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arxiv: 2411.12682 · v3 · submitted 2024-11-19 · 📡 eess.SY · cs.SY· math.OC

Distributed Coordination of Grid-Forming and Grid-Following Inverters for Optimal Frequency Control in Power Systems

Xiaoyang Wang , Xin Chen This is my paper

Pith reviewed 2026-05-23 08:21 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords distributed optimal controlfrequency restorationgrid-forming invertersgrid-following invertersinverter-based resourcesprimal-dual gradient methodpower system dynamics
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The pith

A distributed algorithm coordinates grid-forming and grid-following inverters to restore nominal frequency while minimizing total control cost and enforcing power limits.

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

This paper develops a control strategy for power systems with large shares of inverter-based renewable sources. The strategy coordinates grid-forming and grid-following inverters to return system frequency to its nominal value. Coordination occurs by minimizing aggregate control cost while respecting inverter capacity limits and line thermal limits. The updates rely only on local measurements plus neighbor-to-neighbor messages, and require no communication whatsoever when thermal limits are omitted. Global asymptotic convergence to the optimum is shown through Lyapunov analysis of the closed-loop dynamics.

Core claim

The paper establishes that a projected primal-dual gradient algorithm, constructed by exploiting the physical dynamics of the power network, provides a distributed solution to the optimal frequency control problem. This solution drives the system frequency to its nominal value, minimizes aggregate control cost, satisfies inverter capacity constraints and line thermal limits, and achieves global asymptotic convergence as proven by a Lyapunov function.

What carries the argument

Projected primal-dual gradient method applied to the structure of the power system dynamics, which produces neighbor-only communication updates.

If this is right

  • System frequency returns to its nominal value under the proposed updates.
  • Aggregate control cost reaches its minimum value subject to the active constraints.
  • Inverter power outputs remain inside their capacity limits at all times.
  • Line thermal limits are satisfied whenever those constraints are included in the formulation.
  • When thermal constraints are removed, the same optimality and frequency restoration are achieved with purely local measurements and zero communication.

Where Pith is reading between the lines

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

  • The same structural approach could allow frequency control to scale to transmission networks of arbitrary size without requiring a central coordinator.
  • Exploiting analogous network dynamics might permit similar distributed solutions for related tasks such as voltage regulation.
  • Hardware experiments that introduce realistic communication delays would test whether the proven convergence remains intact outside ideal conditions.

Load-bearing premise

The method assumes that the physical power system dynamics possess a structure that directly yields fully distributed primal-dual updates from local information alone.

What would settle it

Simulating the algorithm on the IEEE 39-bus test case and recording either a steady-state frequency offset from nominal or a total control cost exceeding the known optimum would show the claim does not hold.

Figures

Figures reproduced from arXiv: 2411.12682 by Xiaoyang Wang, Xin Chen.

Figure 2
Figure 2. Figure 2: The fully local IBRs control algorithm. 0, ψ∗ , P ∗ ,σ ∗ ,λ ∗ , µ ∗ ) in the equilibrium set of dynamics (11), which is an optimal solution of the saddle point problem (9), and (P r∗ M , P r∗ L , ω ∗ =0, ψ∗ , P ∗ ) is an optimal solution of the modified OFC problem (7). D. Fully Local Optimal IBR Control Algorithm When the line thermal constraints (6f) or (7h) are not considered, based on the distributed c… view at source ↗
Figure 5
Figure 5. Figure 5: IBRs power setpoint adjustments with OFC. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Power flow in line 3-18 under step power change. (Blue solid line: [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Time-varying power generation from the PV unit at bus 4. [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Frequency dynamics at IBR buses under continuous disturbance. [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
read the original abstract

The large-scale integration of inverter-interfaced renewable energy sources presents significant challenges to maintaining power balance and nominal frequency in modern power systems. This paper studies grid-level coordinated control of grid-forming (GFM) and grid-following (GFL) inverter-based resources (IBRs) for scalable and optimal frequency control. We propose a fully distributed optimal frequency control algorithm based on the projected primal-dual gradient method and by leveraging the structure of the underlying physical system dynamics. The proposed algorithm i) restores the nominal system frequency while minimizing total control cost and enforcing IBR power capacity limits and line thermal constraints, and ii) operates in a distributed manner that only needs local measurements and neighbor-to-neighbor communication. In particular, when the line thermal constraints are disregarded, the proposed algorithm admits a fully local implementation that requires no communication, while still ensuring optimality and satisfying IBR power capacity limits. We establish the global asymptotic convergence of the algorithm using Lyapunov stability analysis. The effectiveness and optimality of the proposed algorithms are validated through high-fidelity, 100% inverter-based electromagnetic transient (EMT) simulations on the IEEE 39-bus system.

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

0 major / 2 minor

Summary. The manuscript proposes a fully distributed optimal frequency control algorithm for power systems integrating grid-forming (GFM) and grid-following (GFL) inverter-based resources (IBRs). Derived from the projected primal-dual gradient method and leveraging the structure of the underlying physical dynamics, the algorithm restores nominal frequency, minimizes total control cost, and enforces IBR power capacity limits and line thermal constraints. It requires only local measurements and neighbor-to-neighbor communication (or no communication when thermal constraints are ignored), establishes global asymptotic convergence via Lyapunov stability analysis, and validates performance through high-fidelity EMT simulations on the IEEE 39-bus system.

Significance. If the Lyapunov argument and simulation results hold, the work offers a scalable, theoretically grounded approach to frequency control in 100% IBR systems. The exploitation of physical dynamics to enable both the distributed updates and the stability proof, together with the fully local implementation option, represents a practical advance over centralized or communication-heavy methods. The EMT validation on a standard test system adds credibility to the optimality and constraint-handling claims.

minor comments (2)
  1. [Abstract and Introduction] The abstract and introduction would benefit from a brief statement of the key structural assumption on the physical dynamics that enables both the fully distributed implementation and the Lyapunov construction; this would clarify the scope without requiring additional technical detail.
  2. [Simulation Results] In the simulation section, specify the exact EMT model parameters, solver settings, and any data exclusion criteria used for the IEEE 39-bus case to allow reproducibility of the reported frequency restoration and cost minimization results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the constructive and positive review of our manuscript on the distributed primal-dual algorithm for optimal frequency control in mixed GFM/GFL systems. The recommendation for minor revision is noted. However, the report lists no specific major comments under the MAJOR COMMENTS section, so we have no individual points requiring detailed rebuttal or revision at this stage. We remain available to address any additional feedback or minor clarifications that may arise.

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard methods to known dynamics

full rationale

The paper derives a distributed projected primal-dual algorithm by applying established optimization techniques to the physical power system dynamics, then proves global asymptotic convergence via Lyapunov analysis. No step reduces by construction to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation whose validity depends on the present work. The structural leverage of the dynamics is an external modeling input, not an internal tautology, and the claims remain independently verifiable against the stated assumptions and standard stability tools.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the central claim rests on unstated modeling assumptions about inverter dynamics and the validity of the Lyapunov construction.

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Works this paper leans on

37 extracted references · 37 canonical work pages

  1. [1]

    Robust capacity assessment of distributed generation in unbalanced distribution networks incorporating anm techniques,

    X. Chen, W. Wu, and B. Zhang, “Robust capacity assessment of distributed generation in unbalanced distribution networks incorporating anm techniques,” IEEE Trans. Sustain. Energy , vol. 9, no. 2, pp. 651– 663, 2017

    work page 2017

  2. [2]

    A Survey of Frequency and V oltage Control Ancillary Services—Part I: Technical Features,

    Y . G. Rebours, D. S. Kirschen, M. Trotignon, and S. Rossignol, “A Survey of Frequency and V oltage Control Ancillary Services—Part I: Technical Features,” IEEE Trans. Power Syst. , vol. 22, no. 1, pp. 350– 357, Jan. 2007

    work page 2007

  3. [3]

    Aggregate power flexibility in unbalanced distribution systems,

    X. Chen, E. Dall’Anese, C. Zhao, and N. Li, “Aggregate power flexibility in unbalanced distribution systems,” IEEE Trans. Smart Grid , vol. 11, no. 1, pp. 258–269, 2019

    work page 2019

  4. [4]

    The role of co-optimization in trading off cost and frequency regulation service for industrial microgrids,

    C. Lyu, W. Wang, J. Wang, Y . Bai, Z. Song, W. Wang, and J. Meng, “The role of co-optimization in trading off cost and frequency regulation service for industrial microgrids,” Appl. Energy , vol. 375, p. 124131, Dec. 2024

    work page 2024

  5. [5]

    Stochastic Model Predictive Control Based Fast-Slow Coordination Automatic Generation Control,

    Y . Shen, W. Wu, and S. Sun, “Stochastic Model Predictive Control Based Fast-Slow Coordination Automatic Generation Control,” IEEE Trans. Power Syst., vol. 39, no. 3, pp. 5259–5271, Oct. 2023

    work page 2023

  6. [6]

    Connecting Automatic Generation Control and Economic Dispatch From an Optimization View,

    N. Li, C. Zhao, and L. Chen, “Connecting Automatic Generation Control and Economic Dispatch From an Optimization View,” IEEE Trans. Control Network Syst. , vol. 3, no. 3, pp. 254–264, Jul. 2015

    work page 2015

  7. [7]

    Design and Stability of Load- Side Primary Frequency Control in Power Systems,

    C. Zhao, U. Topcu, N. Li, and S. Low, “Design and Stability of Load- Side Primary Frequency Control in Power Systems,”IEEE Trans. Autom. Control, vol. 59, no. 5, pp. 1177–1189, Jan. 2014

    work page 2014

  8. [8]

    Distributed Automatic Load Frequency Control With Optimality in Power Systems,

    X. Chen, C. Zhao, and N. Li, “Distributed Automatic Load Frequency Control With Optimality in Power Systems,” IEEE Trans. Control Network Syst., vol. 8, no. 1, pp. 307–318, Sep. 2020

    work page 2020

  9. [9]

    Optimal Load-Side Control for Frequency Regulation in Smart Grids,

    E. Mallada, C. Zhao, and S. Low, “Optimal Load-Side Control for Frequency Regulation in Smart Grids,” IEEE Trans. Autom. Control , vol. 62, no. 12, pp. 6294–6309, Jun. 2017

    work page 2017

  10. [10]

    Distributed Op- timal Frequency Control Considering a Nonlinear Network-Preserving Model,

    Z. Wang, F. Liu, J. Z. F. Pang, S. H. Low, and S. Mei, “Distributed Op- timal Frequency Control Considering a Nonlinear Network-Preserving Model,” IEEE Trans. Power Syst., vol. 34, no. 1, pp. 76–86, Aug. 2018

    work page 2018

  11. [11]

    Distributed optimal load frequency control considering nonsmooth cost functions,

    Z. Wang, F. Liu, C. Zhao, Z. Ma, and W. Wei, “Distributed optimal load frequency control considering nonsmooth cost functions,” Syst. Control Lett. , vol. 136, p. 104607, 2020. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0167691119302178

    work page 2020

  12. [12]

    An operator splitting scheme for distributed optimal load-side frequency control with nonsmooth cost functions,

    Y . Wang, S. Liu, X. Cao, and M.-Y . Chow, “An operator splitting scheme for distributed optimal load-side frequency control with nonsmooth cost functions,” IEEE Trans. Autom. Control, vol. 69, no. 9, pp. 6442–6449, 2024

    work page 2024

  13. [13]

    Source-Load Collaborative Frequency Control With Real-Time Optimality in Power Networks,

    C. Wu, Z. Wu, W. Gu, Z. Yi, C. Xi, and Z. Shi, “Source-Load Collaborative Frequency Control With Real-Time Optimality in Power Networks,” IEEE Trans. Smart Grid , vol. 16, no. 1, pp. 164–172, Sep. 2024

    work page 2024

  14. [14]

    Initialization-free distributed coordination for economic dispatch under varying loads and generator commitment,

    A. Cherukuri and J. Cort ´es, “Initialization-free distributed coordination for economic dispatch under varying loads and generator commitment,” Automatica, vol. 74, pp. 183–193, Dec. 2016

    work page 2016

  15. [15]

    Modeling of Grid-Forming and Grid-Following Inverters for Dynamic Simulation of Large-Scale Distribution Systems,

    W. Du, F. K. Tuffner, K. P. Schneider, R. H. Lasseter, J. Xie, and Z. Chen, “Modeling of Grid-Forming and Grid-Following Inverters for Dynamic Simulation of Large-Scale Distribution Systems,” IEEE Trans. Power Delivery, vol. 36, no. 4, pp. 2035–2045, Aug. 2020

    work page 2035

  16. [16]

    Modeling, Analysis and Testing of Autonomous Operation of an Inverter-Based Microgrid,

    N. Pogaku, M. Prodanovic, and T. C. Green, “Modeling, Analysis and Testing of Autonomous Operation of an Inverter-Based Microgrid,” IEEE Trans. Power Electron., vol. 22, no. 2, pp. 613–625, Mar. 2007

    work page 2007

  17. [17]

    A generic primary-control model for grid-forming inverters: Towards interoperable operation & control

    B. B. Johnson, T. Roberts, O. Ajala, A. D. Dom ´ınguez-Garc´ıa, S. V . Dhople, D. Ramasubramanian, A. Tuohy, D. Divan, and B. Kroposki, “A generic primary-control model for grid-forming inverters: Towards interoperable operation & control.” in HICSS, 2022, pp. 1–10

    work page 2022

  18. [18]

    Reinforcement learning for optimal primary frequency control: A lyapunov approach,

    W. Cui, Y . Jiang, and B. Zhang, “Reinforcement learning for optimal primary frequency control: A lyapunov approach,” IEEE Transactions on Power Systems, vol. 38, no. 2, pp. 1676–1688, 2023

    work page 2023

  19. [19]

    Distributed Cooperative AGC Method for New Power System With Heterogeneous Frequency Regula- tion Resources,

    Z. Li, Z. Cheng, J. Liang, and J. Si, “Distributed Cooperative AGC Method for New Power System With Heterogeneous Frequency Regula- tion Resources,” IEEE Trans. Power Syst., vol. 38, no. 5, pp. 4928–4939, Nov. 2022

    work page 2022

  20. [20]

    Distributed Optimization for Integrated Frequency Regulation and Economic Dispatch in Microgrids,

    Y . Xu, Z. Dong, Z. Li, Y . Liu, and Z. Ding, “Distributed Optimization for Integrated Frequency Regulation and Economic Dispatch in Microgrids,” IEEE Trans. Smart Grid , vol. 12, no. 6, pp. 4595–4606, Jun. 2021

    work page 2021

  21. [21]

    Stable reinforcement learning for optimal frequency control: A distributed averaging-based integral approach,

    Y . Jiang, W. Cui, B. Zhang, and J. Cort´es, “Stable reinforcement learning for optimal frequency control: A distributed averaging-based integral approach,” IEEE Open Journal of Control Systems , vol. 1, pp. 194–209, 2022. 11

    work page 2022

  22. [22]

    Initialization-free distributed algorithms for optimal resource allocation with feasibility constraints and application to economic dispatch of power systems,

    P. Yi, Y . Hong, and F. Liu, “Initialization-free distributed algorithms for optimal resource allocation with feasibility constraints and application to economic dispatch of power systems,”Automatica, vol. 74, pp. 259–269, Dec. 2016

    work page 2016

  23. [23]

    On the Stability of Globally Projected Dynami- cal Systems,

    Y . S. Xia and J. Wang, “On the Stability of Globally Projected Dynami- cal Systems,” J. Optim. Theory Appl. , vol. 106, no. 1, pp. 129–150, Jul. 2000

    work page 2000

  24. [24]

    Exponential stability of globally projected dynamic sys- tems,

    X.-B. Gao, “Exponential stability of globally projected dynamic sys- tems,” IEEE Trans. Neural Networks , vol. 14, no. 2, pp. 426–431, Mar. 2003

    work page 2003

  25. [25]

    Stability of primal–dual gradient dynamics and applications to network optimization,

    D. Feijer and F. Paganini, “Stability of primal–dual gradient dynamics and applications to network optimization,” Automatica, vol. 46, no. 12, pp. 1974–1981, Dec. 2010

    work page 1974

  26. [26]

    Asymptotic convergence of constrained primal–dual dynamics,

    A. Cherukuri, E. Mallada, and J. Cort ´es, “Asymptotic convergence of constrained primal–dual dynamics,” Systems Control Lett. , vol. 87, pp. 10–15, Jan. 2016

    work page 2016

  27. [27]

    On the exponential stability of primal-dual gradient dynamics,

    G. Qu and N. Li, “On the exponential stability of primal-dual gradient dynamics,” IEEE Control Systems Letters, vol. 3, no. 1, pp. 43–48, 2019

    work page 2019

  28. [28]

    Semi-Global Exponential Stability of Augmented Primal-Dual Gradient Dynamics for Constrained Convex Optimization,

    Y . Tang, G. Qu, and N. Li, “Semi-Global Exponential Stability of Augmented Primal-Dual Gradient Dynamics for Constrained Convex Optimization,” arXiv, Mar. 2019

    work page 2019

  29. [29]

    Model-Free Feedback Constrained Op- timization Via Projected Primal-Dual Zeroth-Order Dynamics,

    X. Chen, J. I. Poveda, and N. Li, “Model-Free Feedback Constrained Op- timization Via Projected Primal-Dual Zeroth-Order Dynamics,” arXiv, Jun. 2022

    work page 2022

  30. [30]

    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, Feb. 2021

    work page 2021

  31. [31]

    F. H. Clarke, Optimization and Nonsmooth Analysis , ser. Wiley- Interscience Series in Discrete Mathematics and Optimization. Philadel- phia: Society for Industrial and Applied Mathematics, 1990, originally published by Wiley, reprinted by SIAM

    work page 1990

  32. [32]

    Discontinuous dynamical systems,

    J. Cortes, “Discontinuous dynamical systems,” IEEE Control Syst. Mag., vol. 28, no. 3, pp. 36–73, May 2008

    work page 2008

  33. [33]

    An introduction to variational inequalities and their applications (d. kinderlehrer and g. stampacchia),

    L. C. Evans, “An introduction to variational inequalities and their applications (d. kinderlehrer and g. stampacchia),” SIAM Review, vol. 23, no. 4, pp. 539–543, 1981. [Online]. Available: https://doi.org/10.1137/1023111

    work page doi:10.1137/1023111 1981

  34. [34]

    Applied Nonlinear Control,

    J. Slotine and W. Li, “Applied Nonlinear Control,” 1991, [Online; accessed 15. Apr. 2025]. [Online]. Available: https:// www.semanticscholar.org/paper/Applied-Nonlinear-Control-Slotine-Li/ 1ae0d9625f9f580a3b8d8e92a0edbc2087a1cc0e

    work page 1991

  35. [35]

    H. K. Khalil and J. W. Grizzle, Nonlinear Systems , 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2002

    work page 2002

  36. [36]

    Ruszczynski, Nonlinear Optimization

    A. Ruszczynski, Nonlinear Optimization . Princeton University Press, 2011

    work page 2011

  37. [37]

    An Extension of Pseudolinear Functions and Variational Inequality Problems,

    M. Bianchi and S. Schaible, “An Extension of Pseudolinear Functions and Variational Inequality Problems,” J. Optim. Theory Appl. , vol. 104, no. 1, pp. 59–71, Jan. 2000

    work page 2000