This paper is a generalization of the models considered in [J. Stat. Phys.128,1365 (2007)]. Using an analogy with free fermions, we compute exactly thelarge deviation function (ldf) of the energy injected up to time $t$ in aone-dimensional dissipative system of classical spins, where a drift isallowed. The dynamics are T=0 asymmetric Glauber dynamics driven out of rest byan injection mechanism, namely a Poissonian flipping of one spin. The driftinduces anisotropy in the system, making the model more comparable toexperimental systems with dissipative structures. We discuss the physicalcontent of the results, specifically the influence of the rate of the Poissoninjection process and the magnitude of the drift on the properties of the ldf.We also compare the results of this spin model to simple phenomenologicalmodels of energy injection (Poisson or Bernoulli processes of domain wallinjection). We show that many qualitative results of the spin model can beunderstood within this simplified framework.