This paper presents a new approach on robust current control of LCL-type grid-tied voltage source converters. This control method takes advantage of grid impedance nature to manage its unavoidable uncertainty at every point of common coupling. The Passivity-based Robust Control framework minimizes the converter impact over grid stability based on the passivity index of the impedance uncertainty, which leads to less conservative designs. System robustness is improved by means of a passivity constraint while keeping a performance goal. In order to overcome the non-convex behavior of the proposed multi-objective problem, we present a two-step approach design to find an acceptable solution. The methodology is initialized obtaining a controller via H-infinity synthesis procedure, which achieves the performance objective. Then, if the passivity constraint is not fulfilled, the non-smooth optimization technique will enforce it while taking care of the trade-off with the performance objective. Numerical and experimental results verify the achievement of the proposed objectives in time and frequency domain.