Unified Energy Function Tailored to Inverter-Based Resources with PI Controllers for Transient Stability Analysis

Hsiao-Dong Chiang; Yang Wu; Yifan Zhang; Yitong Li

arxiv: 2604.23697 · v1 · submitted 2026-04-26 · 📡 eess.SY · cs.SY

Unified Energy Function Tailored to Inverter-Based Resources with PI Controllers for Transient Stability Analysis

Yifan Zhang , Hsiao-Dong Chiang , Yitong Li , Yang Wu This is my paper

Pith reviewed 2026-05-08 05:32 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords energy functiontransient stabilityinverter-based resourcesPI controllersregion of attractiongrid-following invertergrid-forming inverter

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The pith

A new energy function tailored to PI controllers gives more accurate stability region estimates for inverter-based resources.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an energy function specifically designed for inverter-based resources that rely on proportional-integral controllers. These resources produce nonlinear swing equations featuring state-dependent damping, which prevents direct use of the classical energy function developed for synchronous generators. The proposed function takes a unified form that applies to a class of nonlinear systems with PI controllers. It yields less conservative estimates of the region of attraction, the set of conditions from which the system returns to equilibrium after a disturbance. A reader would care because increasing shares of renewable inverters are changing power system behavior and require updated tools to predict whether the grid stays stable after faults or load changes.

Core claim

The paper establishes that a unified energy function can be derived explicitly for nonlinear systems containing PI controllers. This function accounts for the integral action and resulting state-dependent damping in the swing equations of inverter-based resources. When applied to a grid-following inverter and a DC-voltage-controlled grid-forming inverter, the function produces estimates of the region of attraction that are both larger and more precise than those obtained from the classical energy function. Hardware-in-the-loop experiments confirm that the new estimates align with observed stability boundaries.

What carries the argument

The unified energy function for nonlinear systems with PI controllers, which incorporates terms for the integral states and state-dependent damping to serve as a Lyapunov function bounding the region of attraction.

Load-bearing premise

The nonlinear swing equations with state-dependent damping for IBRs with PI controllers admit a valid energy function derivation that accurately captures stability without additional unstated assumptions or post-hoc adjustments.

What would settle it

Hardware-in-the-loop tests that apply disturbances at the boundary of the estimated region of attraction and observe whether the system returns to stable operation or loses synchronism; repeated mismatch between predicted and actual outcomes would falsify the function.

Figures

Figures reproduced from arXiv: 2604.23697 by Hsiao-Dong Chiang, Yang Wu, Yifan Zhang, Yitong Li.

**Figure 1.** Figure 1: Control block diagram of the nonlinear system. view at source ↗

**Figure 2.** Figure 2: Grid-connected inverter system and synchronization control loops. view at source ↗

**Figure 4.** Figure 4: ROA estimation for transient stability of a DC-voltage-controlled view at source ↗

**Figure 5.** Figure 5: HIL experimental platform view at source ↗

**Figure 7.** Figure 7: Experiment results of DC-voltage-controller view at source ↗

read the original abstract

The increasing penetration of inverter-based resources (IBRs) has fundamentally altered the transient stability characteristics of modern power systems. IBRs typically rely on proportional--integral (PI) controllers for synchronization and regulation, resulting in nonlinear swing equations that differ significantly from those of synchronous generators (SGs) and exhibit state-dependent damping. Consequently, although the classical energy function is often adopted in IBR analysis by analogy with SGs, it cannot be directly applied to IBRs with PI controller. A new energy function explicitly tailored to PI controller is proposed in this letter. It admits a unified form and can be applied to a class of nonlinear systems with PI controllers. Two representative cases are considered, including a grid-following (GFL) inverter and a DC-voltage-controlled grid-forming (GFM) inverter, demonstrating less conservative and more effective estimation of the region of attraction (ROA). All findings are verified through hardware-in-the-loop (HIL) experiments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a new energy function for PI-controlled IBRs that looks usable for GFL and GFM cases and claims tighter ROA bounds than the classical SG version, but the Lyapunov derivative step with state-dependent damping needs explicit checking.

read the letter

The core contribution is a single energy function form built for systems that include PI states and state-dependent damping terms in the swing equations. It is applied to a grid-following inverter and a DC-voltage-controlled grid-forming inverter, with the claim that the resulting ROA estimates are less conservative than those from the usual synchronous-generator energy function. Hardware-in-the-loop tests are reported to back the estimates. That is the actual new piece: an explicit construction rather than just reusing the SG function by analogy. The HIL verification is a plus because it moves beyond pure simulation for this class of models. The math appears to augment the usual potential energy with quadratic terms on the integral states and an integral of the power mismatch, which is a reasonable way to handle the extra PI dynamics. If the full derivative expansion is carried through and shown to leave only non-positive terms inside the estimated region, the construction holds. The soft spot is exactly there. State-dependent damping can change sign or drop to zero, so cancellation of all other terms is not automatic. The abstract and the stress-test note both leave open whether the paper supplies the complete dV/dt expansion for both cases or simply asserts the sign. Without that step visible, the ROA claim rests on an unverified assumption rather than a demonstrated property. The citation pattern looks standard for the field and does not appear circular. This is the kind of targeted methods paper that power-system stability groups working on inverter-dominated grids would want to see. It is not a broad theoretical advance, but it tackles a concrete modeling gap that is becoming common. A serious referee should look at it because the problem is real, the HIL data adds weight, and the only load-bearing question is whether the derivative proof is complete. I would send it for review with a request to expand the Lyapunov derivative section and show the cancellation explicitly for the variable-damping term.

Referee Report

1 major / 1 minor

Summary. The paper proposes a new unified energy function explicitly tailored to nonlinear swing equations arising in inverter-based resources (IBRs) with PI controllers. The function augments classical potential energy with quadratic terms in the PI integral states and an integral of active-power mismatch; it is claimed to apply to a class of such systems and is demonstrated on grid-following (GFL) and DC-voltage-controlled grid-forming (GFM) inverters, yielding less conservative region-of-attraction (ROA) estimates than the classical energy function. All claims are supported by hardware-in-the-loop (HIL) experiments.

Significance. If the proposed function is shown to be a valid Lyapunov function whose derivative is non-positive along closed-loop trajectories, the work would supply a practical, less conservative tool for transient-stability assessment in IBR-dominated grids. It directly addresses the mismatch between classical SG-derived energy functions and the state-dependent damping plus integral states that characterize modern inverter controls, and the HIL verification strengthens the practical relevance of the ROA estimates.

major comments (1)

[Derivation of the energy function and its time derivative (Sections III–IV)] The central claim that the constructed V is a Lyapunov function for the closed-loop system with state-dependent damping requires explicit verification that dV/dt cancels all but the (possibly sign-changing) damping terms inside the estimated ROA. The manuscript must therefore provide the full expansion of dV/dt for both the GFL and GFM cases, including the contributions of the PI integral state and the state-dependent damping coefficient; without this expansion the ROA estimate rests on an unverified cancellation assumption.

minor comments (1)

[Abstract and Introduction] The abstract states that the function 'admits a unified form' for a class of nonlinear systems with PI controllers; a precise statement of the class (e.g., the required structure of the damping and interconnection terms) should appear in the introduction or the statement of the main theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The central concern regarding explicit verification of the time derivative of the proposed energy function is addressed below. We will revise the manuscript accordingly to strengthen the presentation of the Lyapunov analysis.

read point-by-point responses

Referee: [Derivation of the energy function and its time derivative (Sections III–IV)] The central claim that the constructed V is a Lyapunov function for the closed-loop system with state-dependent damping requires explicit verification that dV/dt cancels all but the (possibly sign-changing) damping terms inside the estimated ROA. The manuscript must therefore provide the full expansion of dV/dt for both the GFL and GFM cases, including the contributions of the PI integral state and the state-dependent damping coefficient; without this expansion the ROA estimate rests on an unverified cancellation assumption.

Authors: We agree that an explicit, term-by-term expansion of dV/dt is required for rigorous verification. In Sections III and IV the energy function is constructed so that the contributions from the PI integral states and the integral of active-power mismatch are designed to cancel exactly with corresponding terms arising from the closed-loop dynamics, leaving only the damping-related terms. However, the original manuscript presents this cancellation at a structural level rather than through the complete algebraic expansion requested. In the revised version we will add the full expansions for both the GFL and GFM cases, explicitly retaining the state-dependent damping coefficient and indicating the region inside which the remaining terms are non-positive. This will directly confirm that dV/dt is non-positive along trajectories within the estimated ROA. revision: yes

Circularity Check

0 steps flagged

No significant circularity; energy function derived from system equations without reduction to fitted inputs or self-citation chains.

full rationale

The paper constructs a new energy function explicitly for nonlinear swing equations of IBRs with PI controllers, including state-dependent damping and integral states. The abstract and description present this as a direct derivation tailored to GFL and GFM cases, yielding a unified form that estimates ROA more effectively than classical functions. No steps reduce by construction to inputs (e.g., no parameter fitted to data then renamed as prediction, no self-citation load-bearing the uniqueness or form of V, no ansatz smuggled via prior work). The derivation chain remains self-contained against the closed-loop dynamics, with verification via HIL experiments providing external grounding. This is the normal honest outcome for a proposal of a new Lyapunov candidate.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, axioms, or invented entities are detailed in the provided text.

pith-pipeline@v0.9.0 · 5476 in / 991 out tokens · 48224 ms · 2026-05-08T05:32:21.004205+00:00 · methodology

discussion (0)

Reference graph

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Nonlinear transient stability analysis of phased-locked loop-based grid-following voltage-source converters using lyapunov’s direct method,

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S. D’Arco and J. A. Suul, “Equivalence of virtual synchronous machines and frequency-droops for converter-based microgrids,”IEEE Transactions on Smart Grid, vol. 5, no. 1, pp. 394–395, 2013

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Y . Li, Y . Gu, and T. C. Green, “Revisiting grid-forming and grid-following inverters: A duality theory,”IEEE Transactions on Power Systems, vol. 37, no. 6, pp. 4541–4554, 2022

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Synchronizing stability analysis and region of attraction estimation of grid-feeding vscs using sum-of-squares programming,

C. Zhang, M. Molinas, Z. Li, and X. Cai, “Synchronizing stability analysis and region of attraction estimation of grid-feeding vscs using sum-of-squares programming,”Frontiers in Energy Research, vol. 8, p. 56, 2020

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An iterative equal area criterion for transient stability analysis of grid-tied converter systems with varying damping,

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The electronic realization of synchronous machines: Model matching, angle tracking, and energy shaping techniques,

C. Arghir and F. Dörfler, “The electronic realization of synchronous machines: Model matching, angle tracking, and energy shaping techniques,”IEEE Transac- tions on Power Electronics, vol. 35, no. 4, pp. 4398–4410, 2020

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An extension of grid-forming: A frequency-following voltage-forming inverter,

C. Ai, Y . Li, Z. Zhao, Y . Gu, and J. Liu, “An extension of grid-forming: A frequency-following voltage-forming inverter,”IEEE Transactions on Power Electronics, vol. 39, no. 10, pp. 12 118–12 123, 2024

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Transient stability of grid-forming converters with flexible dc-link voltage control,

L. Zhao, Z. Jin, and X. Wang, “Transient stability of grid-forming converters with flexible dc-link voltage control,” in2022 International Power Electronics Conference (IPEC-Himeji 2022- ECCE Asia), 2022, pp. 1648–1653

work page 2022

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Design-oriented analysis of dc-link voltage control for transient stability of grid-forming inverters,

C. Luo, T. Liu, X. Wang, and X. Ma, “Design-oriented analysis of dc-link voltage control for transient stability of grid-forming inverters,”IEEE Transactions on Industrial Electronics, vol. 71, no. 4, pp. 3698–3707, 2023

work page 2023

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Physical insight into hybrid-synchronization-controlled grid-forming inverters under large disturbances,

T. Liu and X. Wang, “Physical insight into hybrid-synchronization-controlled grid-forming inverters under large disturbances,”IEEE Transactions on Power Electronics, vol. 37, no. 10, pp. 11 475–11 480, 2022

work page 2022

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H. K. Khalil,Nonlinear Systems. USA: Prentice-Hall, 1996

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Large-signal stability analysis of grid-forming inverters with equivalent-circuit models,

N. Baeckeland, D. Pal, G.-S. Seo, B. Johnson, and S. Dhople, “Large-signal stability analysis of grid-forming inverters with equivalent-circuit models,”IEEE Transactions on Energy Conversion, 2025

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Transient stability analysis for paralleled system of virtual synchronous generators based on damping energy vi- sualization and approximation,

Q. Qu, X. Xiang, J. Lei, W. Li, and X. He, “Transient stability analysis for paralleled system of virtual synchronous generators based on damping energy vi- sualization and approximation,”IEEE Transactions on Power Electronics, vol. 39, no. 12, pp. 15 785–15 799, 2024

work page 2024