Movable antennas (MAs) represent a promising paradigm to enhance the spatial degrees of freedom of conventional multi-antenna systems by dynamically adapting the positions of antenna elements within a designated transmit area. In particular, by employing electro-mechanical MA drivers, the positions of the MA elements can be adjusted to shape a favorable spatial correlation for improving system performance. Although preliminary research has explored beamforming designs for MA systems, the intricacies of the power consumption and the precise positioning of MA elements are not well understood. Moreover, the assumption of perfect CSI adopted in the literature is impractical due to the significant pilot overhead and the extensive time to acquire perfect CSI. To address these challenges, we model the motion of MA elements through discrete steps and quantify the associated power consumption as a function of these movements. Furthermore, by leveraging the properties of the MA channel model, we introduce a novel CSI error model tailored for MA systems that facilitates robust resource allocation design. In particular, we optimize the beamforming and the MA positions at the BS to minimize the total BS power consumption, encompassing both radiated and MA motion power while guaranteeing a minimum required SINR for each user. To this end, novel algorithms exploiting the branch and bound (BnB) method are developed to obtain the optimal solution for perfect and imperfect CSI. Moreover, to support practical implementation, we propose low-complexity algorithms with guaranteed convergence by leveraging successive convex approximation (SCA). Our numerical results validate the optimality of the proposed BnB-based algorithms. Furthermore, we unveil that both proposed SCA-based algorithms approach the optimal performance within a few iterations, thus highlighting their practical advantages.