Strictly speaking the word 'refinance' means moving to another lender and it's no possible to do this with a fixed loan unless you're willing to pay the penalties associated with breaking the fixed loan contract.
The variable loan is quite flexible, you can 'top up' or 'restructure' this loan with the existing to access your equity quite easily.
It actually is possible to get a second mortgage over your own property with another lender and keep the fixed loan with the first lender. Practically this tends to be expensive, a hassle and neither lender likes it. It's much easier to keep it with the original lender in this situation.