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Dynamic Slack Bus

Federico Milano

TL;DR

The paper addresses slack-bus modeling by introducing a generalized dynamic slack-bus framework that recasts the static angle constraint as a set of differential-algebraic equations with an energy source, enabling distributed, local slack behavior. The main approach defines local slack capability through $\mathbf{T}_h \boldsymbol{\sigma}'_h = \boldsymbol{f}_h(\boldsymbol{\sigma}_h, \theta_h)$ and $p_h = p_{s,h}(\boldsymbol{\sigma}_h) + p_{t,h}(\boldsymbol{\sigma}_h, \boldsymbol{\sigma}'_h)$, with steady-state convergence $\lim_{t\to\infty} \boldsymbol{\sigma}_h = \hat{\sigma}$. Key contributions map the swing equations, AGC, and converter-based resources (grid-following and grid-forming) into this framework, showing that transient power $p_{t,h}$ captures stored energy and decays over time, while steady-state behavior reproduces a distributed slack condition. Case study on WSCC 9-bus demonstrates that grid-forming converters can supply inertial-like support and maintain stability, whereas grid-following converters lacking energy storage fail to provide adequate slack under disturbance. The framework offers a principled path to design converter-based resources that actively participate in power balance and frequency regulation.

Abstract

This letter proposes a general dynamic formulation of slack bus. With this aim, the angle constraint imposed by the slack bus is redefined as a set of differential equations and an energy source. The existence and role of the transient component of this source is also discussed in the letter. Based on this framework, the letter shows that the swing equations of synchronous machines can be interpreted as distributed, dynamic, multi-variable, local slack buses. Other relevant cases, including primary and secondary frequency regulation, passive loads as well as grid following and grid forming converters are discussed.

Dynamic Slack Bus

TL;DR

The paper addresses slack-bus modeling by introducing a generalized dynamic slack-bus framework that recasts the static angle constraint as a set of differential-algebraic equations with an energy source, enabling distributed, local slack behavior. The main approach defines local slack capability through and , with steady-state convergence . Key contributions map the swing equations, AGC, and converter-based resources (grid-following and grid-forming) into this framework, showing that transient power captures stored energy and decays over time, while steady-state behavior reproduces a distributed slack condition. Case study on WSCC 9-bus demonstrates that grid-forming converters can supply inertial-like support and maintain stability, whereas grid-following converters lacking energy storage fail to provide adequate slack under disturbance. The framework offers a principled path to design converter-based resources that actively participate in power balance and frequency regulation.

Abstract

This letter proposes a general dynamic formulation of slack bus. With this aim, the angle constraint imposed by the slack bus is redefined as a set of differential equations and an energy source. The existence and role of the transient component of this source is also discussed in the letter. Based on this framework, the letter shows that the swing equations of synchronous machines can be interpreted as distributed, dynamic, multi-variable, local slack buses. Other relevant cases, including primary and secondary frequency regulation, passive loads as well as grid following and grid forming converters are discussed.
Paper Structure (11 sections, 33 equations, 1 figure)

This paper contains 11 sections, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Slack Bus as a Set of Differential Equations
  3. Examples
  4. Synchronous Machines
  5. Primary Frequency Control
  6. Automatic Generation Control
  7. Passive Load
  8. Grid-Following Converters
  9. Grid-Forming Converters
  10. Case Study
  11. Conclusions

Figures (1)

  • Figure 1: Trajectories of the voltage phase angle at bus 1 ($\theta_1$) of the WSCC 9-bus system for different generation scenarios.