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Neutrino self-interactions in a dense neutrino gas can induce collective neutrino flavor conversions. Fast neutrino flavor conversions (FFCs), one of the collective neutrino conversion modes, potentially change the dynamics and observables in core-collapse supernovae and binary neutron star mergers. In cases without neutrino-matter interactions (or collisions), FFCs are essentially energy-independent, and therefore the single energy treatment has been used in previous studies. However, neutrino-matter collisions in general depend on neutrino energy, suggesting that energy-dependent features may emerge in FFCs with collisions. In this paper, we perform dynamical simulations of FFCs with iso-energetic scatterings (emulating nucleon scatterings) under multi-energy treatment. We find that cancellation between in- and out-scatterings happens in high energy region, which effectively reduces the number of collisions and then affects the FFC dynamics. In fact, the lifetime of FFCs is extended compared to the single-energy case, leading to large flavor conversions. Our result suggests that the multi-energy treatment is mandatory to gauge the sensitivity of FFCs to collisions. We also provide a useful quantity to measure the importance of multi-energy effects of collisions on FFCs.