  Cell-free (CF) massive multiple-input multiple-output (MIMO) is a promising
solution to provide uniform good performance for unmanned aerial vehicle (UAV)
communications. In this paper, we propose the UAV communication with wireless
power transfer (WPT) aided CF massive MIMO systems, where the harvested energy
(HE) from the downlink WPT is used to support both uplink data and pilot
transmission. We derive novel closed-form downlink HE and uplink spectral
efficiency (SE) expressions that take hardware impairments of UAV into account.
UAV communications with current small cell (SC) and cellular massive MIMO
enabled WPT systems are also considered for comparison. It is significant to
show that CF massive MIMO achieves two and five times higher 95\%-likely uplink
SE than the ones of SC and cellular massive MIMO, respectively. Besides, the
large-scale fading decoding receiver cooperation can reduce the interference of
the terrestrial user. Moreover, the maximum SE can be achieved by changing the
time-splitting fraction. We prove that the optimal time-splitting fraction for
maximum SE is determined by the number of antennas, altitude and hardware
quality factor of UAVs. Furthermore, we propose three UAV trajectory design
schemes to improve the SE. It is interesting that the angle search scheme
performs best than both AP search and line path schemes. Finally, simulation
results are presented to validate the accuracy of our expressions.
