Sparsity and structure exploiting diagonally dominant relaxation of the OPF problem

The Optimal Power Flow (OPF) is an optimization problem which tackles both the economy and physics of power systems operation. Due to high non-linearity in the power flow equations, the OPF problem is non-convex. Consequently, optimally solving for the OPF problem at a reasonable computational time presents a serious challenge. Several approaches were presented to solve the OPF problem. These include local solvers, heuristic methods and the approximation of non-linear equations. However, these approaches either do not bound the true value of the objective function or are lacking in the trade-off they provide between solution time and quality. As an alternative, convex relaxation techniques could be used to address this challenge. A convex relaxation is obtained by means of finding a convex representation of the problem’s feasible space. As a natural byproduct of the convexity of the resulting problem, a wide array of convex optimization techniques could be utilized. Furthermore, the solution obtained presents a lower bound on the global solution of the original non-convex problem. Several factors influence the tightness and scalability of convex relaxations. Those include the number and type of constraints used in the relaxation of the original non-convex problem. Most relaxations of the optimal power flow problem are based on second order conic or positive semidefinite type of constraints. Alternatively, in this dissertation we address the utilization of the linearly representable diagonally dominant cone in relaxing the optimal power flow problem. First, we investigate the diagonally-dominant-sum-of-squares relaxation of the problem. We evaluate the reasons behind its poor optimality gaps and scalability issue. We demonstrate that diagonal dominance could be utilized in creating a similar, yet tighter relaxation. The relaxation we propose is based on the semidefinite relaxation of the problem. This dissertation then follows to improve the tractability of the aforementioned relaxation. We achieve that by an investigation into the optimal exploitation of the sparsity and structure of the OPF problem. Several methods exist for the exploitation of sparsity in semidefininte programming. Specifically, chordal decomposition has been applied with great success to improve the tractability of the semidefinite relaxation of the optimal power flow problem. Accordingly, we investigate the utilization of chordal decomposition in improving the diagonal dominance based relaxation proposed in this thesis. We find that the direct exploitation of sparsity requires a number of linear inequalities that scales linearly with the size of the problem. Alternatively, chordal decomposition introduces equality and inequality constraints into the problem which needlessly increases its computational demand. We prove the direct exploitation of sparsity to be more beneficial in the case of a relaxation similar to that of this dissertation. Additionally, we exploit the structure of the problem in further reducing the number of linear inequalities by half. We further suggest two more relaxations based on the empirical results of the improved relaxation proposed

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