  We consider the cell-free massive multiple-input multiple-output (MIMO)
downlink, where a very large number of distributed multiple-antenna access
points (APs) serve many single-antenna users in the same time-frequency
resource. A simple (distributed) conjugate beamforming scheme is applied at
each AP via the use of local channel state information (CSI). This CSI is
acquired through time-division duplex operation and the reception of uplink
training signals transmitted by the users. We derive a closed-form expression
for the spectral efficiency taking into account the effects of channel
estimation errors and power control. This closed-form result enables us to
analyze the effects of backhaul power consumption, the number of APs, and the
number of antennas per AP on the total energy efficiency, as well as, to design
an optimal power allocation algorithm. The optimal power allocation algorithm
aims at maximizing the total energy efficiency, subject to a per-user spectral
efficiency constraint and a per-AP power constraint. Compared with the equal
power control, our proposed power allocation scheme can double the total energy
efficiency. Furthermore, we propose AP selections schemes, in which each user
chooses a subset of APs, to reduce the power consumption caused by the backhaul
links. With our proposed AP selection schemes, the total energy efficiency
increases significantly, especially for large numbers of APs. Moreover, under a
requirement of good quality-of-service for all users, cell-free massive MIMO
outperforms the colocated counterpart in terms of energy efficiency.
