[set fair (OP)]

I've heard suggestions that hydrogen cars are the future. More generally, power stations could electrolyse water into hydrogen into hydrogen and oxygen when demand is low and burn them to produce electricity when demand is high. So how much energy is lost in this cycle as a percentage? Does anyone know the cost of the extra equipment needed?

[alancalverd]

The argument is that conversion efficiency is irrelevant. You have to build your unreliable sources llike wind and solar, and your inconvenient sources like tidal, with several times the anticipated peak demand capacity since on average they only deliver 10 - 20% of their rated power. Now when the wind is blowing and the sun is shining, like today, you have a potential free surplus of up to 4 times actual demand. At present, we pay the windmill companies a subsidy for not generating power  when it is available but not needed (pity it doesn't apply to any other industry - imagine the taxpayer having to bail out overstocked greengrocers!) so it would be more sensible to store the energy by any means than not to generate it at all.   

That said, the primary efficiency of electrolysis can be over 80% and the overall efficiency including delivery of compressed hydrogen to a car, over 50%.

It's a very sensible idea because we already have a national gas grid that used to run on 50% hydrogen made in Britain, and now handles 100% methane, imported from unreliable sources like Russia. My preference would be to investigate

4H₂ + CO₂ →2H₂O + CH₄

which is easier to handle as a liquid, loses less in grid distribution as a gas, works in  cars with very little modification, actually sucks some CO₂out of the atmosphere, gives you really clean water from the sewage, sea or river gunge you electrolysed, and makes windmills useful.

[syhprum]

Sounds like too good an idea to actually be used I am sure some ingenious ways will be found to ignore it   

[chiralSPO]

Splitting water into hydrogen and oxygen using electrolysis is quite easy to do (and scalable), with reasonably efficiency. It doesn't necessarily have the same round trip efficiency as pumped hydro, but electrolysis also takes up several orders of magnitude less space than pumped hydro to store similar amounts of energy.

To alancalverd's point--hydrogen is annoying to handle, but can be reacted with CO₂ to make hydrocarbon fuels. Alan mentioned methane, but it is also possible to use Fisher-Tropsch type chemistry to make essentially any alkanes with between 1 and 16 carbons.

These liquid or gaseous hydrocarbons are useful as portable forms of stored energy, but if we are just thinking of stationary power storage, then another way to avoid the hydrogen problem is to split something other than water. Zinc oxide is a leading contender: 2 ZnO → 2 Zn + O₂
• No membranes required to keep the O₂ and Zn separated, because one is a gas, and the other is a solid (a significant amount of the expense of water splitting devices is the membrane that keeps the O₂ and H₂ apart while allowing the ions to flow efficiently)
• Zinc metal has an energy density of about 38 MJ/L (40 kWh/gal), which is comparable to gasoline, while even liquified hydrogen only has an energy density of about 8.5 MJ/L (8.9 kWh/gal).
• Zinc is abundant, cheap, nontoxic, and easily recycled.

But zinc is also very heavy, so it would not work so well for powering vehicles...

[Bored chemist]

It's a very sensible idea because we already have a national gas grid that used to run on 50% hydrogen made in Britain

Really?
Where's that then?
Because all the big pipes of the high pressure network these days are not designed for hydrogen.

There are moves to try introducing some hydrogen.
https://hydeploy.co.uk/
but obviously, in the same way that the change to North Sea Gas required changes to equipment- fires cookers, boilers etc, a change to a largely hydrogen fuel would require a similar conversion.
It's not impossible but...
Who's paying?

[Bored chemist]

The argument is that conversion efficiency is irrelevant. You have to build your unreliable sources llike wind and solar, and your inconvenient sources like tidal, with several times the anticipated peak demand capacity since on average they only deliver 10 - 20% of their rated power.

Not really.
Once you decide that "unreliable sources llike wind and solar, and your inconvenient sources like tidal, with several times the anticipated peak demand capacity since on average they only deliver 10 - 20% of their rated power. " are better than trashing the world, you more or less have to build the storage to make up for shortfalls.
So it's hardly reasonable to say

The argument is that conversion efficiency is irrelevant.

Electrolysis is unusual in that it's a thermodynamically reversible process and in principle operates at near 100% efficiency.

As you say "That said, the primary efficiency of electrolysis can be over 80% " and the efficiency of fuel cells is up to 60% (much higher than that if they are part of a CHP system)
So the overall efficiency is something like 48%

And I suspect that's the answer to the question the OP was asking.

[alancalverd]

Here's the argument about required efficiency.

Assume you want to power everything by windmills. On average, they produce about 10 - 20% of their maximum rated power, so in order to keep the lights on, the trains running, and replace all industrial heating and road transport with electrical power (never mind

who's paying?

) we need to install about 5 times more  generating capacity than we actually need. But some (warm, windy, weekend) days will have above-average generating potential - maybe 6 times demand. On these occasions it really doesn't matter what we do with 80% of the electricity: we can switch off the windmills and pay the owners a subsidy, or store the energy with minimal efficiency, which would clearly be more sensible, or with reasonable efficiency, which would be very sensible indeed.

For this reason I have always advocated electrolytic storage as an essential part of any energy system based on unreliable generation of electricity, and instead of subsidising windmills by whatever means of tariffs or direct subsidy, we should require every windmill to be built with at least 5 days' storage capacity at average output before it can be  connected to the public grid.

Apropos transmission: It's true that the gas grid now uses mainly plastic pipes for bulk transmission but distribution at the point of use is still through steel and copper pipes, and the storage and process plant is mostly steel, so the economic equations are not obvious: should we run everything on hydrogen and replace the bulk transmission pipes (or maybe allow substantial losses of "free" hydrogen? - remember that the fractional loss per meter decreases with increasing pipe diameter), or increase the capital plant, lose a bit of overall thermal efficiency, and transmit synthetic methane?   

I'm very much in favor of synthesising methane and higher alkanes because we already have the technology to use hydrocarbon fuel, it works very well, and if road vehicles are anything to go by, you have to burn as much energy to make a new  car as it will use in its life, so it's better to use what you have rather than make new ones. And not just road transport. I regularly go to work in a 40-year-old plane  (piston aero engines are rebuilt every 300,000 miles or so, not scrapped, and the aluminum bits just need repainting every 15 years - the only maintenance problem is whether veganism will kill the supply of replacement seat leather!) and have booked a holiday on a 50 year old arctic cruise ship. Synthetic octane and dodecane will keep the transport system running long after all the lithium batteries have caught fire.

[Bored chemist]

Synthetic octane and dodecane

Synthesised from what?
Plant/ algae derived fuels may make more sense.

[alancalverd]

As discussed above, synthesised from "free" hydrogen and any atmospheric CO2 you want to get rid of, either direct from the air (as Audi have done recently), from a smokestack, or from any waste organic gunge you have to hand.

Problem with most direct production from living cells is that fermentation releases CO2, which is supposed to be the problem, and all current biofuels are hygroscopic which prevents their use in most existing engines unless well diluted (at least 1:5) with fossil hydrocarbons. Brand new "biobuses" are OK if you are developing whole a new urban transport system, but there are millions of ordinary diesel buses and trucks out there with a good 500,000 miles left in the engine and a worldwide distribution network for heavy alkanes.

Plus, of course, a necessary and growing surplus of free electricity with no means of storage, which we are currently paying people not to produce. It's as mad as "set aside", though slightly less immoral.