Your use of "tremendously" is correct. I liked the commenter's statement that in a formal context such as research papers, "significantly" seems more appropriately bland.
I think the common figure of speech is "applications in [fields]." But the phrase "many applications to various problems in numerous areas" seems repetitive with little added information. In describing the range of applications, what are you trying to say? If you just want to describe the quantity, I would say something like:
"Smart grid research has increased tremendously in the last years, since the applications are numerous."
If you want to describe the quantity, diversity, and perhaps profitability of the applications, you might say:
"Smart grid research has increased tremendously in the last years, since it has many lucrative applications in different fields."