What power generation method is the best to sustain a hovering city?

The Technology

We're in the year 2250 on an Earth-like planet. All the space on the earth's surface has been used up by the population of 15 billion, so we're moving into the air. Cities, along with all the farming and other capabilities they need to thrive, are being built on hovering platforms.

The Situation

The construction project is almost complete. Wiring, services, roads, buildings, all have been put into place. There's just one thing left to go in: something on the end of the wires to squirt power juice down them. And nobody can decide what to use.

Supplies

This hovering city will come over another major city once approximately every six months. The city has batteries: total capacity 4GAh. At this point, cables and pipes are dropped, and supplies and power can be sent up to the city. After 24 hours, it disconnects again and moves on.

The Question

What power source is the best to plug into the end of the wires? It has to satisfy the following conditions:

• Sustain a city the size of London (7 million people) for 6 months at a time without refuelling.
• Can supply the city with a deficit of 4GAh over the six months; the deficit will be made up by the batteries.
• BONUS: Can keep the city hovering indefinitely.

I have tagged this because I am looking for a numerical comparison of power sources, since that is the only way I see to qualify a "best" power source to avoid this being too subjective.

• Well, if you're gonna be all hard-science about it, you'd better use Amp hours rather than Watts for your battery capacity. You've got voltage, energy capacity and electric charge rating. All different. All matter. – Serban Tanasa Apr 17 '15 at 11:11
• What kind of system is used to keep the city afloat? Considering the weight of the entire system, I'm not sure an engine large enough is even possible. Do you have an estimation of how much energy we're talking here? – Erik Apr 17 '15 at 13:22
• It can't be done. To lift yourself up, you need to send energy down. The amount of energy being sent down by a city to stay in the air is enough to obliterate anything below it. Your floating city would quickly be the ONLY thing left on the planet. – Erik Apr 17 '15 at 13:42
• I'm not sure I understand the question? I have a cordless drill battery that is 4Ah. That doesn't seem like enough power to do anything? – AndyD273 Apr 17 '15 at 14:00
• I figure you need on the order of 100 GAh to supply your needs, so I think we can safely discount the batteries. – 2012rcampion Apr 17 '15 at 15:41

The only possible answer to your question is nuclear (fusion or fission, I'm not picky) and that still isn't reasonable.

I'm afraid "hard science" answers won't satisfy your criteria. Some critiques of your question.

How Much Power?!
A large city uses GW of power not MW.

According to Wikipedia per capita energy usage in the UK is 244 TWhr / year and this site claims average energy consumption of 35.8 GW. Since London's population is about 11% of the UK total, this makes London's power requirements about 3.9 GW of operating electrical generation. This is about 100 times the energy storage you planned to provide.

City Power
Renewables don't work for you they are energy diffuse - meaning you needs lots of heavy things to generate the necessary power. One of those giant high efficiency wind turbines might have a rated power generation of 2 MW but likely generated power of less than 20% of that. So your city might require 70,000 or more of these to generate its power needs for all things excluding hovering and that's a lot of weight - which we'll find out later, is a bad thing (TM).

4/23/15 correction
London requires about 3.9 GW of operating capacity not the previously quoted 47 GW.

Wind Turbine Rated Power Generation

Just how much might such renewable energy source weigh? I ask because weight will be a hugely important factor, you need something energy dense (lots of energy per unit weight) as well as something that doesn't require fuel $\rightarrow$ so you're talking nuclear power (fusion or fission).

Here's an interesting article about the energy efficiency of various power generation schemes.

Solar Power
Mass produced solar cells can produce up to $175 \frac {W}{m^2}$. However, if you plug inefficiencies into their operation (such as 50% of the time, they generate no power at all and most of the day they produce far less than the advertized peak generation), you'll get about $44 \frac {W}{m^2}$. In order to completely supply the necessary power for such a city (ignoring the hovering), you'd need about 893,141,945 $m^2$ or a square about 30 km on a side worth of solar panel.

Energy Storage
You mentioned story your power in batteries in between "refueling" stops and using it when intermittent energy sources (like solar and wind) aren't available. For a test case, let's think about storing a night's worth of power. A night is 12 hours * 3.9 GW power generation = 47 GWh of power storage.

Battery Specific Power

If we take our storage needs of 47 GWh and divide by the best storage shown on the graph of 200 Wh / kg we get:

$$M_{batteries} = \frac {47 GWh}{200 Wh/kg} \rightarrow = 234,449,761 kg = 234,449 tonnes$$

Batteries are limited by the energy of molecular bonds. However, nuclear power comes from the energy of the strong nuclear force and this is a 1,000,000 times more energetic per unit weight. Even though we suck at extracting this energy in an efficient manner (we can only extract about 0.25% of the nuclear reaction in our power plants), it's extremely high energy density makes it something like 10,000 better for energy storage and extraction than the next best option.

Hovering
Hovering introduces all sorts of issues of its own. The first of which is we generate lift by means of a propellant via conservation of momentum. Aircraft engines and wings use the atmosphere as a propellant but must supply the energy to move it.

The only propulsive scheme ever proposed to produce the kind of thrust you would need is the Project Orion nuclear pulse propulsion. It uses nuclear bombs to provide the energy, detonating about 1/sec. This engine could lift 8,000,000 tons (about the size of a small city into orbit) using only 800 bombs with a yield of about 272 kt each and do it in about 14 minutes.

You'll need something even more energetic than this to lift itself and keep your London sized city floating for 6 months or more.

If we assume a "small city" is one of 70,000 people and London is a city of 7,000,000, then we need to scale this huge craft up by a factor of 100x. The Project Orion designers didn't include any planning for something of that size so I have no idea if that's practical (the engineer in me says we'll run into problems) but let's assume we can. This means we need to boost our bomb yields up by 100x - putting them at 27 mt each.

Scale test of Nuclear Pulse Propulsion

Even this highly energetic propulsion scheme could not carry its fuel and keep a city floating for 6 months. So your city must use a propulsion scheme even more energetic than detonating nuclear bombs under it to keep it floating. Detonating nuclear bombs introduces a host of other issues (radiation protection and variable apparent gravity chief among them). Although we could make life in your city survivable, it won't be comfortable.

But imagine the conditions on the world underneath your floating city with something more energetic than a 27 mt nuclear bomb detonating every second for a 6 month period. We're talking about total annihilation of all life on the planet in very short order.

If you exclude the power required to hover, you could reasonably power your city with either sort of nuclear power.

If you include the energy costs of hovering, then hard science isn't your friend. You'll have to introduce a lot of handwavium to make your dream a reality. The combine requirements for energy storage, propulsion, power generation, etc. all don't fit with the world as we know it - not even in theory. You need fiction to make this work and plausible fiction won't do it in this case.

A Possible Out
One possible out to the above problems is this: physics says energy isn't expended if the object doesn't move in the gravity field. Putting this another way; if you can find a way to levitate this, other than through conventional propulsion, then you aren't actually expending energy.

One possible way to do this is magnetic levitation. Create a track that generates a (hugely powerful) magnetic field and place superconductors in your city. Superconductors expel magnetic fields from the material and this results in levitation.

Although the physics say this is possible, I don't know what the strength of the magnetic field would have to be (certainly many orders of magnitude stronger than the Earth's) and we'll also require a new family of superconductors which don't break down under such powerful magnetic fields.

There are 3 properties that cause superconductors to lose superconductivity; they are temperature, current, and magnetic field field strength. This is WAY above the critical magnetic field strength for anything that we know about.

Chart of Maximum Critical Magnetic Field Strengths for various superconductors.

• Yeah, the energy output to keep the city in the air will destroy the rest of the planet, no matter how you do it. Imagine standing under a helicopter, only one that weighs a few billions tonnes. It would flatten everything for miles around it just by the amount of energy it's putting into the enviroment below it in order to stay up. – Erik Apr 17 '15 at 13:41
• Interestingly, the physics says that if you're not changing the thing's height, you're not actually expending any energy (e.g. it's F*d). So as @SF said, if we could rig something with magnets and superconductors he might actually get this to work without the stupendous expenditure of energy. But that would have to be very powerful magnets and superconductors of a type we haven't seen yet. – Jim2B Apr 17 '15 at 13:49
• I might write up something about the magnetic levitation later but I've got to do work now :( – Jim2B Apr 17 '15 at 13:53
• I keep thinking of doing this in some heavy-gas atmosphere that acts almost like a sea, and having a lighter-than-air gas inside the thing (giant blimp style) - also, I don't think you will be able to "generate" energy from the wind if you then plan on using that energy to move around.. – user2813274 Apr 18 '15 at 20:41
• You read that chart entirely wrong. If "per capita energy usage in the UK were 62 TWhr / year" then each person would be essentially burning 1.85 Billion gallons of gasoline every year. The chart says that 62.7 million people are using 2,187 terawatt hours per year. That means per capita use is 34.88 megawatt hours per year, three orders of magnitude smaller than what you wrote. Did that number not seem insane when you wrote it down? That's all energy. Electricity only is significantly smaller, 47GW power production is higher than the average demand for the entire country. – Samuel Apr 19 '15 at 2:02

Since the easy answer (lots and lots of helium/hydrogen balloons plus pretty much any power source, like nuclear or solar) would be boring, let us reach for something far more exotic:

Geomagnetic.

The city isn't so much hovering, as orbiting the planet. It has a Space Elevator of its own, plus a bunch of enormous sails that "anchor it" against the air.

The space elevator is long and heavy enough on the end extending past the (semi-)geostationary orbit, that it keeps the city afloat, in a precarious balance between falling to the ground and flying into space.

As it travels through the magnetic field of the planet, it acts as an Electrodynamic Tether delivering all the needed electricity for the city - including power to Air Ionisers that stabilize the city's altitude and speed, negating effects of unpredictable winds and loss of altitude due to electrodynamic drag.

In effect, the -actual- energy source is the wind power - not of natural, weather wind but of air drag against the atmosphere preventing the city from stopping and falling; propelling it along the planet surface; the huge sails return whatever is lost through electrodynamic drag of the tether. The tether itself acts as the generator of the wind-powered power plant and supplies energy for adjustments for the air ionisers (acting as jet engines) whenever just adjustment of sails would be insufficient. Ultimately, the original source of power is the rotation of the planet, transferred to the city through air drag.

Edit: The Wikipedia article on Electrodynamic Tether has enormous amounts of scientific analysis.

"In 1996, NASA conducted an experiment with a 20,000-meter conducting tether. When the tether was fully deployed during this test, the orbiting tether generated a potential of 3,500 volts." The voltages available would be absolutely enormous, although the current would be limited by ability to discard ions - the Air Ionisers would play a critical role at the bottom end while the top would need powerful plasma emitters to provide the amperage - they wouldn't so much use the energy as provide it by turning the high static charge into current flow. Unfortunately I can't find any numbers on efficiency of these devices; still, most of their complexity, cost and losses nowadays come from need to provide enormous voltages - in our case we have more than enough voltage. (and if you want a simpler way or more current, lower a cable until it starts spewing thunders at the ground... the people below do have lightning rods, don't they?)

Dissipation of ions on the other end of the cable would need plasma emitters, but there's just ludicruous amount of space for them there.

Essentially the power available is regulated by ability of the sails and ion propellers to catch up with electrodynamic draw, with optional restriction of efficiency of the propellers at dissipating the ions, with the "trailing storm cables" option to remove this restriction.

• This is possible but only as long as the city is stationary. If you start moving it about or put the tether in an orbit which drags the city through the atmosphere, that drag will rapidly pull the whole thing out of orbit where upon the city will crash to the ground in a giant heap while the tether will wrap around the equator multiple times and generally destroy everything and make a huge mess. In order to be plausible such a levitating city will have to be anchored at the equator and not budge an inch. But otherwise an elegant solution. – Jim2B Apr 17 '15 at 16:09
• The problem gets a whole lot easier to solve on a lower mass planet - like Mars. – Jim2B Apr 17 '15 at 16:10
• @Jim2B: The tether synchronous to the ground would need to be with the "0g point" (center of mass) at geosynchronous orbit. There were plans for much shorter tethers skyhooks that move at a much lower orbit, at speed far exceeding Earth rotation speed. This one would trail behind Earth rotation speed, so it would need to have the center of balance outside GEO. Air drag, instead of slowing it down, would act to speed it up - which would lead to increase of altitude, reducing the drag), and allow to recuperate losses of speed. – SF. Apr 17 '15 at 16:17
• @Jim2B: Also, no worries about it wrapping around. If, say, due to lifting a way too heavy load, or some other stupid engineering failure the city crashes, it would likely break apart and the tether devoid of the weight would float into space. – SF. Apr 17 '15 at 16:20
• @Jim2B: It is self-regulating in a small margin. Electrodynamic drag backwards, air drag forwards (as the planet + its air overtakes it). Electrodynamic drag can be regulated by power usage (even wasting it). Air drag is lower in lower, denser atmosphere, increasing speed and raising the orbit. City in upper atmosphere gets less wind propulsion and falls lower. Thus altitude -> wind propulsion strength -> orbital speed -> altitude, closed stable loop; also the "average" air propulsion cancels out with electrodynamic drag. – SF. Apr 17 '15 at 21:24

The power source that's best for a hovering city is the one that has the highest power to weight ratio.

There's a nifty new solar panel that's essentially printed on kapton plastic, and provides about 150 MW of power if spread over an area the size of London (1,572 $km^2$). That's not a very big power plant, but it weighs very, very little. The downside, however, is that you're shading your city, and increasing your citizen's needs for artificial lighting. However if you instead put the panels on wings away from the city you can increase the size of the panels exponentially while preserving the direct light for the city. People below might complain. Also keep in mind that these are fragile - more weight might be needed for heavier plastics that can withstand the winds and storms you fly around in - but my guess is you'll actively avoid turbulence anyway so it might not be an issue.

Beyond that, you're really looking at nuclear power. You can produce tremendous amounts of energy using nuclear reactors and it weighs significantly less than coal, oil, gas, and other energy sources. This will probably supply the bulk of your power. If you correctly harness the waste heat you won't need the huge cooling towers, you'll instead use it to provide for all the heating needs in the city (hot water, hot air, cooking, etc). Consider nuclear submarines and air craft carriers as examples - two smaller nuclear reactors designed to support a small city worth of people for months away from port at a time. You'll just have to scale it up further.

• If we use your numbers to generate 47 GW, you'll need 320x this amount or 492,560 $km^2$ – Jim2B Apr 17 '15 at 22:57
• @Jim2B Exactly - solar could only be one small part of the power supply. – Adam Davis Apr 17 '15 at 23:54

I can't even begin to answer the Bonus of how to make that city hover. But for operating the city...

TL;DR:

You can power your city with rooftop solar. But then you won't have rooftop recreation and farming available.

I worked on a demonstration project of rooftop solar power, just plain-jane flat solar panels, but it was in Abu Dhabi, where there is lots of sun. It was much smaller in scale than what is needed to power London, but it gives me a good idea. Sorry I don't know how to do the pretty math things, but I'm just scaling up proportionately.

I had calculated that you need 2.8m residential units and also the associated Ground Floor Areas (GFAs) for a 7million person compact urban development.

I also mentioned I received approvals on all the utility requirements for a complete standalone city I project managed for 100,000 people and the facilities, etc. associated.

For a city with 100,000 people (and all the other 'buildings'), the total demand load was 319MVA. You're a city in the future, so much more efficient, so we're going to say a thousand people and their other facilities, homes, retail, etc.: 1,000pax:1MVA. Your seven million require 7,000MVA.

Our rooftop solar project calculated that for about 1,000 people (and the offices, clinics, hotels, schools, etc. that come with it) we needed the rooftops of 4 hectares (ha): a lot, but if roofs are 80% of the urban portion (4,500ha), that means you can have 35,000ha of solar, and I'm sure you will have advanced solar by the year your story is happening. Enough to power your city for sure.

The problem here is that we used a lot of your rooftops as recreational and farming uses in a previous question, in order to keep the city compact.

I'm happy to calculate your demand loads, stormwater, domestic water, wastewater, TSE, etc. if you like. :)

• Powering a city with roof top solar power is doubtful. At 150 W/sqm, onyxsolar.com/power-per-unit-area.html, peak power generation, we're talking 1/2 that for daytime operation and 1/4 for daily operation, so approximate it at 37.5 W/sqm. 47 GW / ~47W/sqm -> 1 billion+ square meters of solar panels is a square 32 km on a side. These cells lose efficiency over a 10-20 year lifespan so it'd be a never ending task to replace the failing cells. – Jim2B Apr 17 '15 at 20:53
• @Jim2B, many current cities are covered with high densities of street lights, whose rated life spans are on the order of a decade or two. Replacing these is, as you said, a never-ending task, but we do it without much fuss. energyexperts.org/EnergySolutionsDatabase/… – Vectornaut Apr 18 '15 at 16:05
• @Jim2B, replacing failing nuclear power plants is also a never-ending task, although it's carried out in giant bursts over long time periods rather than in small bursts over somewhat shorter time periods. – Vectornaut Apr 18 '15 at 16:07
• @Vectornaut, true but the power plants in nuclear vessels only need "refueling" once every 20-25 years (I understand commercial plants refuel more frequently than that). I just imagine maintain 1 billion square meters of solar panels would require more effort than maintain 47 GW of nuclear capacity but I base this on gut feeling more than intimate knowledge. – Jim2B Apr 18 '15 at 20:10
• @Jim2B, I guess our guts just say different things. It's not obvious to me that maintaining a nuclear power plant, and then retiring it and building a new one when its time comes, should be easier than maintaining an equivalent rooftop photovoltaic installation, especially if folks in 2250 use things like printable solar cells (gm.dk/es/Solar-SP). – Vectornaut Apr 18 '15 at 23:04

Never going to happen unless you change your definition of hover to include hovering at a height of 35,786 km, a.k.a. geosynchronous orbit. Drop in an O'Neill cylinder there and you can maintain self-supporting colonies of 7 million people. O'Neill cylinders could be scaled up to support much larger populations too. And you could deploy large numbers of them in geosynch, Lagrange points, or orbits in general.

You will need to master mining the asteroids too, but that should be realistic by 2250. You probably also want a space elevator to make getting up and down cheap.

Also, by 2250 there would be no technical problem supporting a population much larger than 15 billion people on the earth. You adopt widescale fusion power, solar power satellites, or whatever combination of power sources that make the planet energy rich and clean and you can grow all the food you need with hydroponics or suchlike. Asteroid mining will also be useful for getting resources that are relatively rare on the planet.

However you eventually reach the point where you want to move population off planet if the population grows without bound.

This is a supplement to @SF.'s answer just because.

Atmospheric electricity can be used to provide you some power, and technically could be done even without a space elevator.

In 'fine weather', the potential, aka 'voltage', increases with altitude at about 30 volts per foot (100 V/m), when climbing against the gradient of the electric field.[3] This electric field gradient continues up into the atmosphere to a point where the voltage reaches its maximum, in the neighborhood of 300,000 volts. This occurs at approximately 30–50 km above the Earth's surface.

You could also put some capacitors and stuff to capture the lightning strikes that would be attracted by this giant structure too for a quick power boost.

As always, direct matter conversion ($e=mc^2$) yields the "best" power source and one does not need to have batteries or wires to charge them. Just suck in air and convert it to energy. Using @jim2b's calculation of 47GW for a city you only need half a kilogram of material a year to power your city.

Given our pace of progress it is entirely plausible that we will know how to safely harness a technology of that sort in 200+ years.

As others have said, levitation is your best known way to achieve a hovering city and again, the pace of discovery and innovation cannot be underestimated - especially if we devise an ASI.

However, Isaac Asimov (I think?) came up with a 'gravitational lens' that focused a more distant body's gravitational attraction onto an object that allowed it to overcome 'local' gravity - focusing the Sun's or Jupiter's gravity on a car for example. How plausible this is is unknown but we can focus light as well as other frequencies of the EM spectrum. Gravity has the same drop off characteristics in its 'field strength' $1/r^2$ - perhaps that is a tenable enough connection to believe that one can focus gravity just as we do light.

• We already have that "battery" technology of converting matter into energy. But we need anti-matter, which takes energy to make... Still a great potential battery would be some anti-matter container. – akaltar Apr 19 '15 at 0:43

The answer is very simple, but you need to throw away your assumptions about how the city hovers in the air. It would not need to have any sort of power plant or batteries, but simply be built as a hot air balloon of enormous size.

Buckminister Fuller described this as a "Cloud 9", after inventing the geodesic dome, noting that when the radius of a sphere doubles, the internal volume increases 8x. For very large spheres built on geodesic dome principles, once you get beyond a diameter of about a kilometre, the mass of air contained within is so vast it massively outweighs the structure of the dome. At this point, heating the air within by as little as 1 degree F compared to the outside air would cause the dome to float.

A "city" capable of containing all the people of a major metropolitan city like London might not be possible as a single sphere, but a cluster of spheres, each several kilometres in diameter and linked together by cables, walkways etc. might be feasible. This has the added bonus of allowing "suburbs" to cast off for various reasons, as well as preventing a singular disaster from bringing the entire city down at once. Urban renewal is also much easier, since old bubbles can be detached and new once installed in their place.

The heat energy needed to keep these huge hot air balloons aloft can be collected by solar energy during the day, and venting waste heat from internal activities and machinery into the interior at night. Moving farther north where the average temperature is cooler makes floating these bubbles much easier, while moving south to the tropics would require much more heat input into the domes to maintain a temperature differential between the inside of the dome and the outside air.

One issue with any flying city, especially large ones, would be to minimize the unwelcome effects of shadowing huge areas of land underneath. "Cloud 9" balloons could be largely transparent, allowing some sunlight to pass through to the people below.

• Heat flows from hottest to lowest, so you would have to keep heating the spheres. – Jimmy360 May 25 '15 at 21:33
• Since Fuller calculated that a temperature differential of 1 degree F could lift a half mile diameter dome off the ground, the waste heat of machinery and human activity in the dome should be enough to keep it aloft under normal circumstances. – Thucydides May 25 '15 at 23:06
• Using waste heat is difficult, and one must remember that temperature is average heat. The heat energy to do this is massive. – Jimmy360 May 25 '15 at 23:07

It would greatly depend on how high you want to hover. The density of air reduces as you go too much higher. And using air flow (or resistance of certain shapes to air flow) can provide lower energy solutions at lower atmospheric levels.

This is not dissimilar to buoyancy. At lower depths, the buoyancy exerts more force than gravity (with density of material determining what the equilibrium point is).

Certainly, egg-drop competitions provide a lot of insight into how to resist gravity with as little energy as possible.

But if you want to float too high up, the atmosphere will be of little help.

Some things to consider:

• Nuclear fission is high-output, but also very heavy not good for floating city.
• Nuclear fusion is even higher output but the weight is same so not good.
• Renewables such as solar,wind, might be good for you because they are slightly lighter but also they don't produceas much energy
• Fossil fuels plants are similar to nuclear, they produce less energy but are still heavy - not good.
• Hydroelectric or tidal is impossible/impractical that high in the air.

Out of these option I think the best is probably mix - some renewables for less weight and maybe small scale normal power plant for more power.

Batteries can be charged but will not produce energy, only distribute.

• Which weighs more 47 mid-sized nuclear plants, 120,000 wind turbines, or 1 billion square meters of solar cells? Renewables are energy diffuse. Any single one is much lighter than a conventional plant but you need lots, and Lots, and LOTS of them to make the same amount of power. They won't be lighter weight than nuclear. – Jim2B Apr 17 '15 at 22:54
• ok @Jim2B well it's not like this is complete useless answer – cflat Apr 18 '15 at 8:32

There is a way to pull this off without blowing up a 27 megaton nuke every second!

For the city to be in orbit is obviously the best way to have it 'hover'. But in orbit there is no air! So what can we do?

Well, for an object to be in orbit it's center of mass has to be at that elevation where its in balance between gravity and the centrifugal force.

We also need air on the city, so the best solution I can think of is to have a counterweight on the city, above it, at a point that is far up enough to have the entire thing's center of mass at that perfect balance point.

You will also probably need many carbon nanotube ropes of considerable diameter to hold these to massive objects together.

A problem is that the city wont be able to stop to refuel. This can be solved by having ships fly to deposit the goods on the actual city, which would be expensive, but it would still be less expensive than having some sort of engine on 24/7.

The distance would probably have to be adjustable because of the refueling (mass changes). This can be overcome if it is guaranteed that the waste removed has a similar mass to the new goods that are coming onboard. Another solution to prevent having to adjust the distances would be to have small correction thrusters or movable weights that move about the ropes. (Elevators)

A benefit is that the carbon nanotube ropes could serve as space elevators, and then you can have a space suite on the counterweight, where people can hang out and see the stars.

Additionally, you can have vast solar arrays on the counterweight, to power the city below. Or you can have nuclear generators up there, at a safe distance from the city. Solar panels would have direct light (no atmosphere to worry about, so the light is more intense).

• I don't think the city has to be in orbit. – HDE 226868 Apr 18 '15 at 23:42

Why do you want a single type of power? We as a society combine multiple types for a very good reason.

Assumption - 7 million people = 3 million households. Power consumption from the hovering is a significant power draw.

Now, a 4 GAh battery is pitiful for this city. The average British household uses 4 MWh/year. As I am lazy, I am equating 1 GWh = 1 GAh.

That means that the battery can power 1000 households for a year, or 3M households for 2.67 hours. Less as we have to power the hoverdevice, and also maybe power industrial/commercial load.

Any single power source is unlikely to be able to meet our needs.

Solar will lead to the city crashing on the first night. Wind will lead to the city crashing when the wind dies down for several hours. Coal/Gas would be too heavy and would lead to the city crashing when the fuel runs out. Nuclear would either be too heavy (needs a lot of water as shielding) or would ether explode or leak radiation (can't remember off hand which is likely to be the biggest issue).

Power would come via a hierachy.

1) Solar/Wind.

2) Hydro.

3) Waste to energy.

4) Demand Response.

5) Battery Storage.

6) Fossils.

Power would need to come from a variety of sources, and demand response would be essential. Solar and wind would be the best in terms of total power produced over a year/weight required at the start, and by using both, you reduce how 'intermitant' they are. You would then need to have some contingency plans for when the sun and wind were both low.

You could stretch the renewables a bit further through the use of hydro. The basic premise would be'catch water in the city (wastewater and rain), empty it out into pipes over the side, and at the bottom of these pipes, have a turbine. The higher the city floats, the more turbines you can have.' This also can be used to mitigate the risks of intermittent power, as you could store the water in the city until it is needed.

Waste to energy solves two problems in one. Do it.

Demand Response/Storage. Obviously, there is the battery as storage, we also have a great storage type, similar to 'pumped hydro' storage systems. We have a pumped city storage system. When the wind is blowing and the sun is shining, have the city slowly ascend to the highest point it can by using more power for the hover device. At night, when the wind drops, take power out of the hoverdevice and the city gradually sinks. You can also use more normal demand response and simply cut off people's power if the wind and the sun both drop - people would prefer the TV to cut off than a horrible fiery death in a city crash.

Then, you would want some fossil fuels for when these options aren't sufficient. Gas would likely be the best, as it can be turned on/off quickly to react to sudden drops in wind/solar. I really haven't looked up energy output per weight required, but that doesn't matter, as this is expected to be a low % of total annual power consumption, this is just "the city is about to crash, turn on the backup power". Coal takes hours/a day to spool up.

Edit: I should make a comment on realisticness, and over 200 years of technological progress. I am assuming that all technology increases in efficiency by roughly the same rate, and that this rate is sufficient to make the question answerable. The evolution of batteries is the one exception, which I am assuming have evolved to the point where we can afford the 1 battery mentioned in the OP, and no others. These assumptions may involve some handwaving, such as assuming floating cities can be done without annihilating everything below.