We consider the problem of minimizing the energy of
$N$ points repelling each other on curves in
${{\mathbb{R}}^{d}}$ with the potential
${{\left| x\,-\,y \right|}^{-s}},\,s\,\ge \,1$, where
$\left| \cdot \right|$ is the Euclidean norm. For a sufficiently smooth, simple, closed, regular curve, we find the next order term in the asymptotics of the minimal s-energy. On our way, we also prove that at least for
$s\,\ge \,2$, the minimal pairwise distance in optimal configurations asymptotically equals
$L/N,\,N\to \,\infty $, where
$L$ is the length of the curve.