  The DC network security constraints have been extensively studied in numerous
power system problems, such as optimal power flow (OPF), security-constrained
economic dispatch (SCED), and security-constrained unit commitment (SCUC).
Linear shift factors, i.e., power transfer distribution factors (PTDFs), are
widely applied to replace DC power flow constraints. However, the PTDF matrix
is extremely dense, making it difficult to solve security-constraint
optimization problems. This paper analyzes/investigates the computational
inefficiency of PTDF-based security constraints from the sparse structure
perspective of the primal-dual interior point method(IPM). Additionally, a
matrix transformation method is proposed for restoring the sparsity of the
linear system during IPM iterations. It turns out that the transformation
method is equivalent to solving the original optimization problem expressed in
pure voltage angle, which preserves the sparsity structure but introduces
additional variables and constraints proportional to one to two times the total
number of buses. The regular B-$\theta$ formulation is also a variant of the
proposed transformation. Numerical studies show that sparsity rather than the
size of variables and constraints is the key factor impacting the speed of
solving convex quadratic problems (QP), i.e., OPF and SCED problems. In
contrast, sparsity is less desirable when solving a mixed integer problem
(MIP), such as the SCUC problem, where reoptimization techniques are
significantly more critical and the dual simplex method is typically employed
rather than IPM.
