Extension of ARC can apply from Online or Paper procedure,  the below procedure introduces the Online ARC extension.

Online Extension of ARC
Require documents (scan and upload to the website of online application)

• Passport
• Alien Resident Certificate (ARC)
• For the current student: a student ID card with the current semester registration stamp or a school attendance certificate / For the new student: a school Admission Letter of the new semester.
• The 2-inch color photo is front-facing, above the waist, and without wearing a hat and taken within 2 years.

To apply for and collect the certificate

1. The applicant shall sign up for a new account to apply for an alien residence certificate at “Online Application System for Foreign Students, Overseas Chinese Students, Mainland China, Hong Kong, and Macau Students without Registered Household in the Taiwan Area” of the National Immigration Agency(link: https://coa.immigration.gov.tw/coa-frontend/student/entry?lang=en )
2. After the applicant fills out the required fields, scans and uploads the required documents, he/she shall wait for the approval. Once the application has been approved, the applicant shall pay the certificate fee by online ATM, virtual account payment the e-government payment platform, or other ways within 5 working days. 
3. The Alien Residence Certificate  (ARC) will be ready for the applicant to collect at a National Immigration Agency service center three working days after the applicant pays the fees online. Present the payment receipt when collecting the certificate (the old certificate must be returned).

Certificate fees
Application/extension fees are NT$1,000 (each one year) for foreign students and NT$500 (one year) for overseas Chinese students. The one-year residence restriction does not apply to university scholarship recipients approved by the Ministry of Education. 

Review period
5 working days under normal circumstances where no correction is necessary. 

Note
In the case of extension for alien residence certificates, such extension application shall be made within 30 days prior to the residence certificate expiration. If an alien overstays his/her permitted residence for less than 30 days and the original purpose for residence application remains unchanged, then he/she may reapply for a residence subject to penalties for violations pursuant to Subparagraph 4 of Article 85. If the alien has overstayed for more than 30 days, he/she should leave the country after paying the fine. He/she may reapply for residence and reenter the country after the approval has been given.

Resources
"Notices for the submission of online application for foreign and overseas Chinese student residence certificate extension or modification"