The first part of this two-part paper developed the framework for characterizing the feasibility regions of power systems in the demand space. This characterization, however, leads to the generation of a large number of extraneous constraints as an unwelcome byproduct. This shortcoming motivates the second part of this paper series, whose objective is the achievement of a minimal representation for loadability sets. Thus, in this paper we set forth to eliminate as many of those extraneous constraints with the proposal of an enhanced umbrella constraint discovery (E-UCD) problem formulation. The use of E-UCD in this paper are fourfold: (1) It serves to identify redundant line flow constraints not potentially shaping the feasibility regions of power systems in the generation-demand space. (2) It serves to determine the maximum number of line flow limits that could ever become active simultaneously in a given power system. (3) It pinpoints generators who have the ability to become pivotal in relieving network congestion. (4) It is used to identify redundant constraints generated while generation dispatch variables are projected from generation-demand space onto the demand space. Experiments are carried out on standard IEEE test systems to show that the computational burden associated with the calculation of the minimal representation of loadability sets is reasonable for planning and operations planning applications. We discuss also potential applications of loadability sets for power system operation and planning.