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Distributed Droop-Free Control of Grid-Forming Inverters with Dynamic Line and Load Modeling

Hiya Akhil Gada, Marija Ilic

arXiv:2609.07777v1eess.SYcs.SY

TL;DR

Existing droop-free approaches often use quasi-steady-state models that omit fast transients and assume decoupled power interactions. This paper proposes dynamic network modeling with distributed optimization for GFM inverters. In simulation, it improves transient performance in weak grids while preserving steady-state operating conditions.

  • Problem

    Existing droop-free methods rely on QSS network models that neglect fast electromagnetic transients and assume decoupled active- and reactive-power dependence.

    Source: §I · p. 1

  • Method

    The framework uses dynamic line and load models in a gradient-based distributed optimization update, with projection enforcing voltage-magnitude constraints.

    Source: §I · p. 2, C. Distributed gradient computation · p. 3, D. Projected gradient descent · p. 4

  • Results

    Approximately 10% reduction in the global objective function at peak demand compared to the QSS-based controller.

    Source: B. Study 2: Transient performance in a… · p. 5

  • Takeaways & Limitations

    Four-inverter microgrid simulations show improved transient performance under load changes, particularly in weak grids, while preserving the same steady-state operating conditions.

    Source: §V · p. 6

Abstract

from arXiv · show

Droop-free distributed control has emerged as a promising alternative to conventional linear droop control for coordinating inverter-based resources in AC microgrids. However, existing droop-free methods typically rely on quasi-steady state network models that neglect fast electromagnetic transients and assume a decoupled dependence of active and reactive power on frequency and voltage, respectively. In this paper, we propose a transient-aware droop-free distributed primary/secondary control framework for grid-forming (GFM) inverters. The control objective is to achieve proportional active and reactive power sharing, frequency regulation, and voltage regulation within prescribed bounds. The proposed approach incorporates dynamic models of network lines and loads into the control design, enabling accurate representation of transient behavior while capturing the coupled nature of power flow interactions. A gradient-based distributed control update is derived from an optimization formulation, in which voltage magnitude constraints are enforced through a projection operator. To validate the performance of the proposed control, real-time simulation studies are conducted on a four-inverter microgrid under load changes in both strong and weak grid conditions. The results demonstrate that the proposed method improves transient performance, particularly in weak grids, while preserving the same steady state operating conditions as existing QSS-based droop-free approaches.

I. INTRODUCTION

The introduction motivates droop-free control by limitations in droop and existing quasi-steady-state methods, then introduces a transient-aware framework for GFM inverters. The GFM control architecture uses cascaded voltage and current loops to track references.

I. INTRODUCTION

  • GFM inverters establish the voltage waveform and frequency, while GFL inverters synchronize to an externally imposed waveform and frequency.Primary control regulates sharing and stability; secondary control corrects steady-state voltage and frequency deviations.Source: §I · p. 1
  • Linear droop can mis-share power under nonlinear loads, introduce frequency and voltage errors as loads vary, and rely on inductive-network decoupling assumptions.Reactive-power sharing is also sensitive to line-impedance mismatches and unequal bus voltages.Source: §I · p. 1
  • Existing droop-free methods use QSS models that neglect fast electromagnetic transients and may miss coupled active- and reactive-power dynamics.Earlier approaches also regulate average voltage or relax reactive-power sharing to enforce voltage limits.Source: §I · pp. 1–2
  • The proposed framework models line and load currents dynamically and derives distributed control from optimization, with voltage limits enforced by projection.Its gradient incorporates current sensitivities to local voltage; the stated design uses local measurements and limited neighbor communication.Source: §I · p. 2

II. ZERO-LEVEL CONTROL OF GFM INVERTERS

  • The zero-level GFM architecture uses cascaded inner voltage and current loops in a dq frame to track references and control the grid-terminal voltage.Primary and secondary control provide voltage-magnitude and frequency references.Source: §II · p. 2

III. TRANSIENT-AWARE DROOP-FREE DISTRIBUTED CONTROL OF GFM INVERTERS

The communication model distinguishes electrical connections from directed information links and specifies the measurements exchanged among inverters.

A. Distributed communication model

  • The electrical network is an undirected graph whose nodes are inverter buses and whose edges connect electrically adjacent inverters.Each inverter’s electrical-neighbor set contains the directly connected units.Source: A. Distributed communication model · p. 2
  • The directed communication graph is assumed to contain a spanning tree so information can propagate across the network.Each inverter’s in-neighbors are the units from which it receives information.Source: A. Distributed communication model · p. 2
  • Inverters receive neighbors’ active and reactive power injections and dq-frame voltages, while other control quantities are available locally.The voltage signals v_di and v_qi are the references generated by the proposed controller.Source: A. Distributed communication model · pp. 2–3

B. Problem formulation

The formulation targets proportional power sharing, nominal-frequency regulation, and terminal voltages within prescribed bounds through an optimization problem.

B. Problem formulation

  • The control objective is proportional active and reactive power sharing, nominal-frequency regulation, and terminal-voltage regulation within prescribed bounds.The voltage and frequency targets are stated as ω◦ and prescribed voltage limits.Source: B. Problem formulation · p. 3
  • The control objective is posed as an optimization problem with a local objective function for each inverter.The supplied passages do not specify the objective’s full mathematical form.Source: B. Problem formulation · p. 3
  • The formulation refers to normalized active and reactive power injections and identifies P_i and Q_i as inverter rated capacities.Source: B. Problem formulation · p. 3

C. Distributed gradient computation

The controller computes power gradients using dynamic line and load models rather than quasi-steady-state algebraic approximations. Current sensitivities make these gradients usable with local measurements and limited neighbor communication.

C. Distributed gradient computation

  • Current sensitivities to local voltage are obtained from the line and load current dynamics, which are linear time-varying systems solved recursively.With constant line and load parameters, their steady-state solutions reduce to the algebraic expressions used in the QSS model.Source: C. Distributed gradient computation · p. 3
  • Inverter output current is represented as the sum of currents through neighboring lines and the local load current.Both current components are expressed in the dq frame.Source: C. Distributed gradient computation · p. 3
  • Line and load currents are modeled with differential equations rather than QSS algebraic relations, accounting for fast electromagnetic transients.The paper identifies those transients as significant in inverter-dominated microgrids.Source: C. Distributed gradient computation · p. 3
  • The power expressions and current sensitivities yield gradients for inverter i's and neighboring inverters' active and reactive power injections.The section obtains the sensitivities by differentiating the active and reactive power expressions with respect to local voltage.Source: C. Distributed gradient computation · pp. 3–4
  • The resulting gradients use local inverter quantities and communicated neighboring voltage and power measurements, preserving the distributed control structure.The required measurements include local voltage and current, line and load RL parameters, and neighbors' voltage and power measurements.Source: C. Distributed gradient computation · p. 4

D. Projected gradient descent

Each inverter updates its voltage reference using a projected gradient-descent step. Projection enforces the prescribed voltage-magnitude constraints, and the discrete-time update also admits a continuous-time limit.

D. Projected gradient descent

  • The controller substitutes the power gradients into the control law and applies projected gradient descent to update each inverter's voltage reference.The update is stated in discrete time.Source: D. Projected gradient descent · p. 4
  • The projection maps the updated voltage reference onto the feasible set, ensuring prescribed voltage-magnitude constraints are satisfied.The projection onto the feasible set has a closed-form expression.Source: D. Projected gradient descent · p. 4
  • The discrete-time voltage-reference update admits a continuous-time limit with projection onto the feasible set.Source: D. Projected gradient descent · p. 4

E. Steady state performance analysis

At steady state, the proposed control regulates frequency to nominal and voltage within prescribed bounds. It achieves proportional active and reactive power sharing when voltage constraints are inactive, and the closest feasible sharing otherwise.

E. Steady state performance analysis

  • When voltage constraints are inactive, normalized active and reactive power injections agree across neighboring inverters, and a communication-graph spanning tree extends this agreement network-wide.Source: E. Steady state performance analysis · p. 4
  • The steady-state frequency of every inverter equals the nominal angular frequency ω◦.Source: E. Steady state performance analysis · p. 4
  • At steady state, the controller achieves proportional active and reactive power sharing when voltage constraints are inactive, and the closest feasible sharing otherwise, while regulating frequency and voltage within prescribed bounds.The voltage-magnitude constraints are satisfied at all times.Source: E. Steady state performance analysis · p. 4

IV. RESULTS

The experiments use a radial microgrid with four GFM inverters, local loads, and a communication graph. The system starts from a feasible steady state satisfying equal power sharing and voltage constraints.

IV. RESULTS

  • The evaluation comprises two experiments on a radial microgrid with four GFM inverters and a communication graph.Each inverter supplies a local load.Source: §IV · p. 4
  • The inverter setup specifies RLC-filter parameters, unequal capacities for inverter 4, and objective weights βp = 10 and βq = 10.Tables I and II identify the line and load parameters, respectively.Source: §IV · p. 5, Table I · p. 5, Table II · p. 5

A. Study 1: Performance in a strong grid

In the strong-grid study, the controller restores nominal frequency and meets sharing and voltage objectives after the first load change. At the later disturbance, voltage limits bind and constrain equal power sharing.

A. Study 1: Performance in a strong grid

B. Study 2: Transient performance in a weak grid

In the weak-grid study, the transient-aware controller lowers the peak-demand global objective relative to a QSS-based controller while retaining comparable steady-state performance. The study uses relaxed voltage bounds to ensure optimization feasibility.

B. Study 2: Transient performance in a weak grid

  • Figure 3 presents normalized active and reactive power, voltage magnitude, and frequency responses under load changes at t = 2 s and t = 10 s.Source: Figure 3 · p. 5
  • The weak-grid test applies a step load increase at t = 0.2 s and uses relaxed bounds of 1.2V◦ and 0.7V◦ for feasibility.The gradient-descent step size is αi = 6.5 for all inverters.Source: B. Study 2: Transient performance in a… · p. 5
  • Approximately 10% reduction in the global objective function at peak demand is achieved by transient-aware control versus QSS-based control, with comparable steady-state performance.Source: B. Study 2: Transient performance in a… · p. 5

V. CONCLUSION

The conclusion presents a transient-aware distributed control framework for GFM inverters that models dynamic lines and loads and uses optimization-based voltage-reference updates. Four-inverter simulations show improved peak-demand transient performance versus QSS-based methods, with the same steady-state operating conditions.

V. CONCLUSION

  • The framework models line and load dynamics to represent electromagnetic transients without assuming decoupled active-power/frequency and reactive-power/voltage relationships.Source: §V · p. 5
  • Voltage references are updated through a projection operator that enforces magnitude constraints within a distributed optimization framework.The architecture uses local measurements and limited information exchange among neighboring units.Source: §V · p. 6
  • Four-inverter simulations show up to a 10% peak-demand improvement over QSS-based droop-free methods while preserving the same steady-state operating conditions.Source: §V · p. 6
  • Future work includes explicit line-current constraints, broader microgrid topologies, more flexible voltage-regulation and power-sharing trade-offs, and systems with both GFM and GFL inverters.Source: §V · p. 6
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