Low-order Moment Relaxation of ACOPF via Algorithmic Successive Linear Programming

Nowadays, there is a critical and urgent need for developing smart and robust OPF solvers since the conventional options currently available for OPF problems are quite limited. This research is based on AC Optimal Power Flow (ACOPF) with active and reactive quadratically constrained quadratic programming optimization problems of a form that arises in operation and planning applications in power systems. Besides being non-convex, these problems are identified to be NP-hard. This paper first utilized semi-definite programming (SDP) relaxation to convexify the original ACOPF problems and then solve the SDP relaxation problem with “moment-based” algorithm to get the rank-1 solutions of the $W$ matrix. However, the computation time will increase exponentially with higher order of the moment matrix. To improve the computation efficiency, we added some penalty terms in the objective function to push the rank of the moment matrix reach to 1 by using the proposed SLP(SLPBB) algorithms. The proposed algorithm is verified by simulating on small scale test cases and NP-hard topologies in MATLAB. Also, the results were compared with the ones obtained by only using SLP (SLPBB) algorithms and the local solutions (The SLP and SLPBB algorithms were denoted as SLP(BB) afterwards). Numerical simulations illustrate that the SDP moment-based SLP(BB) algorithm can obtain the global optimal solutions which can guarantee the rank-1 solutions of the moment and $W$ matrices.