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A Parallelized, Adam-Based Solver for Reserve and Security Constrained AC Unit Commitment

Samuel Chevalier

TL;DR

A parallelized Adam-based numerical solver to overcome one of the most challenging power system optimization problems: security and reserve constrained AC Unit Commitment.

Abstract

Power system optimization problems which include the nonlinear AC power flow equations require powerful and robust numerical solution algorithms. Within this sub-field of nonlinear optimization, interior point methods have come to dominate the solver landscape. Over the last decade, however, a number of efficient numerical optimizers have emerged from the field of Machine Learning (ML). One algorithm in particular, Adam, has become the optimizer-of-choice for a massive percentage of ML training problems (including, e.g., the training of GPT-3), solving some of the largest unconstrained optimization problems ever conceived of. Inspired by such progress, this paper designs a parallelized Adam-based numerical solver to overcome one of the most challenging power system optimization problems: security and reserve constrained AC Unit Commitment. The resulting solver, termed QuasiGrad, recently competed in the third ARPA-E Grid Optimization (GO3) competition. In the day-ahead market clearing category (with systems ranging from 3 to 23,643 buses over 48 time periods), QuasiGrad's aggregated market surplus scores were within 5% of the winningest market surplus scores. The QuasiGrad solver is now released as an open-source Julia package: QuasiGrad.jl. The internal gradient-based solver (Adam) can easily be substituted for other ML-inspired solvers (e.g., AdaGrad, AdaDelta, RMSProp, etc.). Test results from large experiments are provided.

A Parallelized, Adam-Based Solver for Reserve and Security Constrained AC Unit Commitment

TL;DR

A parallelized Adam-based numerical solver to overcome one of the most challenging power system optimization problems: security and reserve constrained AC Unit Commitment.

Abstract

Power system optimization problems which include the nonlinear AC power flow equations require powerful and robust numerical solution algorithms. Within this sub-field of nonlinear optimization, interior point methods have come to dominate the solver landscape. Over the last decade, however, a number of efficient numerical optimizers have emerged from the field of Machine Learning (ML). One algorithm in particular, Adam, has become the optimizer-of-choice for a massive percentage of ML training problems (including, e.g., the training of GPT-3), solving some of the largest unconstrained optimization problems ever conceived of. Inspired by such progress, this paper designs a parallelized Adam-based numerical solver to overcome one of the most challenging power system optimization problems: security and reserve constrained AC Unit Commitment. The resulting solver, termed QuasiGrad, recently competed in the third ARPA-E Grid Optimization (GO3) competition. In the day-ahead market clearing category (with systems ranging from 3 to 23,643 buses over 48 time periods), QuasiGrad's aggregated market surplus scores were within 5% of the winningest market surplus scores. The QuasiGrad solver is now released as an open-source Julia package: QuasiGrad.jl. The internal gradient-based solver (Adam) can easily be substituted for other ML-inspired solvers (e.g., AdaGrad, AdaDelta, RMSProp, etc.). Test results from large experiments are provided.
Paper Structure (20 sections, 1 theorem, 41 equations, 5 figures, 1 table, 4 algorithms)

This paper contains 20 sections, 1 theorem, 41 equations, 5 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Motivation for Reformulation
  4. MINLP Reformulation
  5. Further refinements: constraint penalization, variable clipping, and integer relaxation
  6. Integer projection
  7. Contingency gradients
  8. Contingency evaluation
  9. Contingency backpropagation
  10. Contingency backpropagation summary
  11. Implementation of the Adam solver
  12. QuasiGrad
  13. Copper plate economic dispatch with LP relaxed binaries
  14. Successively Linearized Power Flow Approximations
  15. Reserve variable cleanup
  16. ...and 5 more sections

Key Result

Theorem 1

Assume eq: ramps initially holds. By enforcing eq: ramp_con_alt, the power balance problems eq: pfs can be solved in parallel while maintaining ramp feasibility eq: ramps across all devices.

Figures (5)

  • Figure 1: Illustrated is an Adam solve on a 617-bus, 18 time period, real-time market clearing test case (integers relaxed); this system is initialized with a copper pate economic dispatch solution (LP), whose upper bound is given as the orange dashed line. Within several thousand iterations, Adam finds an AC network solution to within 1% of this global bound. A single back-propagation (i.e., gradient calculation) through this entire system, include all $18\times562$ contingencies, takes $\sim$24ms when parallelized on 6 CPU threads.
  • Figure 2: The parallel nature of devices constraints and power flow constraints are portrayed. Devices can be projected feasible in parallel (via Proj. \ref{['bin_proj']}), and power flow solves can be performed in parallel (via Proj. \ref{['proj_parallel_pf']}).
  • Figure 3: Adam step size decay ($\alpha$, left) and soft-abs/soft-ReLU tightening ($\epsilon$, right). Step size decay "leads" soft-abs/soft-ReLU tightening.
  • Figure 4: Illustrated is the successive tightening of the soft-abs function $\beta\sqrt{x^{2}+\epsilon^{2}}$ as wall clock time increases. The value of epsilon is decreased in concordance with the red curve illustrated in Fig. \ref{['fig:decay']}.
  • Figure 5: Ramp-constrained power flow planes.

Theorems & Definitions (2)

  • Theorem 1
  • proof