Paul Burridge wrote:

The power company run a line to my house. They supply me with
electricity. This amounts to a 230V, 65A facility at the
distribution
board in a cupboard under the stairs. I run all my stuff
from that
board. The board contains several RCBOs that trip-out in
the event of
any leakage current being sensed. If current in = current
out; they're
happy and won't trip. Because they don't trip out, I
conclude I don't
use any current.

No, you're simply not permitting it to go to ground.
Neatness counts.

The voltage supplied is 230VAC RMS. Since this is
alternating between
equal positive and negative half-cycles, the average
level of this
voltage supply is zero.

Yup. It better be zero.

I use no current and they effectively supply no voltage.
Why do I get
billed for electricity usage when I clearly can't have
used any?

Except you slowed down the electrons on their way through
your appliances (made them do work). Speed them back up (do
work on them) and you'll see a zero bill except for the
minimum line-maintenance fees, of course.

Mark L. Fergerson