DC-based Security Constraints Formulation: A Perspective of Primal-Dual Interior Point Method

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.