Learning Optimal Power Flow: Worst-Case Guarantees for Neural Networks

Abstract

This paper introduces for the first time a framework to obtain provable worst-case guarantees for neural network performance, using learning for optimal power flow (OPF) problems as a guiding example. Neural networks have the potential to substantially reduce the computing time of OPF solutions. However, the lack of guarantees for their worst-case performance remains a major barrier for their adoption in practice. This work aims to remove this barrier. We formulate mixed-integer linear programs to obtain worst-case guarantees for neural network predictions related to (i) maximum constraint violations, (ii) maximum distances between predicted and optimal decision variables, and (iii) maximum sub-optimality. We demonstrate our methods on a range of PGLib-OPF networks up to 300 buses. We show that the worst-case guarantees can be up to one order of magnitude larger than the empirical lower bounds calculated with conventional methods. More importantly, we show that the worst-case predictions appear at the boundaries of the training input domain, and we demonstrate how we can systematically reduce the worst-case guarantees by training on a larger input domain than the domain they are evaluated on.

A Convex Neural Network Solver for DCOPF with Generalization Guarantees

“…Since Algorithm 1 only relies on getting µ to be in the correct range, the active constraints would be identified correctly for all of these cases. This theorem also formalizes the empirical observation in [31], where the error of neural network-based OPF is reduced if training points are on the boundary of the feasible region. of the solutions obtained from our algorithm are optimal, while less than half of the solutions from the end-to-end model are feasible.…”

“…However, unlike LU factorization, it is hard to guarantee that the machine learning methods would always recover the right answer. The approach in [31] can bound the worst case errors for a trained neural network with fixed parameters. But these types of ex post analysis is hard to generalize and do not shed light on why a method may perform better or worse.…”

A Convex Approximation for the Tertiary Control of Unbalanced Microgrids

“…Conic relaxations Heuristics AI Energy Management Proposed Reference [6] to [9] [11] to [12] [15] to [16] [17] to [19] approach Convergence guarantee Global optimum Energy storage Practical oriented Implementation Stochastic model on conic approximations such as semidefinite programming and second-order cone optimization (see [7] and the references therein for a complete review of this subject). These relaxations transform the non-convex problem into a convex thereof with theoretical and practical advantages related to global optimal, uniqueness of the solution, and fast convergence rate [8].…”

“…Other methods based on artificial intelligence have also been proposed, especially reinforcement learning [14] and neural networks [15]. These types of algorithms are promising for real-time implementation, although they may be improved by theoretical analysis of the optimization problem [16].…”

“…For the sake of completeness, the entire model is presented in Table 2 including the type of equation (notice that quadratic equality constraints are also non-convex). (11) load flow non-convex (12) power balance affine (13) power loss quadratic equality (14) grid power quadratic equality (15) load model non-convex (16) solar generation affine (17) wind generation affine (18) energy balance affine (19) battery charging quadratic equality (20) battery discharge quadratic equality (21) battery power affine (22) converter capacity second-order cone (23) voltage limits second-order cone (24) power deficit affine (25) static reserve affine…”