Here's a link to your fuel pump relay. BMW E39 Fuel pump relay locations Tech Bentley Publishers Support
SOURCE: bmw 540i fuel pump
You must check to see if you have power at the fuel pump relay which is probable in a fuse box under the hood.
SOURCE: 1999 bmw 540i automatically shuts off
It sound dumb, but check if both battery posts are OK. Pull on both of them one after the other to make sure they wont budge at all. Cause i had this problem once where the negative post had broken, but inside the battery where the post and the plates that make up the battery's cells become one. But since the post was still sitting in the battery, it touched just enough to give the car juice to start and all, but when i hit a big enough bump, the post jumped up so the car had no more ground, and with no ground to work with, the spark plugs can't make any fire so the engine dies apparently for no reason. So even if it sound weird or dumb, check that out first cause it would be the thing that makes you search the longest. If everything regarding the battery is fine, then i'm guessing a problem with a body ground or grounding strap somewhere. Might either be the grounds regarding the engine that give it the possibility to fire and such, or might be the grounds and/or fuses and connectors that make the car's onboard computer work.
SOURCE: I have a 1995 540I bmw..fuel pump doesn't
if you have 12 volts at the pump and it doesn't run it's the pump.
If you have 12 volts at the pump it means the relay is doing its job and the fuse is ok
remove the pump and run 12v to it directly from the battery to test the pump.
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