We have investigated the electrical double layer (EDL) structure at an interface between ionic liquid (IL) and charged surface using molecular dynamics simulations. We show that for three different models of ILs the EDL restructuring, driven by surface charging, can be rationalized by the use of two parameters--renormalized surface charge (κ) and charge excess in the interfacial layers (λ). Analysis of the relationship between the λ and κ parameters provides new insights into mechanisms of over-screening and charge-driven structural transitions in the EDL in ionic liquids. We show that the restructuring of the EDL upon charging in all three studied systems has two characteristic regimes: (1) transition from the bulk-like (κ(Ion) = 0) to the multilayer structure (κ(Ion) ≈ 0.5) through the formation of an ionic bilayer of counter- and co-ions; and (2) transition from the multilayer (κ(Ion) ≈ 0.5) to the crowded (κ(Ion) > 1) structure through the formation of a monolayer of counter-ions at κ(Ion) = 1.