  Improving renewable energy resource utilization efficiency is crucial to
reducing carbon emissions, and multi-parametric programming has provided a
systematic perspective in conducting analysis and optimization toward this goal
in smart grid operations. This paper focuses on two aspects of interest related
to multi-parametric linear/quadratic programming (mpLP/QP). First, we study
degeneracy issues of mpLP/QP. A novel approach to deal with degeneracies is
proposed to find all critical regions containing the given parameter. Our
method leverages properties of the multi-parametric linear complementary
problem, vertex searching technique, and complementary basis enumeration.
Second, an improved critical region exploration (CRE) method to solve
distributed LP/QP is proposed under a general mpLP/QP-based formulation. The
improved CRE incorporates the proposed approach to handle degeneracies. A
cutting plane update and an adaptive stepsize scheme are also integrated to
accelerate convergence under different problem settings. The computational
efficiency is verified on multi-area tie-line scheduling problems with various
testing benchmarks and initial states.
