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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.

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