Fast QC Relaxation of the Optimal Power Flow Using the Line-Wise Model for Representing Meshed Transmission Networks

In this paper, we investigate the recently introduced McCormick-based Quadratic Convex (QC) relaxation of the Optimal Power Flow (OPF) where Line-Wise Model (LWM) is used for representing meshed transmission systems (QC-LW OPF) for the sake of decisively determining its relationship to other available convex relaxations in the literature. We also extend the recently introduced convex envelope of the tangent function so it would be suitable for test cases with any voltage angle difference range. A computational study where the recently proposed QC-LW OPF formulation is compared to an equivalent McCormick based QC relaxation of the OPF where Bus Injection Model (BIM) is used for representing meshed transmission systems (QC-BI OPF) is presented in this paper. This computational study was conducted using test cases that belong to different operational categories using 123 test cases from the PGLib-OPF library with a bus size range between 3 up to 6515 buses for the sake of understanding the effect of the change of operating conditions on the quality of solutions obtained using the QC-LW OPF and QC-BI OPF formulations. Results are compared using several metrics that testify to the obtained solution’s quality and the problem’s computational complexity. Comparison of results shows that the QC-LW relaxation neither dominates nor is dominated by the QC-BI relaxation in terms of solution quality. Therefore, it dominates the Second Order Cone (SOC) relaxation and neither dominates nor is dominated by the Semidefinite Programming (SDP) relaxation. Furthermore, it is shown that the QC-LW OPF has reduced the number of relaxed trigonometric functions and McCormick envelopes needed when compared to the QC-BI OPF, leading to a faster solution time for more than 84% of the test cases in the range of 2% up to 67%.