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Learning-Augmented Primal-Dual Control Design for Secondary Frequency Regulation

Yixuan Yu, Rajni K. Bansal, Yan Jiang, Pengcheng You

TL;DR

A systematic framework to embed learning in the design of a primal-dual controller that provides provable stability and steady-state optimality, while simultaneously improving key transient metrics, including frequency nadir and control effort, in a data-driven manner is presented.

Abstract

Frequency stability is fundamental to the secure operation of power systems. With growing uncertainty and volatility introduced by renewable generation, secondary frequency regulation must now deliver enhanced performance not only in the steady state but also during transients. This paper presents a systematic framework to embed learning in the design of a primal-dual controller that provides provable (potentially exponential) stability and steady-state optimality, while simultaneously improving key transient metrics, including frequency nadir and control effort, in a data-driven manner. In particular, we employ the primal-dual dynamics of an optimization problem that encodes steady-state objectives to realize secondary frequency control with asymptotic stability guarantee. To augment transient performance of the controller via learning, a change of variables on control inputs, which will be deployed by neural networks, is proposed such that under mild conditions, stability and steady-state optimality are preserved. It further allows us to define a learning goal that accounts for the exponential convergence rate, frequency nadir and accumulated control effort, and use sample trajectories to enhance these metrics. Simulation results validate the theories and demonstrate superior transient performance of the learning-augmented primal-dual controller.

Learning-Augmented Primal-Dual Control Design for Secondary Frequency Regulation

TL;DR

A systematic framework to embed learning in the design of a primal-dual controller that provides provable stability and steady-state optimality, while simultaneously improving key transient metrics, including frequency nadir and control effort, in a data-driven manner is presented.

Abstract

Frequency stability is fundamental to the secure operation of power systems. With growing uncertainty and volatility introduced by renewable generation, secondary frequency regulation must now deliver enhanced performance not only in the steady state but also during transients. This paper presents a systematic framework to embed learning in the design of a primal-dual controller that provides provable (potentially exponential) stability and steady-state optimality, while simultaneously improving key transient metrics, including frequency nadir and control effort, in a data-driven manner. In particular, we employ the primal-dual dynamics of an optimization problem that encodes steady-state objectives to realize secondary frequency control with asymptotic stability guarantee. To augment transient performance of the controller via learning, a change of variables on control inputs, which will be deployed by neural networks, is proposed such that under mild conditions, stability and steady-state optimality are preserved. It further allows us to define a learning goal that accounts for the exponential convergence rate, frequency nadir and accumulated control effort, and use sample trajectories to enhance these metrics. Simulation results validate the theories and demonstrate superior transient performance of the learning-augmented primal-dual controller.
Paper Structure (20 sections, 7 theorems, 28 equations, 4 figures, 1 table)

This paper contains 20 sections, 7 theorems, 28 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Power System Model
  4. Optimal Steady-State Objectives
  5. Transient Performance Objectives
  6. Controller Design and Analysis
  7. Primal-Dual Interpretation
  8. Stability Analysis
  9. Learning for Transient Improvement
  10. Transient Performance Metrics
  11. Strictly Monotone Neural Network Training
  12. Numerical Results
  13. Training of Monotone Neural Network
  14. Performance of the Learned Controller
  15. Conclusion
  16. ...and 5 more sections

Key Result

Lemma 1

Suppose Assumption ass: f holds. The optimization problem eq: general optimization problem with precondition admits a unique global optimal solution $(\tilde{\theta}^*, \omega^*, s^*, \tilde{\phi}^*)$. Furthermore, there exists a unique Lagrange multiplier $(\nu^*, \lambda^*)$ such that $(\tilde{\th

Figures (4)

  • Figure 1: Loss over epochs in training
  • Figure 2: Frequency deviation of generator buses under a disturbance $p_i = 3$ p.u. on buses $\{14,22,28,36,38\}$: learned controller (left); traditional primal-dual based controller (right).
  • Figure 3: Cost of generators under a disturbance $p_i = 3$ p.u. on buses $\{14,22,28,36,38\}$: learned controller (left); traditional primal-dual based controller (right).
  • Figure 4: Identical marginal cost among generators

Theorems & Definitions (12)

  • Remark 1: Precondition
  • Lemma 1: Uniqueness
  • Remark 2: Hidden convexity
  • Lemma 2: Equivalence of optimization problems
  • Theorem 1: Closed-loop equilibrium
  • Theorem 2: Asymptotic stability
  • Remark 3: Change of variables
  • Theorem 3: Potentially exponential stability
  • Lemma 3
  • proof
  • ...and 2 more