Have you looked at gravity assists for spacecraft? They do this.

Can't write hard science answer, so just a hint: tidal power comes not from mysterious "gravitational energy". Tides gets their energy from moon's kinetic energy. That's why moon is slowly flying away.

@CortAmmon It seems to me that spacecraft use gravity assist to move around. I want to stay in orbit!

@Mołot my bad. I'm terrible at physics and think both of those would be cool to do, hence I'm asking for both in a terribly clumsy way. If a knowledgeable person may want to edit/suggest correction I would be more than happy to change my phrasing.

Probably related: worldbuilding.stackexchange.com/questions/52725/…

@PatJ It's not an edit suggestion. This clarification probably should be an answer :) And you don't want to generate electricity by degrading your own orbit, right?

@dot_Sp0T A very good question! The answers lack numerics though (or even orders of magnitude).

I'd bet that a specially-designed and really long spacecraft could generate a bit of energy from irregularities in Jupiter's gravity (perhaps caused by storms or something happening below.) I don't have the physics on me right now to figure out precisely how/whether that would work, though.

@Thucydides I would encourage you to post the link to answer, last time you almost convinced me, it is a good option when no onther sources are available, probably in comments in this answer if I remember correctly, but looks like comments are goon now, so I have lost the link.

This wouldn't really work, but I had a thought to lock yourself in an orbit between the sun and Jupiter so you always have solar power. My back-of-the-napkin math says your orbital radius would be ~.5 AU, which is a bit far to have a nice view.

The key difference you've missed between the idea of the tides on earth taking energy from the moon and your situation, is that Jupiter is not orbiting you

Actually, my own comment got me thinking a bit more. You might consider harnessing power from Jupiter's moons, rather than Jupiter itself. Not sure how it would work, but if our tides can do it...

It is probably easier to turn Jupiter into Lucifer.

Depends on how u arrange your ocean of piezo crystals around your enormous space station, you can get free electricity by having tides in your colossal Death Star III...

Forget about gravity -- you're in orbit and already tied to Jupiter gravitationally; you're not going to be able to use its gravity without breaking your orbit one way or another. Instead, use its radiation. Jupiter outputs a crazy amount of it (but you're in orbit so you already know that!). Draw your power needs from the radiation and you should be well set for even the most power-hungry cat video session.

Decrease the radius of your orbit. $E_{total} = -\frac{GMm}{2r}$. This won't give you a steady power but nothing will, so this is the best you can do.

I am not going to do the math...because I don't know it. But conceptually, you have a spherical domicile that rotates like the earth. between and outer and inner shell there is a fluid medium of some sort, as you spin the fluid is pulled toward Jupiter and some sort of hydroelectric system could generate power. I have no idea if this is practical or can be done...just an idea.

@Mołot That's actually not correct. If tides were using the Moon's kinetic energy, the Moon would be moving towards Earth rather than away from it. The actual energy source is the Earth's rotation.

@kasperd sorry. It's much more complicated than both your or my comment. But what matters is - it's kinetic energy used, not gravitational one.

Related meta post: meta.worldbuilding.stackexchange.com/q/4060/627

@Devsman He'll likely be dead shortly from the radiation so that's probably as good a method as any.