No there is not, and there is no reason to expect BA to open award seats on empty flights.
Let's just be clear that award seats are different from free upgrades. Award seats cost me a currency (Avios) whereas the later cost me nothing. I argue that it is in the airline's favor to get me to spend my Avios balance down, especially if it is at a less favorable exchange rate to the £ than I might otherwise get from whole reward flights.