The Plusnet billing system prepares the bills in advance of the billing date. I have seen seven days in advance postulated! Therefore, if you renewed your contract after the bill was prepared you will have received a bill reflecting your out of contract price for your expiring contract.
As has been said, your next bill should sort things out with a refund for the overpayment this time (comprising a refund of that money and a correct charge based on your ne contract for the same period). If your new contract starts before your existing billing date then there will be other adjustments for those "overlapping" days (a refund of the charges for those days based on your old contract rate and a charge based on your new contract rate). The bill will be quite complex and long.
