  In this paper, we show that the standard semidefinite programming (SDP)
relaxation of altering current optimal power flow (AC OPF) can be equivalently
reformulated as second-order cone programming (SOCP) relaxation with maximal
clique- and cycle-based SDP feasibility constraints. The formulation is based
on the positive semi-definite (PSD) matrix completion theorem, which states
that if all sub-matrices corresponding to maximal cliques in a chordal graph
are PSD, then the partial matrix related to the chordal graph can be completed
as a full PSD matrix. Existing methods in [1] first construct a chordal graph
through Cholesky factorization. In this paper, we identify maximal cliques and
minimal chordless cycles first. Enforcing the submatrices related to the
maximal cliques and cycles PSD will guarantee a PSD full matrix. Further, we
conduct chordal relaxation for the minimal chordless cycles by adding virtual
lines and decompose each chordless cycle to 3-node cycles. Thus, the entire
graph consists of trees, maximal cliques, and 3-node cycles. The submatrices
related to the maximal cliques and 3-node cycles are enforced to be PSD to
achieve a full PSD matrix. As majority power grids having the size of maximal
cliques limited to 4-node, this graph decomposition method results in a
low-rank full PSD matrix. The proposed method significantly reduces computing
time for SDP relaxation of AC OPF and can handle power systems with thousands
of buses.
