I ranked first in my university in my undergrads, which had many affiliated colleges. How can I specify that in an essay in a compelling yet precise manner?
Making a note of the number of affiliated colleges might be fine. But I will suggest including the total number of candidates (students) in your batch that includes all affiliated colleges (if available) within which you've established your rank.
From your comments, it is perfectly fine to state that you ranked 1 in the university. In my case I had been asked to provide the number of students. I had no difficulty since mine was a National institute which is a university by itself, all I had to do is to count the number of students in my class. But in your case you might have to give a brief explanation about the affiliation, possibly an attestation from the University and college.