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This work aims to improve the energy efficiency of cloud radio access network (C-RAN). Power consumption is an important issue in deploying dense small cell networks such as C-RAN, especially for millimeter wave (mmWave) band where systems generally consume large power due to having massive phase shifters. To address this issue, we propose three different solutions to design the C-RAN beamforming to minimize power consumption under fixed user target rates. The first and second solutions are respectively based on the second-order cone programming (SOCP) and the semidefinite programming (SDP). Inspired by the results, we propose the third solution, which has a closed-form expression. The numerical results of these three solutions are extremely close. Since the first two solutions are obtained by standard optimization methods SOCP and SDP, which require high computational complexity, in most cases one can use the third solution to achieve comparable performance with significantly lower complexity. Based on the proposed beamforming solutions, the phase shifters of the array antennas can be selected and turned off to further significantly reduce the power consumption. Simulation results show that overall power consumption of C-RAN can be greatly decreased using the proposed schemes.