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Spatial-Frequency Control Authority of Grid-Forming Inverters over Voltage Waveforms in Distribution Feeders

Amir H. Abolmasoumi, Hamed Mohsenian-Rad

arXiv:2610.28841v1eess.SYcs.SY

TL;DR

Waveform controllability depends on how GFM actions propagate with frequency and electrical distance, a relationship existing work has largely left unquantified. The paper derives authority and relative-dominance measures and tests them with EMT simulations. Its analysis and simulations show spatial decay, frequency-dependent behavior in less-inductive feeders, and local authority amplification near resonance.

Abstract

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This paper develops a spatial-frequency framework to quantify the impact of distributed energy resources with grid-forming inverters (GFMs) on voltage waveforms at different locations along a feeder. An analytical authority function is formulated to characterize how controllability varies with both electrical distance from the point of common coupling (PCC) and frequency. It is shown that the GFM control authority generally decays with both distance and frequency, reflecting the limited ability of inverter actions to influence voltage waveforms at distant points. To better capture controllability near resonance, a relative dominance metric is introduced to compare the influence of inverter-induced and grid-induced voltages at a given location and frequency. These results highlight that waveform controllability is fundamentally a spatial-frequency phenomenon along the feeder. Simulation studies validate the analytical findings by demonstrating both the spatial decay of control effectiveness along the feeder and the frequency-dependent trends in control authority in a realistic electromagnetic transient setting. The proposed framework provides insights into the placement and coordination of GFMs to enable effective waveform-level voltage control across electrical distance along the feeder.

I. INTRODUCTION

Fast electromagnetic dynamics challenge monitoring and control in converter-based grids because they span frequencies that conventional phasor models do not capture. The paper motivates waveform-level control with GFMs and identifies spatial distance and frequency as unresolved determinants of controllability.

I. INTRODUCTION

  • Fast electromagnetic phenomena, including sub-cycle oscillations, span frequencies that conventional phasor models cannot effectively capture.The paper notes these phenomena are becoming more common, especially in weak grids –.Source: §I · p. 1

A. Motivation: A Practical Challenge

  • Wide-area control relies on synchronized phasor measurements, which filter out high frequencies and limit its use for waveform-level disturbances.The passage describes WAC as effective for inter-area oscillation damping and secondary voltage control, but operating on electromechanical time scales –.Source: A. Motivation: A Practical Challenge · p. 1
  • GFMs can directly impose voltage waveforms across multiple time scales, offering a potential actuation mechanism for wide-area electromagnetic control.The paper links this potential to the fast inner-loop dynamics of GFMs –.Source: A. Motivation: A Practical Challenge · p. 1
  • The influence of a GFM at a target point varies with frequency and electrical distance, but this waveform-controllability dependence remains largely unexplored.The paper presents quantifying this dependence as a prerequisite for waveform-level control and GFM placement.Source: A. Motivation: A Practical Challenge · p. 1

B. Summary of Contributions

The paper addresses the lack of explicit analysis of how voltage-control actions propagate spatially. It develops a spatial-frequency framework, analyzes authority across filter cases, and uses EMT simulations to verify authority decay.

B. Summary of Contributions

  • A spatial-frequency framework quantifies GFM influence on voltage waveforms, with control authority varying by electrical distance and frequency.The contributions include analysis of L and LC filter cases, resonance in the LC case, and EMT verification of authority decay.Source: B. Summary of Contributions · p. 1

C. Related Literature

  • Prior work emphasizes stability, local behavior, or monitoring, leaving spatial propagation of voltage-control actions largely unquantified.Impedance-based methods and impedance shaping primarily address stability or PCC voltage stabilization [1]–,,.Source: C. Related Literature · p. 2
  • The framework targets the unexplored dependence of voltage waveform controllability on frequency and electrical distance along a feeder.The paper frames this as quantifying GFM control authority over voltage waveforms.Source: C. Related Literature · p. 2

II. GFM OPEN-LOOP CONTROL AUTHORITY ALONG THE FEEDER

This section derives the inverter’s open-loop influence on network voltage in the Laplace domain. The formulation is intended to quantify authority at any feeder distance and frequency.

II. GFM OPEN-LOOP CONTROL AUTHORITY ALONG THE FEEDER

  • The open-loop inverter influence is derived in the Laplace domain to produce a spatial- and frequency-dependent authority metric.The metric is intended to analyze limitations of electromagnetic control in power systems.Source: §II · p. 2
  • The modeled GFM voltage connects to the PCC through filter impedance, while grid voltage connects through feeder admittance.A shunt capacitor is connected at the PCC in the described system.Source: §II · p. 2
  • A pilot point is parameterized by fractional electrical distance x from the PCC, with x ranging from 0 to 1.Source: §II · p. 2

A. General Expression of Control Authority Function

The section uses nodal analysis and a reduced representation of local GFM control to express pilot-point voltage contributions. Local voltage and current loops reshape the effective contributions, while virtual impedance modifies the filter impedance.

A. General Expression of Control Authority Function

  • Nodal analysis is used to obtain the network expression, followed by solving for PCC voltage and reducing the local GFM control representation.Source: A. General Expression of Control… · p. 2
  • The resulting closed-loop pilot-point voltage is expressed as contributions from the reference voltage and grid voltage.The expression uses effective coefficients α_eff and β_eff.Source: A. General Expression of Control… · p. 2
  • Local voltage and current loops reshape the effective contributions, while virtual impedance further modifies them through Yf.The passage defines Dc = 1 + KvH + KiYf(1 − H) and H = Yf/Δ.Source: A. General Expression of Control… · p. 2

B. Spatial–Frequency Dependence of GFM Authority

Without shunt capacitance, GFM authority depends on electrical distance and the feeder/filter impedance division. Its frequency dependence varies with feeder inductiveness: sufficiently inductive feeders are nearly frequency independent in the middle band, while less inductive feeders lose authority with both frequency and distance.

B. Spatial–Frequency Dependence of GFM Authority

  • The authority function combines the spatial attenuation factor (1−x) with the electromagnetic interaction between feeder and inverter-filter impedances.Here x is electrical distance from the PCC; frequency dependence reflects the relative resistive and inductive terms.Source: B. Spatial–Frequency Dependence of GFM… · p. 2
  • For sufficiently inductive feeders, authority depends on distance and inductive impedance division, and is nearly frequency independent in the middle-frequency band.The passage identifies f ∈ [200, 1000] Hz as an example middle-frequency range when ωL ≫ R.Source: B. Spatial–Frequency Dependence of GFM… · p. 2
  • For less inductive feeders, particularly when L/R < Lf/Rf, authority decreases with both frequency and distance, potentially requiring more effort to control high-frequency oscillations at distant points.The contrast is with highly inductive lines, where the decay is mostly spatial.Source: B. Spatial–Frequency Dependence of GFM… · p. 2

C. Intermediate-Frequency Behavior with Shunt Capacitance

Shunt capacitance introduces resonant behavior that can reshape frequency-dependent authority without removing its spatial attenuation. A relative dominance ratio distinguishes inverter-induced from grid-induced voltage contributions, including their local behavior near resonance.

C. Intermediate-Frequency Behavior with Shunt Capacitance

  • For typical GFM filter and feeder parameters, the predicted resonance range is approximately 0.8–1.8 kHz, near the upper end of the 200 Hz–1 kHz waveform-control band.The example with Lf = 2 mH, L = 2 mH, and Cf = 30 µF gives fr ≈ 920 Hz.Source: C. Intermediate-Frequency Behavior… · p. 3
  • With shunt capacitance, resonance locally modifies the frequency dependence of |α| while preserving the spatial attenuation factor (1−x).The reactive cancellation near resonance reduces total PCC admittance and can produce a pre-resonant increase in |α|.Source: C. Intermediate-Frequency Behavior… · p. 3
  • Unlike |α|, Γ compares inverter-induced voltage directly with grid-induced voltage at the same location and frequency; |Γ| above 1 indicates inverter dominance.Figure 4 shows a pronounced peak near the resonant region.Source: C. Intermediate-Frequency Behavior… · p. 3, Figure 4 · p. 3
  • The Γ peak arises from resonance in the reduced admittance; for x > 0 its frequency increases as x decreases, while at x = 0 there is no resonance.As x decreases, the denominator approaches Yg, increasing the relative dominance of Yf.Source: C. Intermediate-Frequency Behavior… · p. 3
  • For a target frequency set Ω, AΓ(x) compares candidate GFM locations by their minimum |Γ| across that set; larger values indicate stronger authority over the range.The concept is stated to extend to combined authority for placement and coordination of multiple GFMs and pilot points.Source: C. Intermediate-Frequency Behavior… · p. 3

III. SIMULATION RESULTS

The simulation setup uses MATLAB/Simulink Simscape Electrical Specialized Power Systems to construct and test the circuit diagram shown in Fig. 2.

III. SIMULATION RESULTS

  • The circuit diagram in Fig. 2 is constructed and tested in MATLAB/Simulink Simscape Electrical Specialized Power Systems.The supplied passage describes the simulation platform and setup, but does not report simulation outcomes.Source: §III · p. 3

A. Spatial Decay of GFM Control in Disturbance-free Case

GFM voltage-control effectiveness depends on location: it is strongest at the PCC and decays toward the grid. Remote feedback also changes transient response, and greater effort does not necessarily improve remote tracking.

A. Spatial Decay of GFM Control in Disturbance-free Case

  • At x = 0, the controlled voltage source (CVS) achieves smaller tracking error than the uncontrolled voltage source (UCVS), but their errors become almost equal near the stiff grid.This demonstrates spatial decay of GFM authority regardless of how Vinv is generated.Source: A. Spatial Decay of GFM Control in… · p. 4
  • With feedback at x = 0.9, initial error is smaller because of grid dominance, but error decays more slowly; all locations reach small steady-state errors.At x = 0, the response is well damped; at x = 0.5, error magnitude and decay rate are moderate.Source: A. Spatial Decay of GFM Control in… · p. 4
  • As feedback moves farther from the PCC, inverter current increases, yet at x = 0.9 the greater current accompanies slower error decay.The authority function α(s, x) decreases with distance, reducing inverter influence relative to the grid.Source: A. Spatial Decay of GFM Control in… · p. 4

B. Empirical Authority Decay with Frequency

EMT simulations show that empirical control authority generally declines as frequency increases, with the trend depending on feeder impedance and electrical distance. The analysis uses synchronized Fourier projection to characterize the physical coupling.

B. Empirical Authority Decay with Frequency

  • The Control Authority Identification method is implemented in an EMT simulation to characterize physical coupling empirically.Complex phasors are extracted from time-domain waveforms using synchronized Fourier projection.Source: B. Empirical Authority Decay with… · p. 4
  • For constant x = 0.5, |α| generally decays as frequency increases; the decay is more pronounced on low-inductance lines and nearly constant on highly inductive lines.The results match the theoretical analysis, in which coupling reactance dominates control reach in inductive networks.Source: B. Empirical Authority Decay with… · p. 4, Figure 10 · p. 5
  • The empirical authority decay also depends on impedance, which varies with electrical distance between the PCC and pilot point.This yields a spatial-frequency dependency characterizing control authority.Source: Figure 10 · p. 5

IV. CONCLUSIONS AND FUTURE WORK

The framework characterizes GFM control authority across feeder distance and frequency, with authority generally declining over distance and, in low L/R feeders, frequency. Resonance can locally amplify authority, while EMT simulations validate the spatial-frequency analysis and point to challenges for extending it to multi-GFM networks.

IV. CONCLUSIONS AND FUTURE WORK

  • Control authority generally decays with distance and, for low L/R feeders, with frequency, while shunt capacitance can locally amplify authority through resonant dynamics.The relative dominance metric further characterizes controllability near resonance.Source: §IV · p. 5
  • EMT simulations confirm spatial decay of control effectiveness, increased control effort at distant points, and frequency-dependent authority trends.These results support treating waveform-level voltage controllability as a spatial-frequency phenomenon.Source: §IV · p. 5
  • The framework establishes a practical foundation for analyzing the reach and limitations of GFMs in waveform-level voltage control.The paper identifies addressing remaining challenges as essential for practical electromagnetic voltage control in IBR-dominated systems.Source: §IV · p. 5
  • Future work will extend the analysis to multi-GFM networks, multiple pilot points, and interacting feeders to characterize controllability across frequency and electrical distance.Challenges include scalable modeling, converter coordination, and communication and synchronization for remote feedback using WMUs.Source: §IV · p. 5
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