Analysis

This question examines the application of piecewise functions and the method of finding function values, which is a basic question.

Solution

Given $f(x)=\begin{cases} 2x+a, & x\leqslant 1 \\ \log_2x, & x > 1 \end{cases}$, we know $f(\frac{1}{2})=a+1$.

If $a\leqslant 0$, then $f(f(\frac{1}{2}))=2a+3=4$, we get $a=\frac{1}{2}$ (discard this option);

If $a > 0$, then $f(f(\frac{1}{2}))=\log_2(a+1)=4$, we get $a=15$.

Therefore, the correct answer is $\boxed{\text{B}}$.