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How Long Will It Take To Break Even With Sustainable Energy?

Fukushima I reactor unit 3 by Digital Globe

Since watching the preview for the movie The 4th Revolution - Energy Autonomy I wondered what it would take to reach a break even, i. e. generate 100 % renewable energy. Personally I am sceptical that decentralised energy production will equally empower people as it will benefit environment and society, but I did and do believe that it is possible to switch from fossile power sources to renewable energy. The movie didn't show anything groundbreakingly new, but it documents many important projects and people.

It didn't answer my question how long it would take to make the transition, which Wikipedia tells me is called the energetic amortisation time (the English version doesn't currently seem to have nice tables, so I used the German one for my estimates). According to that source, it takes 1.7 to 5.8 years for solar panels and 2.1 to 6.1 months for wind turbines to output as much energy as necessary for the erection. As far as I know maintenance is a lot higher for wind wheels, but that still makes a world of a difference.

So, how long would it hypothetically take to replace the entire energy production with renewable energy? I'm not a mathematician, so please correct me where I am embarrassingly wrong.

E
annual energy usage
ta
energetic amortisation time for a specific alternative energy source
fa
fraction of (that) renewable energy source in relation to original E
Ei
annual energy investment into producing renewable energy power plants
fi
Ei/E (fraction of total energy invested in renewable energy)

Using compound interest calculations I get to the following naive formula for the fraction of renewable energy after t years when a fraction of fi of the total energy consumed is invested in renewable energy:

fa[t] = (1-(Ei/E + 1)**t)/ta = (1-(fi + 1)**t)/ta

I'll assume that the investment is 1 % of the annual energy transformation/usage (fi = 0.01) - I have no idea whether that is reasonable, but it seems low enough not to inconvenience anyone too much.

For a world of solar panels in northern Europe (ta = 5.8 years) that gives us break even in 193 years (with an investment of 3 % only 65 years, and in southern Europe where the sun shines a lot more (so ta = 1.7 years) 100 % of the Energy usage can be saturated by photovoltaics panels after only 34 years if 3 % of the energy budget are invested ... at 1 % it's still 100 years. That is not good, considering most solar panels are not guaranteed for more than 30 years. Now I understand why everyone is talking about solar breeders that are supposed to self-sustainedly produce solar panels without cannibalising energy from other sources. The way I see it, they'd still need some energy for bootstrapping, but as the next paragraph will show some technologies are vastly more efficient than others.

As mentioned above, wind power is supposed to amortize after 2.1 to 6.1 months. Again, assuming only 1 % of the annual energy budget is invested in building wind parks, how long would it hypothetically take to be able to supply 100 % of the original demand with wind parks? In the optimum case it would only be 17 years ((((0.01+1)**17-1)/(2.1/12))=1.05), in the worst case 42 years. (My naive calculation also says that achieving the 40 % wind energy necessary in addition to the 60 % water power already available in Austria should be possible in 19 years).

So, in effect it would appear that solar power is great for urban fa├žades but problematic in terms of energy amortisation. If solar breeders were to be established (or other means of converting cheap renewable energy to solar panels in places where demand is low), the outlook might be different, but for the moment the more efficient technology for breaking even with alternative energy appears to be wind power.

CAVE: fossile energy is everywhere, from transport to food (fertiliser, farm automation, transport) to construction (cement production, machines) to health sciences ... so the big E has to include all of those consumers.

NB: I have also been pointed to Volker Quaschning's page who qualifies those amortisation time estimates for solar panels. For the production alone, he calculates 7 to over 100 months depending on where the panels are used, a range which is more or less compatible with above estimates.

The image above is licensed under the Creative Commons Attribution Share-Alike 3.0 Unported License from Digital Globe via Wikimedia.