Distributed generation, despite not being a new concept, is assuming a leading role in the field of energy conversion, as it should contribute to the enhancement of efficiency, flexibility, and reliability of national energy systems. However, it also noted that the effective performances of small and flexible power plants is critically influenced by their actual control strategy. Moreover, it is not trivial to identify a univocal parameter to evaluate the plant performance. For instance, cost evaluation clearly responds to an industrial view of the energy supply problem, while energy consumption or polluting emissions comply with a socio economic approach. In this scenario, the optimization of the plant management is a valuable instrument to gain insight on their behavior as the control strategy is varied, as well as to promote the distributed generation development, by maximizing the plants performances. In this paper, we further develop a graph based optimization methodology to optimize the set-point of an internal combustion engine based plant used to satisfy a hospital energy load, under different seasonal load conditions (winter, summer, and transitional seasons) and energy prices. Specifically, in order to dissect the effects of the objective function selection, two different optimization criteria are considered, namely economical optimization and primary energy consumption minimization. In particular, we focus on the features of the prime mover (i.e., the internal combustion engine) control strategy and on its drivers, as a function of the prescribed objective function. Results demonstrate that in the actual Italian energy market, cost minimization does not match primary energy consumption minimization, because the latter is only influenced by energy demand time series, and equipments performance, while the former is fundamentally driven by the electricity prices time series.