

    1. elkement says:

      Thanks, Peter! I suppose that the input solar radiation would then be equivalent to the mass of zinc?
      One difference I see is that on principle all solar energy could be harvested if the panel consisted of materials with different band width. So I think the limitation of efficiency is not that fundamental. Solar cells have already been built from layers – each layer from a different material, with a different bandwidth. This increased efficiency a lot, but the layered panels were not yet economical (as far as I know… there is so much innovation in solar energy; I am not aware of all the leading edge research).
      I guess, one could model some hypothetical solar cell coating (mathematically) that results in a gradual variation of bandwidth – so that all photons in the spectrum can be absorbed. Of course, it is another challenge to manufacture this at reasonable costs.

  2. Ed Davies says:

    Personally, I don’t like the “p” in “kWp”. Two reasons:

    1) it’s not a peak as you can get more output as you’ve discovered so might be better called “nominal” output and

    2) SI/BIPM, etc, rule against messing with unit symbols in this way: “Units are never qualified by further information about the nature of the quantity; any extra information on the nature of the quantity should be attached to the quantity symbol and not to the unit symbol.” [http://www.bipm.org/en/publications/si-brochure/section5-3-2.html]. So I think it should be written as something like Pnom = 4.77 kW.

    “There is no rated power comparable to that for solar thermal collectors, so I wonder why the unit has been defined in this way.”

    If you ignore that silly “p” it is a rated power. It’s kW, not m² though, as you say, it can be converted to an equivalent area of an ideal panel.

    If anything it’s solar-thermal panels which tend not to be rated with a nominal power output as they lose a lot of energy as heat back to the local environment (via the usual radiation and conduction/convection) so their output is very dependent on the fluid and environmental temperatures (or at least their difference).

    1. elkement says:

      Thanks for the comment, Ed! Of course I agree!

      I think the complaint about a missing peak power of solar thermal collectors was something I heard mainly from finance guys – who want to benchmark all systems in terms of easy-to-compare indicators (like full-load hours or € per rated MW power…)
      The output of solar thermal panels depends especially on the size of your hot water buffer tank – the bigger the more energy you can harvest. The typical number that’s often given (in Austria) is about 300kWh energy per year, so about 30% of incoming solar energy … for a somewhat typical tank not too oversized in relation to the panels.

  3. Happy New Year! Here’s to another year of leading the way with intelligent energy usage and production :-)

    1. elkement says:

      Thanks, Maurice – I hope so yes, in particular because the oil price is rising again and fossil fuel is again less attractive ;-)

      1. elkement says:

        …. and Happy New Year :-)

  4. Erik Brown says:

    Hi Elke, I suppose I told you that I worked for a startup in Europe on solar wafer cutting between 2007 and 2012, so did a lot of cost analysis back then. I also was working with another startup to bring and advanced anti-reflective coating to market. It has layers of different nanoparticles for a variable index of refraction through the film. This would keep the power from falling so drastically as the sun moves off of normal incident angle. Was planning to test it at Fraunhofer Institute in Boston, but the startup was struggling and I had to move on to another project.

    1. elkement says:

      Very interesting story, Erik, thanks for sharing!

      If I recall correctly it was about that time (2011/2012) when many solar energy businesses in Europe struggled or went out of business. There were some solar panel manufacturers in Austria who could finally not compete with vendors from Asia.

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