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A Tight SDP Relaxation for MIQCQP Problems in Power Systems Based on Disjunctive Programming

An optimization problem considering AC power flow constraints and integer decision variables can usually be posed as a mixed-integer quadratically constrained quadratic program (MIQCQP) problem. In this paper, first, a set of valid linear equalities are applied to strengthen the semidefinite program (SDP) relaxation of the MIQCQP problem without significantly increasing the problem dimension so that an enhanced mixed-integer SDP (MISDP) relaxation, which is a mixed-integer convex problem, is obtained. Then, the enhanced MISDP relaxation is reformulated as a disjunctive programming (DP) problem which is tighter than the former one, since the disjunctions are designed to capture the disjunctive nature of the terms in the rank-1 constraint about the integral variables. The DP relaxation is then equivalently convert-ed back into a MISDP problem the feasible set of whose continu-ous relaxation is the convex hull of feasible region of the DP prob-lem. Finally, globally optimal solution of the DP problem which is the tightest relaxation for the MIQCQP proposed in the paper is obtained by solving the resulting MISDP problem using a branch-and-bound (B&B) algorithm. Computational efficiency of the B&B algorithm is expected to be high since feasible set of the continuous relaxation of a MISDP sub-problem is the convex hull of that of the corresponding DP sub-problem. To further reduce the dimension of the resulting MISDP problem, a compact for-mulation of this problem is proposed considering the sparsity. An optimal placement problem of smart PV inverter in distribution systems integrated with high penetration of PV, which is an MIQCQP problem, is studied as an example. The proposed ap-proach is tested on an IEEE distribution system. The results show that it can effectively improve the tightness and feasibility of the SDP relaxation.