How Many Grid-Forming Converters Do We Need? A Perspective From Small Signal Stability and Power Grid Strength

I. INTRODUCTION

• The motivation is that GFL converters may induce instability in weak, low-SCR grids, while large-scale converter integration is expected as a grid interface for renewable energy.The introduction also notes that GFM voltage-source behavior can support voltage and improve small-signal grid strength; its effect under large disturbances remains to be investigated.Source: §I · p. 1
• The paper studies GFM voltage-source behavior for small-signal dynamics, not its grid-strength effect under large disturbances.The introduction says current limitation makes GFM converters current sources under large disturbances and that their grid-strength enhancement in those circumstances remains to be investigated.Source: §I · p. 1
• Prior work linked GFM converters to increased generalized short-circuit ratio, but did not determine how to configure or size new GFM capacity across different control methods.The cited prior approach considered one GFM type, modeled it as an ideal voltage source, and focused on replacement locations.Source: §I · p. 2
• Only GFM converters provide effective voltage-source behavior for small-signal grid-strength enhancement, including across different GFM implementations.The paper contrasts this with GFL converters using constant AC voltage-magnitude control, which it says cannot behave as effective voltage sources.Source: §I · p. 2
• The paper links GFM voltage-source behavior to increased grid strength in the frequency range of PLL instabilities and improved overall small-signal stability.The analysis considers different GFM implementations.Source: §I · p. 2
• The authors derive a closed-form relationship between GFM-to-GFL capacity ratio and grid strength, then use it to address the GFM capacity needed for a prescribed stability margin.They describe this configuration question as previously unstudied.Source: §I · p. 2

II. VOLTAGE SOURCE BEHAVIORS OF GFL AND GFM CONVERTERS

• The section uses impedance models to characterize the voltage-source behavior of both GFL and GFM converters.This is the section’s stated analytical approach.Source: §II · p. 2

A. Control structures of GFL and GFM converters

• The analysis first uses VSMs without additional control methods as a prototypical GFM implementation, with other GFM methods treated elsewhere.The compared modes achieve the same steady-state equilibrium under the stated setting, while their transient dynamics differ.Source: A. Control structures of GFL and GFM… · pp. 2–3
• GFL and GFM share objectives of regulating active power, q-axis voltage, and d-axis voltage, but connect their control loops differently.In GFL, the q-axis voltage loop (PLL) generates frequency; in GFM, the active-power loop generates frequency.Source: Figure 1 · p. 3, A. Control structures of GFL and GFM… · p. 3
• The GFL structure uses a PLL for grid synchronization, an active-power loop for the d-axis current reference, and an AC-voltage loop for the q-axis current reference.The voltage loop regulates d-axis voltage to its magnitude reference.Source: A. Control structures of GFL and GFM… · p. 3
• The GFM structure derives frequency from active power and uses d- and q-axis voltage loops to set the current references.The synchronization law may use droop control or an emulated swing equation; a typical VSM choice uses virtual inertia and damping coefficients.Source: A. Control structures of GFL and GFM… · p. 3

B. Admittance model and voltage source behaviors

• The dq impedance model represents the grid-side inductor and the converter’s GFL- or GFM-controlled dynamics as 2 × 2 transfer-function matrices.Converter impedance describes the distance between the controlled capacitor voltage and an ideal voltage source behind the impedance.Source: B. Admittance model and voltage source… · p. 4
• The corresponding admittance models are the inverses of the GFL and GFM impedance matrices, and the models capture control-loop and filter dynamics.The difference between the two admittances follows from the different GFL and GFM control structures.Source: B. Admittance model and voltage source… · p. 4
• A large converter impedance indicates behavior farther from a voltage source, but judging whether the matrix is large is nontrivial.The section notes that impedance is a matrix rather than a scalar and defers the comparison to the following subsection.Source: B. Admittance model and voltage source… · p. 4

C. “Dimension” of voltage source behaviors

• The ideal voltage-control case is not implementable because practical control bandwidth is limited.The ideal case nevertheless reflects how control objectives affect the GFL admittance matrix.Source: C. “Dimension” of voltage source… · p. 5
• Only GFM provides effective voltage-source behavior: its 2D-VS holds across current directions, unlike GFL’s 1D-VS, which fails in one direction.The distinction follows from two infinite admittance elements for GFM versus one for GFL; the GFM result depends on sufficiently small YSwing(s).Source: C. “Dimension” of voltage source… · pp. 4–5
• A smaller largest impedance singular value indicates behavior closer to a voltage source, measuring the weakest direction of that behavior.The paper uses the largest singular value of the impedance matrix, equivalently the smallest singular value of the admittance matrix, for this comparison.Source: C. “Dimension” of voltage source… · p. 5
• Around 5–150 Hz, GFM’s largest impedance singular value is much smaller than GFL’s.The comparison is reported for the frequency range of interest; increasing voltage-control bandwidth and virtual inertia can reduce the GFM value further.Source: Figure 3 · p. 5, C. “Dimension” of voltage source… · p. 5
• Droop, power-synchronization, and VSM implementations all show relatively small singular values in the frequency range of interest, though their values differ.The differences indicate different equivalent impedances.Source: Figure 3 · p. 5, C. “Dimension” of voltage source… · p. 6
• The analysis assumes well-maintained DC voltage and enough DC-side energy storage for GFM active-power regulation through an emulated swing equation.Limited storage may affect equivalent impedance and voltage-source behavior.Source: C. “Dimension” of voltage source… · p. 6

III. POWER GRID STRENGTH AND STABILITY

• For a single GFL converter connected to an infinite bus, small-signal instability occurs when SCR falls below its converter-dependent critical SCR.Converter design can lower CSCR and improve tolerance of low-SCR conditions without changing the SCR calculation.Source: §III · p. 6
• In multi-converter systems, gSCR generalizes SCR and reflects grid strength from network parameters and device capacities.The section notes that characterizing grid strength becomes nontrivial in a multi-device system.Source: §III · p. 6
• The illustrative system connects multiple wind farms through an inductive network, with interior nodes eliminated by Kron reduction.The network also includes infinite buses representing large-capacity synchronous generators or other areas.Source: §III · p. 6, Figure 4 · p. 6

• The diagonal capacity matrix S_B scales each wind farm’s capacity by the global per-unit base capacity S_global.Source: §4 · p. 7
• A homogeneous multi-wind-farm system with GFL control is stable if and only if gSCR > CgSCR; larger gSCR means a larger stability margin.Source: §4 · p. 7
• gSCR reflects weighted network connectivity and extends single-converter SCR to multi-converter systems, using network parameters and converter capacities.It also simplifies large-system stability analysis by focusing on the network rather than directly calculating system eigenvalues.Source: §4 · p. 7

Scope

• The section focuses on small-signal stability associated with GFL converters and weak grids; GFM converters can have more complicated transients under large disturbances because of current limits.Source: §IV · p. 7

Modeling assumptions

• The analysis approximates GFM control as an ideal voltage source in series with an equivalent internal inductor, chosen to represent its impedance at the frequency of interest.A larger equivalent reactance means weaker voltage-source behavior, and different GFM methods can have different reactances.Source: §IV · pp. 7–8

Capacity ratio and grid strength

• GFM converters increase gSCR linearly with capacity ratio γ, with slope Ylocal; smaller Ylocal requires a larger γ for a given target gSCR.The equivalent internal inductance—and thus Ylocal—varies by GFM control method.Source: §IV · p. 8

Example 1

• γ ≥17.8% (Zlocal = 0.2 pu) fills the 220 kV gSCR gap from 1.25 to 2.14 for VSMs without reactive power droop control or power synchronization control.VSMs with reactive power droop control require a higher ratio because their equivalent internal inductance is larger.Source: §IV · pp. 8–9

Example 2

• γ ≥4.8% (Zlocal = 0.08 pu) or γ ≥7.4% (Zlocal = 0.12 pu) fills the 35 kV gSCR gap from 1.1 to 1.7 with added GFM converters.The larger threshold applies to VSMs with reactive power droop control; the smaller applies to VSMs without it or power synchronization control.Source: §IV · p. 9

Example 3

• γ ≥4.2% (Zlocal = 0.08 pu) or γ ≥6.1% (Zlocal = 0.12 pu) raises 35 kV gSCR from 1.1 to at least 1.7 when turbines switch from GFL to GFM.These are the requirements for VSMs without reactive power droop control or power synchronization control, and for VSMs with reactive power droop control, respectively.Source: §IV · p. 9

V. SIMULATION RESULTS

• γ = 17.8% or 21.4% produced satisfactory performance in Example 1, with larger GFM shares improving damping and reactive-power-droop VSMs requiring more capacity for similar damping.The tests applied a 5% voltage sag lasting 10 ms at the infinite bus.Source: §V · p. 10, Figure 9 · p. 10, Figure 10 · p. 10
• Replacing the infinite bus with an equivalent-capacity synchronous generator gave similar GFM requirements, but slightly more GFM capacity was needed for comparable damping.The test used a 10 ms short-circuit event.Source: §V · p. 10, Figure 11 · p. 10
• At a common 35 kV bus, γ = 4.8% of VSMs without reactive-power droop gave satisfactory performance, whereas replacing them with constant-voltage GFL converters made the system unstable.The GFL comparison used the same 4.8% capacity ratio and a 5% voltage sag lasting 1 ms.Source: §V · pp. 10–11, Figure 12 · p. 10, Figure 13 · p. 10
• Changing some wind turbines from GFL to GFM mode without new storage gave satisfactory performance at γ = 4.2%; at medium-voltage buses, less than 20% GFM capacity usually sufficed for the desired stability margin.The simulations also show that larger γ improves damping; Fig. 15 reports capacity ratios needed for prescribed equivalent gSCR values at the terminal 690 V bus.Source: §V · p. 11, Figure 14 · p. 11, Figure 15 · p. 11

VI. CONCLUSIONS

• Only GFM converters provide effective 2D voltage-source behavior and small-signal voltage support; modifying GFL converters with constant AC voltage magnitude control does not enhance grid strength.The paper derives a relationship between GFM capacity ratio and grid strength characterized by gSCR.Source: §VI · p. 11
• Installing GFM converters in a multi-converter system improves grid strength and overall small-signal stability, and not all converters need to operate in GFM mode.The paper reports that capacity ratios around 17.8% or 21.4% can significantly increase stability margin in Example 1, depending on the GFM method.Source: §VI · p. 11
• The paper identifies simultaneous consideration of frequency, transient, and small-signal stability when configuring GFM converters as future work.Source: §VI · p. 11
• The paper also raises whether operating a whole wind farm in GFM mode is preferable to implementing GFM control in energy-storage converters.Source: §VI · p. 11

APPENDIX B ADMITTANCE MODEL OF GFL CONVERTERS

• The GFL model starts from the PLL-based converter’s filter equations and control laws for active power, constant AC voltage, and current regulation.The current-control loop uses a PI regulator and a filtered voltage feed-forward signal.Source: Appendix B · p. 12
• Combining the filter and control equations yields a converter-side equivalent admittance in the controller’s rotating dq-frame, which is then transformed to a fixed-frequency global coordinate.The paper sets the global coordinate frequency to ω0 = 100π rad/s.Source: Appendix B · p. 12
• The global-coordinate model assumes aligned d-axes at steady state and approximately zero steady-state reactive current before deriving the simplified admittance and corresponding impedance.Source: Appendix B · pp. 12–13

APPENDIX C ADMITTANCE MODEL OF GFM CONVERTERS

• The derivation begins with the AC voltage-control law and combines it with filter equations to obtain the GFM converter admittance.The voltage reference is real and aligned with the d-axis.Source: Appendix C · p. 13
• The model accounts for the swing equation and assumes aligned steady-state d-axes with approximately zero steady-state reactive current.Source: Appendix C · p. 13
• The corresponding GFM impedance model is the inverse of the derived admittance.The stated main parameters include J = 20 and D = 500.Source: Appendix C · p. 13