The connection between line-connectivity concepts of graphs and indices of network reliability is well-known. Of particular interest in such studies are the circulant graphs because the connected ones have the largest possible value of line-connectivity $\lambda $ of p-point, degree r, regular graphs, namely $\lambda = r$. In this work, we define the higher order line-connectivity measure $N_i $ as the number of line-disconnecting sets of order i. Regular degree r, p-point graphs having $\lambda = r$ satisfy $N_\lambda \geqq p$. Such graphs which attain this lower bound are called super-$\lambda $. In this work we determine the necessary and sufficient conditions for a circulant to be super-$\lambda$. In addition we determine a lower bound on $N_i $ for $\lambda \leqq i\leqq 2r - 3$. It is shown that a special class of circulants, known as Harary graphs, achieve this lower bound for all these values of i.