The power steering is still powered when my Hyundai i30 is turned off. The ESP light is also on.
Various systems randomly turning themselves on or staying powered up when the car is otherwise turned off often point to a fault with the body computer which controls literally hundreds of functions. Obviously, the power steering should shut down with the rest of the car so, as the master control unit, the body computer is where many mechanics would start looking.
Whether the same fault is also causing the ESP light to come on is another matter, but it’s possible both problems are connected. Again, the body computer is a likely culprit. Have the car scanned to see what and how many individual fault codes are thrown up. That’s the best way to proceed in this case.