We combine hydrodynamic and modulated energy techniques to study the large deviations of systems of particles with pairwise singular repulsive interactions and additive noise. Specifically, we examine periodic Riesz interactions indexed by parameter $\mathbf{s}\in[0,d-2)$ for $d\geq 3$ on the torus. When $\mathbf{s}\in(0,d-2)$, we establish a large deviation principle (LDP) upper bound and partial lower bound given sufficiently strong convergence of the initial conditions. When $\mathbf{s}=0$ (i.e., the interaction potential is logarithmic), we prove that a complete LDP holds. Additionally, we show a local LDP holds in the distance defined by the modulated energy.