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> Transmitting that energy from where nobody lives to where people do live becomes the problem with that.

That’s kind of what we do today for pretty much everything. Most of the population on the planet doesn’t live near oil rigs, refineries, solar farms, power plants or wind. In fact most of the population doesn’t live near where we produce our food or most of the things we need for survival.

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Building HVDC lines from North Africa to Europe, for example, wouldn't be a huge feat of civil engineering. Rather standard stuff, really.
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Spain and Morocco already have a 1.4 GW DC interconnect and the XLinks project intends to connect Morocco and the UK.
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we don't need something that long distance at all

EU has enough areas with sparse population and not that much nature which also are south enough to have it work out well with solar panels of the current generations.

And besides that even most EU countries have enough places in them to still put a lot of solar panels without much issues and/or replacing fields.

going as far as North Africa is a bit too far to be convenient for power transport

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"from North Africa to Europe" is, to be clear, ~9 miles in spots.
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I've been watching the math of batteries and cargo ships and we may not be too far from shipping electrons generated in the Sahara to the UK and Europe at a reasonable price. That totally changes the game if you have cargo ships moving to where the power will be needed. I can imagine these ships going to where the weather is predicted to cause an issue to help even out the grid and just in general creating a responsive base load for the world. It sounds like sci-fi, but with the direction batteries have gone it isn't that crazy anymore.
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Does it still work out if you take into account the insurance premiums for a cargo ship stacked with batteries? Can't imagine the fire hazard is pretty.
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Wait until you hear what other stuff gets shipped
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Oh, there's precedent for shuttling freighter size metal fire hazards intercontinentally to top up charge, is there?

How does that work out in cost per kWh? Profitable operation anywhere close?

Crisis relief (as suggested by jmward01 here) may be another matter, but setting up the ability to do this on scale, and maintaining it, can't be anything like easy economically.

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It could possibly be combined with a solution to the storage problem: store the energy in some transportable chemical form like hydrogen, methane or the electrolyte of a redox flow battery.
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Yeah possibly. Synthetic hydrocarbon fuel that's already compatible with transportation infrastructure and energy consumers might be the best bet.
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We will probably be using that in aviation for a long, long time. Turbines are pretty efficient and jet fuel has a remarkably high energy density for something that does not easily explode.
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The only real problems with long distance electricity transmission are political and to a lesser extent financial. Technically it is solved problem.

The Desertec project could have turned a relatively small patch of Libyan desert into a solar farm that could supply all of Europe's electricity except that politics makes it impossible.

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in the distances we speak about we do so all the time with more centralized energy sources (like e.g. nuklear) due to their centralized nature

the issue is less the transport distances but changes in "from where to where" sometimes needing some extensions/improvements to the power grid. Through commonly in ways which anyway make sense and all pretty much "standard" solutions well understood. Through there are some more complicated exceptions to that.

EDIT: "distances we speak about" assumed less many local less dense populate/suitable spots across the EU, not a mega project like a energy pipeline from North Afrika.

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Buckmister Fuller envisioned a worldwide high-voltage transmission network implemented with 1980’s technology, there just isn’t the worldwide political will or cooperation to build it.
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