This two-part paper presents a framework for the characterization and minimal representation of the feasibility regions of power systems in the demand space. These feasibility regions are called loadability sets, and they represent the projection of generation-demand-network space onto the demand space only. Loadability sets have been characterized previously for power systems with either no or a single active line flow constraint. In Part I of this two-part paper, we generalize this characterization to power systems with any number of active line flow constraints. The proposed characterization approach makes use of the Fourier-Motzkin elimination method which leads to the generation of a large number of constraints. We set course to address this shortcoming in the second part of the paper. The notion of umbrella set is revisited in Part II to identify and remove the redundant constraints produced in the loadability set characterization process. The outcome of the proposed framework is the minimal representation of the power system feasibility region in the demand space. We envision multiple applications for the proposed framework in power system planning, operations planning and real-time operation.