Just to give you all an update - and perhaps some hope! - we have resolved the issue with Aydem, and they are refunding us for errors that they made in putting together our bill. It is too complicated to go into what was wrong with the bill, you'll be asleep before I reached the end, the important point is that the Aydem - without admitting responsibility for the errrors - are giving us credits against future bills. Which should mean we do not have to pay any electricity bill for the communal areas, such as pool and lights, until well into next year. Yes it would have been nice to get the cash back, but we are happy with this outcome.
The successful outcome was achieved by our Russian neighbour who through a combination of diplomacy, schmoozing and controlled anger persuaded the Aydem Manager of the errors of Aydem's ways, and to give us the credits
So even if things look bleak when dealing with some Turkish authorities, keep at it and it will probably come good, at least in part, at the end.