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On the Stability of Networked Nonlinear Negative Imaginary Systems with Applications to Electrical Power Systems

Yijun Chen, Kanghong Shi, Ian R. Petersen, Elizabeth L. Ratnam

TL;DR

The paper tackles stability and consensus in networks of nonlinear negative imaginary (NI) systems with applications to power grids undergoing the net-zero transition. It develops a Luré-Postnikov-like Lyapunov framework to prove local stability for the interconnection of NI plants and controllers and to establish output consensus on networked NI systems. The approach extends NI theory to nonlinear direct feedthrough and networked settings, enabling battery-based actuators to progressively enforce NI dynamics on transmission lines. This yields a practical pathway to enhance transient stability and reduce grid expansion by moving the grid toward an NI-controlled regime one line at a time.

Abstract

In the transition to achieving net zero emissions, it has been suggested that a substantial expansion of electric power grids will be necessary to support emerging renewable energy zones. In this paper, we propose employing battery-based feedback control and nonlinear negative imaginary (NI) systems theory to reduce the need for such expansion. By formulating a novel Luré-Postnikov-like Lyapunov function, stability results are presented for the feedback interconnection of two single nonlinear NI systems, while output feedback consensus results are established for the feedback interconnection of two networked nonlinear NI systems based on a network topology. This theoretical framework underpins our design of battery-based control in power transmission systems. We demonstrate that the power grid can be gradually transitioned into the proposed NI systems, one transmission line at a time.

On the Stability of Networked Nonlinear Negative Imaginary Systems with Applications to Electrical Power Systems

TL;DR

The paper tackles stability and consensus in networks of nonlinear negative imaginary (NI) systems with applications to power grids undergoing the net-zero transition. It develops a Luré-Postnikov-like Lyapunov framework to prove local stability for the interconnection of NI plants and controllers and to establish output consensus on networked NI systems. The approach extends NI theory to nonlinear direct feedthrough and networked settings, enabling battery-based actuators to progressively enforce NI dynamics on transmission lines. This yields a practical pathway to enhance transient stability and reduce grid expansion by moving the grid toward an NI-controlled regime one line at a time.

Abstract

In the transition to achieving net zero emissions, it has been suggested that a substantial expansion of electric power grids will be necessary to support emerging renewable energy zones. In this paper, we propose employing battery-based feedback control and nonlinear negative imaginary (NI) systems theory to reduce the need for such expansion. By formulating a novel Luré-Postnikov-like Lyapunov function, stability results are presented for the feedback interconnection of two single nonlinear NI systems, while output feedback consensus results are established for the feedback interconnection of two networked nonlinear NI systems based on a network topology. This theoretical framework underpins our design of battery-based control in power transmission systems. We demonstrate that the power grid can be gradually transitioned into the proposed NI systems, one transmission line at a time.
Paper Structure (12 sections, 9 theorems, 42 equations, 2 figures)

This paper contains 12 sections, 9 theorems, 42 equations, 2 figures.

Table of Contents

  1. introduction
  2. An Initial Stability Result
  3. Definitions of Nonlinear NI Systems
  4. Stability of NI Systems
  5. Output Consensus of Networked NI Systems
  6. Settings for Networked NI Systems
  7. Main Results
  8. Application to Power Transmission Systems
  9. Transmission Network Model
  10. Resilience in Power Transmission Systems
  11. Battery-based Angle Feedback Controllers
  12. Conclusion

Key Result

Theorem 1

Consider a nonlinear OSNI plant $H_{p}$ satisfying Assumptions apt:state_output_consistency, apt:input_state_consistency, and apt:positive_uh. Also, consider a nonlinear controller $H_{c}$, either being a nonlinear static controller or a nonlinear dynamic NI controller. Suppose Assumptions apt:g and

Figures (2)

  • Figure 1: The feedback interconnection of the nonlinear system $H_{p}$ and the nonlinear system $H_{c}$.
  • Figure 2: The feedback interconnection of nonlinear plants $\mathcal{H}_{p}$ and nonlinear edge controllers $\mathcal{H}_{c}$ based on the incidence matrix $\mathbf{Q}$.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Definition 3: Local Output Consensus
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 3
  • Theorem 4
  • ...and 4 more