OP, you're going to struggle finding a place in central for £150 if you're on your own. It's a decent sized budget if you can get 2 or more people together so my advice is get some flatmates, otherwise you're fighting a losing battle.
We managed to find a 4 bed next to battersea park (zone 1, vauxhall); walking distance to uni; for less than half your budget- a mere £72.50 p/w each.
Before that we found plenty of other properties for similar prices (between £70-85 p/w) albeit slightly further afield- manor house, finsbury park, tuffnell park, oval, e&c.
so with a £150 budget and some flatmates, you'd be laughing.