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A Wirtinger Power Flow Jacobian Singularity Condition for Voltage Stability in Converter-Rich Power Systems

Ahmed Mesfer Alkhudaydi, Bai Cui

Abstract

The progression of modern power systems towards converter-rich operations calls for new models and analytics in steady-state voltage stability assessment. The classic modeling assumption of the generators as stiff voltage sources no longer holds. Instead, the voltage- and current-limited behaviors of converters need to be considered. In this paper, we develop a Wirtinger derivative-based formulation for the power flow Jacobian and derive an explicit sufficient condition for its singularity. Compared to existing works, we extend the explicit sufficient singularity condition to incorporate all bus types instead of only slack and PQ types. We prove that the singularity of the alternative Jacobian coincides with that of the conventional one. A bus-wise voltage stability index, denoted $C_{\mathrm{W}}$, is derived from diagonal dominance conditions. The condition $\min_i C_{W,i}$ being greater than one certifies the nonsingularity of the Jacobian and provides a fast, non-iterative stability margin. Case studies in standard IEEE test systems show that the proposed index yields less conservative and more localized assessments than classical indices such as the L-index, the $K_{\mathrm{R}}$ index, and the SCR index.

A Wirtinger Power Flow Jacobian Singularity Condition for Voltage Stability in Converter-Rich Power Systems

Abstract

The progression of modern power systems towards converter-rich operations calls for new models and analytics in steady-state voltage stability assessment. The classic modeling assumption of the generators as stiff voltage sources no longer holds. Instead, the voltage- and current-limited behaviors of converters need to be considered. In this paper, we develop a Wirtinger derivative-based formulation for the power flow Jacobian and derive an explicit sufficient condition for its singularity. Compared to existing works, we extend the explicit sufficient singularity condition to incorporate all bus types instead of only slack and PQ types. We prove that the singularity of the alternative Jacobian coincides with that of the conventional one. A bus-wise voltage stability index, denoted , is derived from diagonal dominance conditions. The condition being greater than one certifies the nonsingularity of the Jacobian and provides a fast, non-iterative stability margin. Case studies in standard IEEE test systems show that the proposed index yields less conservative and more localized assessments than classical indices such as the L-index, the index, and the SCR index.

Paper Structure

This paper contains 25 sections, 1 theorem, 40 equations, 9 figures, 4 tables.

Table of Contents

  1. Introduction
  2. System Modeling and Wirtinger Preliminaries
  3. Thévenin Form and Bus Partitioning
  4. Wirtinger Calculus Preliminaries
  5. Full Wirtinger State and Jacobian Formulation
  6. Unified Tangent-vector Framework for Bus Constraint Modeling
  7. Concept of Tangent Constraints
  8. Voltage Constraint Bus
  9. Current Constraint Bus
  10. Generic Tangent Representation
  11. Reduced Wirtinger Jacobian and Two-Bus example
  12. Reduced Wirtinger Jacobian
  13. Two-Bus Case
  14. Multi-Bus Extension and Unified Stability Index
  15. Unconstrained Bus Dominance
  16. ...and 10 more sections

Key Result

Theorem 1

Let $J_{\mathrm{conv}}=\partial [P_{\mathcal{U}};P_{\mathcal{C}};Q_{\mathcal{U}}]/\partial[\theta_{\mathcal{U}};\theta_{\mathcal{C}};U_{\mathcal{U}}]$ be the conventional Jacobian and $J_{\mathrm{red}}=\partial [S_{\mathcal{U}};S_{\mathcal{U}}^{*};P_{\mathcal{C}}]/\partial[I_{\mathcal{U}};I_{\mathca whenever $L$ and $R$ are nonsingular (which holds generically at regular operating points). $\black

Figures (9)

  • Figure 1: Tangent direction on the constant-$|V|$ space.
  • Figure 2: Tangent direction on the constant-$|I|$ space.
  • Figure 3: Example of a two bus system.
  • Figure 4: Evolution of the proposed Wirtinger Jacobian matrix across loading conditions
  • Figure 5: The 9-bus system
  • ...and 4 more figures

Theorems & Definitions (2)

  • Theorem 1: Identical singular sets
  • Remark 1