It is well known for which gauge functions H there exists a flow in Zd with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows. Instead of Zd we let our (random) graph calCcal∞(Zd,p) be the graph obtained from Zd by removing edges with probability 1−p independently on all edges. Grimmett, Kesten, and Zhang (1993) showed that for d≥3,p>pc(Zd), simple random walk on calCcal∞(Zd,p) is a.s. transient. Their result is equivalent to the existence of a nonzero flow f on the infinite cluster such that the x2 energy ∑ef(e)2 is finite. Levin and Peres (1998) sharpened this result, and showed that if d≥3 and p>pc(Zd), then calCcal∞(Zd,p) supports a nonzero flow f such that the xq energy is finite for all q>d/(d−1). However, for general gauge functions, there is a gap between the existence of flows with finite energy which results from the work of Levin and Peres and the known results on flows for Zd. In this paper we close the gap by showing that if d≥3 and Zd supports a flow of finite H energy then the infinite percolation cluster on Zd also support flows of finite H energy. This disproves a conjecture of Levin and Peres.