We prove global existence of $H^2$ solutions to the Cauchy problem for the generalized derivative nonlinear Schrödinger equation on the 1-d torus. This answers an open problem posed by Ambrose and Simpson (2015). The key in the proof is to extract the terms that cause the problem in energy estimates and construct modified energies so as to cancel them out by effectively using integration by parts and the equation. This talk is based on a joint work with Tohru Ozawa and Nicola Visciglia.