When you go on exchange through SDU International, you will automatically be granted a travel grant either from the Internationalisation Fund, the Erasmus+ programme or Nordplus, as long as you meet the necessary requirements. The grants awarded will typically cover travel expenses and a little extra. More information is available at the subpage SDU Travel Grants.
Since the grants are considered to be used for travel in connection with a study abroad period, they are exempt from taxation in Denmark as long as you don't profit on them.
You should explore if you are eligible for any scholarships in collaboration with SDU through External Grants and Scholarships.
You can also apply for private grants and scholarships, and sometimes this might be necessary to cover all expenses. It may take a long time to get an answer, so make sure you begin searching for grants early in the process. You can search for grants on the internet or use the handbooks (e.g. Legathåndbogen) at the University Library.