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Power System Electromagnetic Transient Stability: an Analysis Based on Convergent Hamiltonian

Xinyuan Jiang, Constantino M. Lagoa, Yan Li

TL;DR

The paper tackles transient stability in power systems with detailed electromagnetic dynamics by introducing a converging Hamiltonian principle derived from contraction analysis of time-varying port-Hamiltonian systems with constant damping and strictly convex Hamiltonians. It develops horizontal contraction in a canonical quotient space and shows that Hamiltonian convergence implies global attractivity of a synchronized limit cycle when such a cycle exists. Applied to an electromagnetic two-machine model, the authors show that if a synchronized limit cycle exists, all trajectories converge to it, with a provable convergence rate tied to system damping and inertia. Numerical simulations validate the theory and reveal that instability phenomena in traditional stability analysis often reflect the nonexistence of a synchronized limit cycle, highlighting implications for future control of inverter-based resources.

Abstract

Transient stability is crucial to the reliable operation of power systems. Existing theories rely on the simplified electromechanical models, substituting the detailed electromagnetic dynamics of inductor and capacitor with their impedance representations. However, this simplification is inadequate for the growing penetration of fast-switching power electronic devices. Attempts to extend the existing theories to include electromagnetic dynamics lead to overly conservative stability conditions. To tackle this problem more directly, we study the condition under which the power source and dissipation in the electromagnetic dynamics tend to balance each other asymptotically. This is equivalent to the convergence of the Hamiltonian (total stored energy) and can be shown to imply transient stability. Using contraction analysis, we prove that this property holds for a large class of time-varying port-Hamiltonian systems with (i) constant damping matrix and (ii) strictly convex Hamiltonian. Then through port-Hamiltonian modeling of the electromagnetic dynamics, we obtain that the synchronized steady state of the power system is globally stable if it exists. This result provides new insights into the reliable operation of power systems. The proposed theory is illustrated in the simulation results of a two-machine system.

Power System Electromagnetic Transient Stability: an Analysis Based on Convergent Hamiltonian

TL;DR

The paper tackles transient stability in power systems with detailed electromagnetic dynamics by introducing a converging Hamiltonian principle derived from contraction analysis of time-varying port-Hamiltonian systems with constant damping and strictly convex Hamiltonians. It develops horizontal contraction in a canonical quotient space and shows that Hamiltonian convergence implies global attractivity of a synchronized limit cycle when such a cycle exists. Applied to an electromagnetic two-machine model, the authors show that if a synchronized limit cycle exists, all trajectories converge to it, with a provable convergence rate tied to system damping and inertia. Numerical simulations validate the theory and reveal that instability phenomena in traditional stability analysis often reflect the nonexistence of a synchronized limit cycle, highlighting implications for future control of inverter-based resources.

Abstract

Transient stability is crucial to the reliable operation of power systems. Existing theories rely on the simplified electromechanical models, substituting the detailed electromagnetic dynamics of inductor and capacitor with their impedance representations. However, this simplification is inadequate for the growing penetration of fast-switching power electronic devices. Attempts to extend the existing theories to include electromagnetic dynamics lead to overly conservative stability conditions. To tackle this problem more directly, we study the condition under which the power source and dissipation in the electromagnetic dynamics tend to balance each other asymptotically. This is equivalent to the convergence of the Hamiltonian (total stored energy) and can be shown to imply transient stability. Using contraction analysis, we prove that this property holds for a large class of time-varying port-Hamiltonian systems with (i) constant damping matrix and (ii) strictly convex Hamiltonian. Then through port-Hamiltonian modeling of the electromagnetic dynamics, we obtain that the synchronized steady state of the power system is globally stable if it exists. This result provides new insights into the reliable operation of power systems. The proposed theory is illustrated in the simulation results of a two-machine system.

Paper Structure

This paper contains 24 sections, 7 theorems, 74 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Standard Contraction Definition
  4. Port-Hamiltonian System
  5. Horizontal Contraction of PH System
  6. The Canonical Quotient Space
  7. Horizontal Contraction in the Quotient Space
  8. Stability of the Electromagnetic Power System Model
  9. Physical Model in $\alpha\beta$-Coordinates
  10. Time-Varying PH Model
  11. Two-Machine System with Constant Impedance Loads
  12. Stability of the Two-Machine System
  13. Numerical Examples
  14. Global Convergence in the Regular Case
  15. Loss of Synchronized Limit Cycle Steady State
  16. ...and 9 more sections

Key Result

Proposition 1

The closed pH system (E:closed) that has a uniformly strictly convex Hamiltonian, i.e., condition (E:strictly_convex), is HC with the contraction rate, where $\lambda_{\min}(\mathbf{A})$ is the smallest eigenvalue of the Hermitian matrix $\mathbf{A}$. $\lozenge$

Figures (5)

  • Figure 1: Single-line diagram of the two-machine system and the underlying graph topology (SG: red, shunt capacitor: blue, R--L line: green)
  • Figure 2: Case where there exists a synchronized limit cycle
  • Figure 3: Case where there only exists an imperfect limit cycle with low-frequency oscillation (existence of two steady-state frequencies) due to the large difference between the input torques of the two SGs
  • Figure 4: Case where the system collapses to close to zero due to the field fluxes of the SGs being too high
  • Figure 5: Illustration of a key step in proving the contraction of the quotient distance: a zigzag approximation of the minimizing curve and to prove contraction of every transverse zigs.

Theorems & Definitions (13)

  • Remark 1
  • Definition 1
  • Proposition 1: Horizontal Contraction with $\mathbf{R} \succ 0$
  • Proposition 2: Weak Horizontal Contraction with $\mathbf{R} = \mathbf{0}$
  • Remark 2
  • Proposition 3: Converging Hamiltonian Difference
  • Remark 3
  • Proposition 4: Hamiltonian Convergence Principle
  • Remark 4
  • Lemma 1
  • ...and 3 more