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An Augmented Lagrangian Method on GPU for Security-Constrained AC Optimal Power Flow

François Pacaud, Armin Nurkanović, Anton Pozharskiy, Alexis Montoison, Sungho Shin

TL;DR

This work tackles large-scale AC-SCOPF with complementarity constraints by applying a GPU-accelerated Nonlinearly Constrained Augmented Lagrangian (NCL) method. Subproblems are solved with interior-point methods and accelerated on GPUs, while the augmented Lagrangian term regularizes Newton systems, enabling pivoting-free factorization. MadNCL demonstrates strong speedups over CPU solvers, solving a 500-bus, 256-contingency instance in under 3 minutes on a GPU and enabling effective contingency screening. The approach also supports large-scale GPU-accelerated polarization of Recourse constraints and demonstrates substantial practical impact for security-constrained planning and infeasibility detection. Overall, the paper shows that GPU-accelerated MPCC solvers can make AC-SCOPF tractable at scales that challenge traditional solvers.

Abstract

We present a new algorithm for solving large-scale security-constrained optimal power flow in polar form (AC-SCOPF). The method builds on Nonlinearly Constrained augmented Lagrangian (NCL), an augmented Lagrangian method in which the subproblems are solved using an interior-point method. NCL has two key advantages for large-scale SC-OPF. First, NCL handles difficult problems such as infeasible ones or models with complementarity constraints. Second, the augmented Lagrangian term naturally regularizes the Newton linear systems within the interior-point method, enabling to solve the Newton systems with a pivoting-free factorization that can be efficiently parallelized on GPUs. We assess the performance of our implementation, called MadNCL, on large-scale corrective AC-SCOPFs, with complementarity constraints modeling the corrective actions. Numerical results show that MadNCL can solve AC-SCOPF with 500 buses and 256 contingencies fully on the GPU in less than 3 minutes, whereas Knitro takes more than 3 hours to find an equivalent solution.

An Augmented Lagrangian Method on GPU for Security-Constrained AC Optimal Power Flow

TL;DR

This work tackles large-scale AC-SCOPF with complementarity constraints by applying a GPU-accelerated Nonlinearly Constrained Augmented Lagrangian (NCL) method. Subproblems are solved with interior-point methods and accelerated on GPUs, while the augmented Lagrangian term regularizes Newton systems, enabling pivoting-free factorization. MadNCL demonstrates strong speedups over CPU solvers, solving a 500-bus, 256-contingency instance in under 3 minutes on a GPU and enabling effective contingency screening. The approach also supports large-scale GPU-accelerated polarization of Recourse constraints and demonstrates substantial practical impact for security-constrained planning and infeasibility detection. Overall, the paper shows that GPU-accelerated MPCC solvers can make AC-SCOPF tractable at scales that challenge traditional solvers.

Abstract

We present a new algorithm for solving large-scale security-constrained optimal power flow in polar form (AC-SCOPF). The method builds on Nonlinearly Constrained augmented Lagrangian (NCL), an augmented Lagrangian method in which the subproblems are solved using an interior-point method. NCL has two key advantages for large-scale SC-OPF. First, NCL handles difficult problems such as infeasible ones or models with complementarity constraints. Second, the augmented Lagrangian term naturally regularizes the Newton linear systems within the interior-point method, enabling to solve the Newton systems with a pivoting-free factorization that can be efficiently parallelized on GPUs. We assess the performance of our implementation, called MadNCL, on large-scale corrective AC-SCOPFs, with complementarity constraints modeling the corrective actions. Numerical results show that MadNCL can solve AC-SCOPF with 500 buses and 256 contingencies fully on the GPU in less than 3 minutes, whereas Knitro takes more than 3 hours to find an equivalent solution.
Paper Structure (19 sections, 20 equations, 2 figures, 3 tables)

This paper contains 19 sections, 20 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Augmented Lagrangian method
  4. Scope and contributions
  5. Model
  6. Recourse constraints
  7. AC-SCOPF
  8. Mathematical programs with complementarity constraints
  9. MPCC in vertical formulation
  10. First-order stationary conditions
  11. Solution methods
  12. Augmented Lagrangian for MPCCs
  13. Algorithm NCL
  14. Newton systems
  15. Numerical results
  16. ...and 4 more sections

Figures (2)

  • Figure 1: Result of contingency screening for ACTIVSg500. The base case solution in \ref{['eq:screening']} is obtained by solving the classical OPF problem.
  • Figure 2: Time per iteration (in seconds) against number of contingencies $K$ when solving ACTIVSg500. We compare Knitro (using JuMP) against MadNCL (using ExaModels). We test MadNCL both on the CPU (using HSL MA57) and on the GPU (using NVIDIA cuDSS).