In energy harvesting communications, the transmitters have to adapt transmission to the availability of energy harvested during communication. The performance of the transmission depends on the channel conditions which vary randomly due to mobility and environmental changes. During this work, we consider the problem of power allocation taking into account the energy arrivals over time and the quality of channel state information (CSI) available at the transmitter, in order to maximize the throughput. Differently from previous work, the CSI at the transmitter is not perfect and may include estimation errors. We solve this problem with respect to the energy harvesting constraints. Assuming a perfect knowledge of the CSI at the receiver, we determine the optimal power policy for different models of the energy arrival process (offline and online model). Indeed, we obtain the power allocation scheme when the transmitter has either perfect CSI or no CSI. We also investigate of utmost interest the case of fading channels with imperfect CSI. Moreover, a study of the asymptotic behavior of the communication system is proposed. Specifically, we analyze of the average throughput in a system where the average recharge rate goes asymptotically to zero and when it is very high.