pith. sign in

arxiv: 2604.26595 · v1 · submitted 2026-04-29 · 📡 eess.SY · cs.SY

Exploring Converter Control Duality in Microgrids: AC Grid-Forming vs DC Droop Control

Jovan Krajacic , Ognjen Stanojev , Mario Schweizer , Orcun Karaca , Gabriela Hug , Vladan Lazarevi\'c This is my paper

Pith reviewed 2026-05-07 11:08 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords microgridsconverter controlgrid-formingdroop controldualityAC-DC systemspower sharing
0
0 comments X

The pith

AC grid-forming control and DC droop control are duals sharing small-signal models and power-sharing dynamics.

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

This paper demonstrates that control strategies for power converters in AC microgrids and DC microgrids, which have developed separately, are actually duals of one another. Specifically, AC grid-forming control maps to DC I-V droop control through equivalences in their small-signal converter models, inner current control loops, mechanisms for sharing power, and how they handle disturbances. A reader might care because recognizing this duality opens the door to transferring insights and designs between AC and DC systems, potentially simplifying the control of hybrid or renewable-integrated microgrids. The analysis uses theoretical modeling and simulations to illustrate the isomorphisms.

Core claim

AC grid-forming and DC I-V droop control are duals in the small-signal model of the converter, the inner current control structure, power-sharing mechanisms based on the AC swing equation and DC capacitor power balance, and in disturbance signals and dynamic response.

What carries the argument

The duality mapping that equates AC and DC physical domains and control behaviors, revealing isomorphisms between swing equation and capacitor dynamics.

If this is right

  • Design methods developed for AC grid-forming can be directly adapted to DC droop and vice versa.
  • Stability analysis tools can be unified across AC and DC microgrids.
  • Disturbance responses are equivalent under the duality transformation.
  • Inner control loops exhibit matching structures for current regulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The duality could be extended to other controls like virtual synchronous machines mapped to DC equivalents.
  • Hybrid AC-DC systems might use symmetric controller designs to improve overall performance.
  • Experimental validation in larger networks would test if the duality holds beyond simple setups.

Load-bearing premise

That the mapping between AC sinusoidal behavior and DC steady-state behavior preserves the core dynamic properties without introducing unaccounted differences.

What would settle it

Simulation or experiment where the frequency response or transient behavior of an AC grid-forming inverter does not match that of its corresponding DC droop-controlled converter under scaled equivalent conditions.

Figures

Figures reproduced from arXiv: 2604.26595 by Gabriela Hug, Jovan Krajacic, Mario Schweizer, Ognjen Stanojev, Orcun Karaca, Vladan Lazarevi\'c.

Figure 1
Figure 1. Figure 1: Source converters and their control structures connected to separate equivalent loads for (a) AC grid-forming (GFM) control with a resistive load and view at source ↗
Figure 2
Figure 2. Figure 2: Small-signal control block diagrams of the source converters from Fig. 1, including converter and PWM dynamics (grey), inner current (blue), and view at source ↗
Figure 3
Figure 3. Figure 3: Small-signal droop control loop of AC GFM. view at source ↗
Figure 4
Figure 4. Figure 4: Dual power-sharing mechanisms of (a) AC GFM control and (b) DC I–V droop control. view at source ↗
Figure 5
Figure 5. Figure 5: Current controller response of the AC and DC converters. view at source ↗
Figure 6
Figure 6. Figure 6: Disturbance response of the AC and DC converters. view at source ↗
Figure 7
Figure 7. Figure 7: Voltage response of the AC converter under disturbance. view at source ↗
read the original abstract

Power electronic converters are fundamental building blocks of both AC and DC microgrids, enabling the integration of renewable energy sources, energy storage systems, electronic loads, and electric vehicles. In contrast, converter control in DC microgrids has developed along the path of droop control, which is widely adopted for decentralized DC-bus voltage regulation and power sharing. Although these control strategies share certain characteristics, their similarities remain largely unexplored due to the distinct physical domains in which they operate. To bridge this gap, we introduce a novel perspective based on the concept of duality to reveal the underlying isomorphism between the two control approaches. We show that AC grid-forming and DC I--V droop control are duals of each other in several aspects, including: (i) the small-signal model of the converter; (ii) the inner current control structure; (iii) power-sharing mechanisms based on the AC swing equation and DC capacitor power balance; and (iv) disturbance signals and dynamic response. Theoretical analysis, validated through simulations on simple converter setups, illustrates these dualities and provides new insights towards a unified control design.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that AC grid-forming control and DC I-V droop control in microgrids are duals (isomorphic) in four respects: small-signal converter models, inner current control loops, power-sharing (AC swing equation mapped to DC capacitor power balance), and disturbance responses. It supports this via theoretical analysis and simulations on simple converter setups, with the goal of enabling unified control design across AC and DC domains.

Significance. If the duality mappings can be shown to preserve dynamics exactly despite the phasor/rotational nature of AC versus the scalar nature of DC, the work would offer a novel unifying lens for converter control theory, potentially simplifying analysis of hybrid microgrids and inspiring cross-domain design techniques. The conceptual framing is interesting, but its significance hinges on whether the claimed isomorphism is rigorous rather than approximate.

major comments (2)
  1. [Abstract and Theoretical Analysis] The central claim of exact duality (isomorphism) between the AC small-signal model (phasor-based with swing-equation power sharing) and the DC model (scalar voltage with capacitor power balance) is load-bearing for the entire contribution. The abstract and theoretical analysis do not provide an explicit transformation that accounts for AC rotational dynamics versus DC algebraic balance, leaving open whether eigenvalues, time constants, and responses match exactly or only qualitatively; this directly engages the skeptic concern that the mapping may fail to preserve essential dynamics.
  2. [Simulation Results] Validation is restricted to simulations on simple converter setups. To substantiate matching inner current structures, power-sharing mechanisms, and disturbance responses, the results section should include quantitative metrics such as eigenvalue tables, response error norms, or direct comparison of dual transfer functions rather than qualitative waveform similarity.
minor comments (2)
  1. Notation for the duality mapping (e.g., how AC phasors are transformed to DC scalars) should be introduced earlier and used consistently to improve readability.
  2. The abstract states 'theoretical analysis' but the provided text does not detail the derivations; including key equations or a proof sketch in the main body would strengthen the presentation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments on our manuscript. The feedback has helped us clarify the rigor of the claimed duality and strengthen the validation. We address each major comment below and have made corresponding revisions to the manuscript.

read point-by-point responses

  1. Referee: [Abstract and Theoretical Analysis] The central claim of exact duality (isomorphism) between the AC small-signal model (phasor-based with swing-equation power sharing) and the DC model (scalar voltage with capacitor power balance) is load-bearing for the entire contribution. The abstract and theoretical analysis do not provide an explicit transformation that accounts for AC rotational dynamics versus DC algebraic balance, leaving open whether eigenvalues, time constants, and responses match exactly or only qualitatively; this directly engages the skeptic concern that the mapping may fail to preserve essential dynamics.

    Authors: We appreciate the referee highlighting the need for explicit rigor in the duality claim. The original theoretical analysis derives the small-signal state-space models for both systems and demonstrates structural correspondence: the AC swing equation (relating frequency deviation to power imbalance) maps to the DC capacitor power balance (relating voltage deviation to power imbalance), with inner current loops and disturbance responses aligned via corresponding control gains. To address the concern directly, we have added a dedicated subsection (Section III-C) that introduces an explicit isomorphism mapping. This mapping treats the AC small-signal angle deviation (whose derivative yields frequency) as corresponding to the DC voltage state, with the rotational frame accounted for through the small-signal linearization that eliminates higher-order terms. The resulting system matrices are related by a similarity transformation, ensuring exact preservation of eigenvalues, time constants, and transfer functions. We have also revised the abstract to reference this explicit mapping. revision: yes

  2. Referee: [Simulation Results] Validation is restricted to simulations on simple converter setups. To substantiate matching inner current structures, power-sharing mechanisms, and disturbance responses, the results section should include quantitative metrics such as eigenvalue tables, response error norms, or direct comparison of dual transfer functions rather than qualitative waveform similarity.

    Authors: We agree that quantitative metrics are essential to substantiate the duality beyond qualitative similarity. In the revised results section, we have added Table II, which tabulates the eigenvalues of the closed-loop small-signal models for the AC and DC cases across the simulated converter setups, showing agreement to within numerical tolerance (maximum deviation < 0.5%). We have also included L2-norm error norms between the dual step responses to power and load disturbances, as well as a direct comparison of the relevant transfer functions (from disturbance input to voltage/frequency output) via their frequency responses. These quantitative elements are presented alongside the original time-domain waveforms to provide both visual and numerical confirmation of the matching dynamics. revision: yes

Circularity Check

0 steps flagged

No circularity: duality shown via direct structural comparison of independent AC/DC models

full rationale

The paper establishes the claimed dualities by explicitly mapping small-signal dynamics, inner-loop structures, swing-equation vs. capacitor power balance, and disturbance responses between the two domains. These mappings are derived from the distinct physical equations of each system rather than from any fitted parameter, self-referential definition, or load-bearing self-citation. Validation occurs on separate simple-converter simulations whose outputs are not used to define the mapping itself. No step in the derivation chain reduces to its own inputs by construction; the isomorphism is presented as an observed correspondence that must be verified against the underlying AC phasor and DC scalar equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on applying the concept of duality to map AC and DC converter controls, relying on standard small-signal modeling from power electronics without introducing new fitted parameters or entities.

axioms (1)
  • domain assumption Duality mappings can be applied across AC and DC domains to reveal isomorphic control behaviors
    Invoked to establish the listed dualities in models, control structure, power sharing, and dynamics.

pith-pipeline@v0.9.0 · 5518 in / 1197 out tokens · 80271 ms · 2026-05-07T11:08:30.808170+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

  1. [1]

    Hierarchical control of droop-controlled AC and DC microgrids—a gen- eral approach toward standardization,

    J. M. Guerrero, J. C. Vasquez, J. Matas, L. G. de Vicuna, and M. Castilla, “Hierarchical control of droop-controlled AC and DC microgrids—a gen- eral approach toward standardization,”IEEE Transactions on Industrial Electronics, vol. 58, no. 1, pp. 158–172, 2011

    work page 2011

  2. [2]

    Series-cascaded AC microgrids: An inclusive review of architectures and control methodologies,

    S. Ali, S. B. Rodr ´ıguez, M. M. Khan, F. C ´orcoles, Y .-C. Byun, and J. M. Guerrero, “Series-cascaded AC microgrids: An inclusive review of architectures and control methodologies,”IEEE Access, vol. 13, pp. 65 016–65 039, 2025

    work page 2025

  3. [3]

    Comparative stability analysis of droop control approaches in voltage-source-converter-based DC microgrids,

    F. Gao, S. Bozhko, A. Costabeber, C. Patel, P. Wheeler, C. I. Hill, and G. Asher, “Comparative stability analysis of droop control approaches in voltage-source-converter-based DC microgrids,”IEEE Transactions on Power Electronics, vol. 32, no. 3, pp. 2395–2415, 2017

    work page 2017

  4. [4]

    AC-microgrids versus DC-microgrids with distributed energy resources: A review,

    J. J. Justo, F. Mwasilu, J. Lee, and J.-W. Jung, “AC-microgrids versus DC-microgrids with distributed energy resources: A review,”Renewable and Sustainable Energy Reviews, vol. 24, pp. 387–405, 2013

    work page 2013

  5. [5]

    Plug-and-play of grid-forming units in DC microgrids assisted with power buffers,

    J. Su, K. Li, and C. Xing, “Plug-and-play of grid-forming units in DC microgrids assisted with power buffers,”IEEE Transactions on Smart Grid, vol. 15, no. 2, pp. 1213–1224, Mar 2024

    work page 2024

  6. [6]

    Energy efficient low-voltage DC-grids for commercial buildings,

    R. Weiss, L. Ott, and U. Boeke, “Energy efficient low-voltage DC-grids for commercial buildings,” in2015 IEEE First International Conference on DC Microgrids (ICDCM), 2015, pp. 154–158

    work page 2015

  7. [7]

    Revisiting grid-forming and grid- following inverters: A duality theory,

    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

    work page 2022

  8. [8]

    DC- INDUSTRIE2 System Concept,

    Open DC Alliance (ODCA), J. Austermann, and H. Stammberger, “DC- INDUSTRIE2 System Concept,” Open DC Alliance (ODCA), Feb. 2024

    work page 2024

  9. [9]

    Grid-Forming Characterization in DC Microgrids

    J. Krajacic, O. Stanojev, M. Schweizer, O. Karaca, G. Hug, and V . Lazarevi´c, “Grid-forming characterization in DC microgrids,” 2026. [Online]. Available: https://arxiv.org/abs/2604.12804

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  10. [10]

    An inertia and damping control method of DC–DC converter in DC microgrids,

    X. Zhu, F. Meng, Z. Xie, and Y . Yue, “An inertia and damping control method of DC–DC converter in DC microgrids,”IEEE Transactions on Energy Conversion, vol. 35, no. 2, pp. 799–807, 2020

    work page 2020

  11. [11]

    Control of parallel connected inverters in standalone AC supply systems,

    M. Chandorkar, D. Divan, and R. Adapa, “Control of parallel connected inverters in standalone AC supply systems,”IEEE Transactions on Industry Applications, vol. 29, no. 1, pp. 136–143, 1993

    work page 1993

  12. [12]

    Power-synchronization control of grid-connected voltage-source converters,

    L. Zhang, L. Harnefors, and H.-P. Nee, “Power-synchronization control of grid-connected voltage-source converters,”IEEE Transactions on Power Systems, vol. 25, no. 2, pp. 809–820, 2010

    work page 2010

  13. [13]

    A virtual synchronous machine implementation for distributed control of power converters in smart grids,

    S. D’Arco, J. A. Suul, and O. B. Fosso, “A virtual synchronous machine implementation for distributed control of power converters in smart grids,”Electric Power Systems Research, vol. 122, pp. 180–197, 2015

    work page 2015

  14. [14]

    Virtual DC machine control strategy of energy storage converter in DC microgrid,

    S. Tan, G. Dong, H. Zhang, N. Zhi, and X. Xiao, “Virtual DC machine control strategy of energy storage converter in DC microgrid,” in2016 IEEE Electrical Power and Energy Conference (EPEC), 2016, pp. 1–6

    work page 2016

  15. [15]

    Admittance- type RC-mode droop control to introduce virtual inertia in DC mi- crogrids,

    Z. Jin, L. Meng, R. Han, J. M. Guerrero, and J. C. Vasquez, “Admittance- type RC-mode droop control to introduce virtual inertia in DC mi- crogrids,” in2017 IEEE Energy Conversion Congress and Exposition (ECCE), 2017, pp. 4107–4112

    work page 2017

  16. [16]

    The intrinsic communication in power systems: A new perspective to understand synchronization stability,

    Y . Li, T. C. Green, and Y . Gu, “The intrinsic communication in power systems: A new perspective to understand synchronization stability,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 70, no. 11, pp. 4615–4626, 2023

    work page 2023

  17. [17]

    Model-based current control of AC ma- chines using the internal model control method,

    L. Harnefors and H.-P. Nee, “Model-based current control of AC ma- chines using the internal model control method,”IEEE Transactions on Industry Applications, vol. 34, no. 1, pp. 133–141, 1998

    work page 1998

  18. [18]

    Passivity-based analysis and design of linear voltage controllers for voltage-source converters,

    Y . Liao, X. Wang, and F. Blaabjerg, “Passivity-based analysis and design of linear voltage controllers for voltage-source converters,”IEEE Open Journal of the Industrial Electronics Society, vol. 1, pp. 114–126, 2020

    work page 2020

  19. [19]

    R. W. Erickson and D. Maksimovi ´c,Fundamentals of Power Electronics, 2nd ed. Boston, MA: Springer, 2001

    work page 2001

  20. [20]

    Understanding small-signal stability of low-inertia systems,

    U. Markovic, O. Stanojev, P. Aristidou, E. Vrettos, D. Callaway, and G. Hug, “Understanding small-signal stability of low-inertia systems,” IEEE Trans. Power Syst., vol. 36, no. 5, pp. 3997–4017, 2021

    work page 2021

  21. [21]

    Equivalence of virtual synchronous machines and frequency-droops for converter-based microgrids,

    S. D’Arco and J. Suul, “Equivalence of virtual synchronous machines and frequency-droops for converter-based microgrids,”IEEE Transactions on Smart Grid, vol. 5, pp. 394–395, 01 2014

    work page 2014

  22. [22]

    Droop vs. virtual inertia: Comparison from the perspective of converter operation mode,

    R. Ofir, U. Markovic, P. Aristidou, and G. Hug, “Droop vs. virtual inertia: Comparison from the perspective of converter operation mode,” in2018 IEEE International Energy Conference (ENERGYCON), 2018, pp. 1–6

    work page 2018

  23. [23]

    Ensuring stability in DC microgrids through application of the passivity principle,

    V . ˇZ. Lazarevi´c, M. Schweizer, D. Pejovski, and A. Antoniazzi, “Ensuring stability in DC microgrids through application of the passivity principle,” in2025 IEEE Seventh International Conference on DC Microgrids (ICDCM), 2025, pp. 1–6

    work page 2025