[Bookman]

Here is another possible answer as derived via the www; it refers:-

"This use of flat for an electric battery was first used in 1951 according to the OED (sense 9c). The others related senses are: 9a) wanting in energy and spirit (1604); 9b) depressed, dull or inactive trade (1831); and 9d) a drink that has lost its flavour or sharpness (1617)."

I particularly like the latter anology