Ok I've tried it with the right informations for my situation. BUT in that case the link send to the subscription form. so people must repeat all the informations as for a new member.
They need to be logged in for the page to display the renewal button. There's really no getting around that. Please see my previous replies for other possible options to display to them that they need to renew (CBSubs module, etc..), but all of them will require the user to be logged in.