Reduced-Space Interior Point Methods in Power Grid Problems

Abstract

Due to critical environmental issues, the power systems have to accommodate a significant level of penetration of renewable generation which requires smart approaches to the power grid control. Associated optimal control problems are large-scale nonlinear optimization problems with up to hundreds of millions of variables and constraints. The interior point methods become computationally intractable, mainly due to the solution of large linear systems. This document addresses the computational bottlenecks of the interior point method during the solution of the security constrained optimal power flow problems by applying reduced space quasi-Newton IPM, which could utilize high-performance computers due to the inherent parallelism in the adjoint method. Reduced space IPM approach and the adjoint method is a novel approach when it comes to solving the (SC)OPF problems. These were previously used in the PDE-constrained optimization. The presented methodology is suitable for high-performance architectures due to inherent parallelism in the adjoint method during the gradient evaluation, since the individual contingency scenarios are modeled by independent set of the constraints. Preliminary evaluation of the performance and convergence is performed to study the reduced space approach.

A Feasible Reduced Space Method for Real-Time Optimal Power Flow

“…The algorithm takes many iterations to converge, leading to total running time being order of magnitude greater than Ipopt (on case9241pegase, Ipopt converges in only 60s). However, our algorithm remains tractable, and is a net improvement comparing to previous implementation of the reduced space methods [10].…”

“…When ( 8) is satisfied, l does not depend on z and ψ. By deriving ˆ and choosing (z, ψ) solutions of the two linear systems (10), we get the expression (9).…”

“…To alleviate this, both Quasi-Newton and constraint aggregation schemes can be used, so far with mixed results. We refer to [10] for a detailed discussion about the resolution of the reduced problem (11) with IPM. By contrast, by moving all inequality constraints into the objective, an AL formulation requires the resolution of only (2n u + 1) linear systems to evaluate the full reduced Hessian.…”

“…(2) the large number of OCs in the problem limits the available degrees of freedom in the reduced space. We refer to the recent report of Kardos et al [10] for further details on the limitations of the algorithm.…”

Batched Second-Order Adjoint Sensitivity for Reduced Space Methods

“…Reduced-space methods have been applied widely in uncertainty quantification and partial differential equation (PDE)-constrained optimization [7], and their applications in the optimization of power grids is known since the 1960s [11]. However, extracting the secondorder sensitivities in the reduced space has been considered tedious to implement and hard to motivate on classical CPU architectures (see [17] for a recent discussion about the computation of the reduced Hessian on the CPU). To the best of our knowledge, this paper is the first to present a SIMD focused algorithm leveraging the GPU to efficiently compute the reduced Hessian of the power flow equations.…”