Properties of Convex Optimal Power Flow Model Based on Power Loss Relaxation

TL;DR

This paper develops a convex optimal power flow model incorporating power loss relaxation, deriving new constraints, analyzing solution recoverability, and validating findings through numerical experiments on benchmark networks.

Contribution

It introduces a branch ampacity constraint based on power loss relaxation and provides theoretical insights into the model's validity and solution properties.

Findings

Loop constraint is a relaxation of the nonconvex model.

Conditions for recovering feasible solutions from convex relaxations.

Existence of tight solutions for large power loads.

Abstract

We derive the branch ampacity constraint associated to power losses for the convex optimal power flow (OPF) model based on the branch flow formulation. The branch ampacity constraint derivation is motivated by the physical interpretation of the transmission line {\Pi}-model and practical engineering considerations. We rigorously prove and derive: (i) the loop constraint of voltage phase angle, required to make the branch flow model valid for meshed power networks, is a relaxation of the original nonconvex alternating current optimal power flow (o-ACOPF) model; (ii) the necessary conditions to recover a feasible solution of the o-ACOPF model from the optimal solution of the convex second-order cone ACOPF (SOC-ACOPF) model; (iii) the expression of the global optimal solution of the o-ACOPF model providing that the relaxation of the SOC-ACOPF model is tight; (iv) the (parametric) optimal value function of the o-ACOPF or SOC-ACOPF model is monotonic with regarding to the power loads if the objective function is monotonic with regarding to the nodal power generations; (v) tight solutions of the SOC-ACOPF model always exist when the power loads are sufficiently large. Numerical experiments using benchmark power networks to validate our findings and to compare with other convex OPF models, are given and discussed.