Unstable Poles Arising in AC Power Grid Subsystem Representations

Technical Summary: Unstable Poles Arising in AC Power Grid Subsystem Representations

Problem Statement
As power systems transition from synchronous generation to grid-forming power electronics, classical small-signal stability analysis using monolithic system-wide models is becoming insufficient. Consequently, recent research has shifted toward analyzing grids as interconnections of distributed subsystems, leveraging input-output properties to derive scalable stability certificates. However, the representation of these subsystems is non-unique. Two primary models dominate the literature: the PQ model, which couples active/reactive power to phase angle and voltage magnitude (common in power systems literature), and the IV model, which couples currents to voltages (favored in power electronics).

The core problem addressed in this paper is that the choice of subsystem representation (PQ vs. IV) significantly impacts the observed stability properties of individual components. Specifically, the authors demonstrate that a subsystem may appear stable in one representation while exhibiting unstable poles in the other. These unstable poles can arise even when the aggregate interconnection of the entire grid is stable and well-behaved. Neglecting high-frequency dynamics or specific device operating points during system identification or stability studies may obscure these poles, leading to incorrect conclusions about the need for network interconnection to stabilize the system.

Methodology
The authors employ a rigorous mathematical framework based on small-signal linearization and frequency-domain analysis:

1. Model Derivation: The paper derives both the IV and PQ small-signal representations of an AC grid without simplifying assumptions regarding bus or line dynamics. The grid is modeled as a directed graph with buses and lines, utilizing synchronously rotating DQ reference frames.
2. Loop Transformation: A key methodological contribution is the derivation of a loop transformation that mathematically relates the IV and PQ models. The authors show that the bus impedance in the IV model ($Z_i(s)$) and the transfer function in the PQ model ($G_i(s)$) are connected via static matrices ($U^\dagger, U^\ddagger, W$) and a feedback structure. Specifically, $Z_i(s)$ is derived from $G_i(s)$ through a negative feedback loop, while $G_i(s)$ is derived from $Z_i(s)$ through a positive feedback loop.
3. Stability Analysis: The stability of these subsystems is analyzed using the Generalized Nyquist Stability Theorem. The authors examine the characteristic loci of the return ratios ($L_{IV}(s)$ and $L_{PQ}(s)$) to determine the number of right-half-plane (RHP) poles. They investigate how the Nyquist contours encircle the critical point $-1$ under different operating conditions.
4. Case Studies: The theoretical findings are validated through numerical examples, including:
   • A droop-controlled bus with simplified dynamics.
   • A voltage source in series with a passive RLC impedance (representing transformers or filters).
   • A full-order synchronous generator model equipped with an exciter, power system stabilizer (PSS), and a step-up transformer.

Key Contributions

1. Unified Derivation: The paper provides a complete derivation of both PQ and IV grid models without simplifying device dynamics and establishes a formal loop transformation to translate between them.
2. Identification of Representation-Dependent Instabilities: The authors analytically prove that unstable poles can emerge in one representation but not the other, depending on specific conditions:
   • IV Model Instability: Unstable poles in the IV representation of a droop-controlled bus can arise primarily due to the device operating point (specifically the sign and magnitude of reactive power $Q^*$).
   • PQ Model Instability: Unstable poles in the PQ representation can arise when a device is placed in series with a passive impedance (e.g., a transformer or filter) that exhibits high-frequency dynamics.
3. Validation with Complex Dynamics: The paper demonstrates that these phenomena persist even when using high-fidelity, full-order models (e.g., a synchronous generator with an exciter and transformer), confirming that the issue is not an artifact of simplified modeling.

Results

• Droop-Controlled Buses: In the IV model, the characteristic loci of the return ratio can encircle the point $-1$ in a clockwise direction as the frequency approaches zero, depending on the equilibrium reactive power. This indicates the presence of an unstable pole in the IV impedance $Z_i(s)$, even if the underlying PQ transfer function $G_i(s)$ is marginally stable.
• Passive Impedances: In the PQ model, a passive RLC component (which is stable in the IV domain) can introduce unstable poles when transformed into the PQ domain. The Nyquist plot of the return ratio for the passive component, when mapped to the PQ frame, can encircle $-1$ clockwise, indicating RHP poles in the PQ transfer function $G_i(s)$.
• Full-Order Generator: In a multi-machine case study, a stable aggregate system was shown to contain a generator that, when isolated and modeled as a subsystem, possessed unstable poles in the IV representation (due to operating point) and additional unstable poles in the PQ representation (due to the series transformer impedance).

Significance and Claims
The paper claims that the choice of subsystem representation is not merely a mathematical convenience but a critical factor in stability analysis and system identification. The authors emphasize that:

• Stability is Context-Dependent: A subsystem that is stable in one representation may be unstable in another. This complicates the derivation of decentralized stability criteria, which often rely on the assumption that individual subsystems are stable.
• Risk of Obscured Dynamics: Neglecting high-frequency dynamics (such as those in transformers or filters) or ignoring specific device operating points can hide unstable poles. These poles are often stabilized by the network interconnection; failing to account for them in subsystem identification can lead to inaccurate models and potential instability in grid-code formulations.
• Need for Careful Modeling: Engineers and researchers must exercise caution when selecting a subsystem representation. The paper argues that one must account for the specific operating point and high-frequency dynamics to ensure that unstable poles, which rely on the network for stabilization, are correctly identified and managed.