Simple compromise. Do it on the 16th just past midnight (effectively, for you, the night of the 15th).
In any case, the strict requirement is that you must be physically present at the start of the three year period. So, if you were to apply on 15th August, that would mean that you must be physically present in the UK on 16th August 2015, which you were.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.