Filtering inductor for power electronic converter design formula
The solution could be to choose the worst case but I don't know what
  value of d coresponds to the worst case


You have derived a formula for L and the worst case scenario is when L is maximum. With inductors, a smaller value is usually preferred for technical reasons so, choose a value of d that makes L large.

However, I doubt that your formula is correct because if I choose d to be zero then the value of L isn't infinite as I would expect it to be.

Alternatively go and down load LTSpice (free from LT) and simulate solutions to double check your formulas.