Cloud computing has become an essential component of our digital society. Efforts for reducing its environmental impact are being made by academics and industry alike, with commitments from major cloud providers to be fully operated by renewable energy in the future. One strategy to reduce nonrenewable energy usage is the "follow-the-renewables", in which the workload is migrated to be executed in the data centers with the most availability of renewable energy. In this paper, we study the indirect impacts on the energy consumption caused by the additional load in the network generated from the live migrations of the "follow-the-renewables" approaches. We then provide an algorithm that thoroughly considers the network to schedule the live migrations and, combined with an accurate estimation model for the duration of the migrations, is able to perform the live migrations without network congestion with the same or even reducing the brown energy consumption in comparison to other state-of-the-art algorithms.