A wide range of real world optimization problems involves continuous decisions and non-linearities. Each non-linear component of such problems can be modeled either linearly or non-linearly, considering or not additional integer variables. In this paper, different modeling alternatives are proposed for a real non-linear optimization problem, the single unit hydro unit commitment problem (1-HUC). The non-linearities of the 1-HUC come from the power produced at each time period. It is defined as a two-dimensional non-convex and non-concave function of the water flow and head decision variables, the latter being itself a one-dimensional convex non-linear function of the turbined volume. A common simplification is also considered, assuming that the water head is fixed and defining consequently the produced power as a one-dimensional non concave and non convex function of the water flow. Seven non-linear and linear models are described for both the 1-HUC and the fixed-head 1-HUC. These models cover multiple families of modeling alternatives, including common models of the literature as well as new models featuring less common functions. Different sets of instances are generated to evaluate the sensitivity of performance with respect to the main characteristics of the 1-HUC. Several available solvers are used for each non-linear model and the best virtual solver is retained to focus on the model capabilities rather than on the solver performance. Based on the numerical experiments, sometimes counter-intuitive recommendations are given so as to help practitioners in selecting the most adequate model and solver depending on the characteristics of the instance.