Certifying Global Optimality of AC-OPF Solutions via the CS-TSSOS Hierarchy

Abstract

In this paper, we report the experimental results on certifying 1% global optimality of solutions of AC-OPF instances from PGLiB with up to 24464 buses via the CS-TSSOS hierarchy -a moment-SOS based hierarchy that exploits both correlative and term sparsity, which can provide tighter SDP relaxations than Shor's relaxation. Our numerical experiments demonstrate that the CS-TSSOS hierarchy scales well with the problem size and is indeed useful in certifying 1% global optimality of solutions for large-scale real world problem; e.g., the AC-OPF problem. In particular, we are able to certify 1% global optimality for an AC-OPF instance with 6515 buses involving 14398 real variables and 63577 constraints.

Certified and accurate SDP bounds for the ACOPF problem

“…Despite these advances, the clique-based SDR remains difficult to solve for large-scale instances: numerical instabilities may arise when solving this convex optimization problem. In [5] and [15], the authors report numerical instabilities with the commercial IP solver MOSEK [9], which raises a warning for many tested instances. In [6], the authors report that the wellknown academic IP solver SeDuMi fails to solve the cliquebased SDR in more than 50% of the test cases; and that the ADMM-based solver CDCS often terminates with sizable dual residuals.…”

A correlative sparse Lagrange multiplier expression relaxation for polynomial optimization

“…The moment-SOS relaxation with correlative sparsity is studied in [27], and the convergence result is given in [9]. Recently, Wang et al developed the software TSSOS [13] that implements correlative and term sparse SOS relaxations for polynomial optimization (see also [29,30]), and it has been used in many applications [14,28].…”