Novel Approach Towards Global Optimality of Optimal Power Flow Using Quadratic Convex Optimization

Abstract

Optimal Power Flow (OPF) can be modeled as a nonconvex Quadratically Constrained Quadratic Program (QCQP). Our purpose is to solve OPF to global optimality. To this end, we specialize the Mixed-Integer Quadratic Convex Reformulation method (MIQCR) to (OPF). This is a method in two steps. First, a Semi-Definite Programming (SDP) relaxation of (OPF) is solved. Then the optimal dual variables of this relaxation are used to reformulate OPF into an equivalent new quadratic program, where all the non-convexity is moved to one additional constraint. In the second step, this reformulation is solved within a branch-and-bound algorithm, where at each node a quadratic and convex relaxation of the reformulated problem, obtained by relaxing the non-convex added constraint, is solved. The key point of our approach is that the lower bound at the root node of the branch-and-bound tree is equal to the SDP relaxation value.We test this method on several OPF cases, from two-bus networks to more-than-a-thousand-buses networks from the MAT-POWER repository. Our first results are very encouraging.

AC Optimal Power Flow: a Conic Programming relaxation and an iterative MILP scheme for Global Optimization

“…Other Global Optimization approaches for the ACOPF problem follow Spatial Branch-and-Bound schemes [4]. To obtain a lower bound at each node of the exploration tree, these algorithms may use a Second-Order Cone Programming (SOCP) relaxation [21], a Quadratically Constrained Programming (QCP) relaxation [12] or a Semidefinite Programming (SDP) relaxation [7]. Piecewise convex relaxations.…”

“…Proposition 7. We assume that the convex Constraints (8)-( 13) and the MILP Constraints (31)-( 38) are satisfied, but with a tolerance ρ ∈ [0, 1] for the nonlinear Constraints (10) and (12). Then, for any nodes j b ∈ J b , j a ∈ J a and j ba ∈ J ba that are active, i.e., s.t.…”

“…For any j ∈ b J b , we recall that D(j) is the depth of j in the (unique) tree J b it belongs to. As x t is the output of the outer-iteration t − 1, it satisfies Constraint (10) and (12) with tolerance ρ t−1 , and we can apply Proposition 7 with ρ = ρ t−1 . This yields…”

“…. Constraint(12) just follows from the definition of R ba = |W ba |. For any (b, a) ∈ E, we define θ ba = ∠W ba ; considering the definition of φ ba and δ ba , we notice that |θ ba − φ ba | ≤ δ ba .…”

Improving Clique Decompositions of Semidefinite Relaxations for Optimal Power Flow Problems

“…SDP relaxations often provide tight lower bounds [12] that are useful to prove global optimality. Some promising global optimization methods are based on these relaxations : Godard et al propose an adaptation of the Mixed-Integer Quadratic Convex Reformulation method to ACOPF problems in [6], Gopalakrishnan et al present a branch-and-bound approach using SDP relaxations in [7] and Josz et al apply the Lasserre hierarchy in [11] to achieve global optimality. All these methods depend on large-scale SDP problems being solved efficiently.…”