Sufficient Conditions for Exact Semidefinite Relaxation of Optimal Power Flow in Unbalanced Multiphase Radial Networks

This paper proves that in an unbalanced multi-phase network with a tree topology, the semidefinite programming relaxation of optimal power flow problems is exact when critical buses are not adjacent to each other. Here a critical bus either contributes directly to the cost function or is where an injection constraint is tight at optimality. Our result generalizes a sufficient condition for exact relaxation in single-phase tree networks to tree networks with arbitrary number of phases.Comment: Accepted by 2019 Conference on Decision and Control (CDC

Pitfalls of Zero Voltage Values in Optimal Power Flow Problems

The existence of strictly positive lower bounds on voltage magnitude is taken for granted in optimal power flow problems. Nevertheless, it is not possible to rely on such bounds for a variety of real-world network optimization problems. This paper discusses a few issues related to 0 V assumptions made during the process of deriving optimization formulations in the current-voltage, power-voltage and power-lifted-voltage variable spaces. The differences between the assumptions are illustrated for a 2-bus 2-wire test case, where the feasible sets are visualized. A nonzero relaxation gap is observed for the canonical multiconductor nonlinear power-voltage formulation. A zero gap can be obtained for the branch flow model semi-definite relaxation, using newly proposed valid equalities.Comment: 5 pages. Submitted to 2023 IEEE PES General Meetin

Centralized OPF in Unbalanced Multi-Phase Neutral Equipped Distribution Networks Hosting ZIP Loads

The Optimal Power Flow (OPF) model for low voltage active Distribution Networks (DNs), which are equipped with neutral conductors, requires an explicit representation of both phases and neutral conductors in its formulation to obtain complete information about the state variables related to these conductors. In this regard, a centralized OPF relaxation based on semi-definite programming is presented in this paper for neutral-equipped DNs hosting ZIP loads and neutral-ground impedance, and contain a significant level of unbalance. The major restriction in the development of an OPF model for these networks is the coupled power injection across the conductors which is successfully handled by deriving the explicit active and reactive power injections for each conductor through a network admittance matrix-based approach. The shortcomings of existing voltage magnitude-based technique for the modelling of ZIP loads are comprehensively reported and a novel complex voltage variable-based approach is proposed which successfully incorporates ZIP loads in the developed multi-phase OPF relaxation. For the handling of constant current load, a modelling approach based on the first-order-Taylor series is introduced as well. Furthermore, the impact of the application of Kron reduction approach on the global optimal solution of single- and multiple-point grounded DNs is discussed in detail. Three metrics, eigenvalue ratio, power mismatch and cumulative normalized constraint violation, are utilized to evaluate the exactness of proposed relaxation. Simulations, carried out on several medium and low voltage DNs, show that the proposed relaxation is numerically exact under several combinations of ZIP load parameters and a reasonable range of grounding impedance value for both time-varying and extreme system loading scenarios irrespective of the degree of unbalance in a network