Grid-Forming Mode-Based Allocation (GFM-BA)

• Grid-Forming Mode-Based Allocation (GFM-BA) is a methodology that optimally assigns grid-forming and grid-following modes to inverters, balancing system stability and economic operation.
• It connects device-level control characteristics with system metrics like gSCR and Forming Index to ensure robust voltage regulation and improved small-signal stability.
• GFM-BA integrates into power system scheduling by dynamically allocating GFM capacities, resulting in enhanced grid strength, cost savings, and adherence to operational constraints.

Grid-Forming Mode-Based Allocation (GFM-BA) is a methodology for determining the optimal deployment of grid-forming (GFM) versus grid-following (GFL) control modes among inverter-based resources (IBRs) in modern electric power networks. GFM-BA methods explicitly link device-level control characteristics to system-level small-signal stability, grid strength, and economic operation, thus providing a systematic framework for converter mode allocation across time, topology, and operating scenarios (Xin et al., 2022, Chatterjee et al., 2024, Cui et al., 2023, Zhuang et al., 30 Oct 2025).

1. Theoretical Underpinnings and Device-Level Behaviors

At the core of GFM-BA is the distinction between GFM and GFL behavior:

• GFL converters synchronize via a phase-locked loop (PLL) and regulate output currents, behaving as a one-dimensional voltage source (1D-VS) where their admittance matrix in the local d–q frame becomes singular only in one direction (i.e., only one “infinite” admittance entry).
• GFM converters generate their own AC frequency reference (e.g., via virtual swing equation or VSM), imposing both frequency and voltage magnitude, resulting in a two-dimensional voltage source (2D-VS) property with an admittance matrix exhibiting “infinite” entries in both axes. Consequently, GFM units enforce stiffer terminal voltages with respect to network events (Xin et al., 2022).

The notion of "grid strength" in this context builds on the generalized short-circuit ratio (gSCR), a network-centric index:

$$\mathrm{gSCR} = \lambda_{\min}\left[S_B^{-1}B_r\right]$$

where $S_B$ is the diagonal matrix of per-unit converter capacities and $B_r$ is the Kron-reduced susceptance matrix of the network. System small-signal stability criterion is $\mathrm{gSCR} > \mathrm{CgSCR}$, with $\mathrm{CgSCR}$ reflecting the minimum critical value required for stability (Xin et al., 2022).

Device GFM ability is rigorously quantified by the Forming Index (FI), the maximum singular value of the closed-loop voltage sensitivity matrix $S_v(j\omega)$. A converter delivers effective grid-forming support if $\mathrm{FI}<1$ at frequencies of interest (Zhuang et al., 30 Oct 2025).

2. System-Level Metrics and Stability Indices

GFM-BA utilizes metrics that bridge device and network levels:

• System Strength ($\kappa(j\omega)$): Minimum singular value of the composite bus admittance, quantifying the voltage stiffness of the system under distributed current disturbances.
• Grid Strength ($\alpha(j\omega)$): Analogous to SCR, but generalized to multi-bus, frequency-dependent, singular-value–based measures.
• Bus Strength ($\kappa_i(\omega)$): Per-bus minimum response, identifying weak buses.

GFM converters (FI$<$1) provably increase both local ($\kappa_i$) and system-wide ($\kappa$) strength, directly translating to improved voltage regulation and enhanced small-signal stability margins. Explicit optimization objectives for converter placement and control design leverage these metrics—for control design, minimizing the $\mathcal{H}_\infty$-norm of the forming index over target bands; for placement, minimizing the closed-loop system impedance subject to capacity constraints (Zhuang et al., 30 Oct 2025).

3. Analytical and Optimization-Based GFM-BA Methods

GFM-BA methodologies fall in two broad categories:

Closed-form allocation rules:

The seminal result (Xin et al., 2022) shows that, for each site $i$ equipped with IBRs of capacity $S_i$, adding a GFM fraction $\gamma$ increases grid strength additively: $$\mathrm{gSCR}(\gamma) = \mathrm{gSCR}_0 + \gamma\,Y_\text{local}$$ This linear relationship enables direct sizing: to achieve a target small-signal damping, solve

$$\gamma \geq \frac{\mathrm{CgSCR} + \Delta\mathrm{gSCR}_\text{des} - \mathrm{gSCR}_0}{Y_\text{local}}$$

Simulation case studies indicate that $\gamma$ values between 5%–25% may suffice depending on voltage level and network configuration (Xin et al., 2022).

Optimization-based algorithms:

Recent frameworks (Chatterjee et al., 2024, Cui et al., 2023) embed the GFM/GFL mode decision within network-constrained dispatch and dynamic stability optimization:

• In (Chatterjee et al., 2024), the inverter droop gain vector $\mathbf{K_P}$ (frequency–power droop) is assigned to each IBR, with GFM allocation corresponding to large $K_{P,i}$, and GFL to near zero. The algorithm alternates Lyapunov-based small-signal stability constraints and droop gain box constraints to maximize transient energy dissipation.
• In (Cui et al., 2023), GFM-BA is incorporated into day-ahead unit commitment via mixed-integer second-order cone programming. Decision variables include GFM penetration levels $\alpha_{j,t}$ for each wind farm and trajectory of GFM headroom and reserves, subject to system stability, reserve, RoCoF, and frequency-nadir constraints.

4. Integration with Power System Operation and Economic Scheduling

GFM-BA directly integrates into system-level scheduling models:

• Dynamic allocation of GFM versus GFL modes allows the IBR fleet to adaptively satisfy grid stability and ancillary service requirements at minimum operational cost, accounting for the economic trade-off between wind curtailment (due to reserved headroom for GFMs) and thermal unit commitment (needed when GFM penetration is inadequate) (Cui et al., 2023).
• Mathematical constraints ensure that each GFM has sufficient active and reactive power headroom for synthetic inertia and fault ride-through, with these reserves expressed explicitly in the optimization and appearing as operational costs via curtailed wind penalties.
• Results from IEEE 30-bus studies report ~10% average cost savings and continuous enforcement of gSCR and frequency stability constraints with GFM-BA, compared to any static or heuristically fixed GFM allocation (Cui et al., 2023).

5. Case Studies and Quantitative Validation

Extensive time-domain and eigenvalue simulations substantiate the theoretical and optimization-based GFM-BA rules:

• Adding a modest share (5–25%) of GFM capacity—either as new converters or by retuning a subset of GFLs—substantially increases gSCR and the damping ratio of the least-stable system mode (Xin et al., 2022).
• In a three-bus system, optimal allocation of droop gains via unified controllers yields lower frequency and voltage deviations and higher damping ratio ($\approx 0.25$) compared to either pure GFM or pure GFL implementations (Chatterjee et al., 2024).
• Bus-strength mapping supports the prioritization of GFM converter placement at the weakest buses, as confirmed by improvements in system and bus strength metrics and observed voltage suppression during faults (Zhuang et al., 30 Oct 2025).

6. Practical Implementation Considerations

Best-practice recommendations for GFM-BA include:

• Compute Kron-reduced susceptance and baseline gSCR from existing network and device data.
• Select an appropriate $Y_\text{local}$ from the dominant GFM impedance frequency.
• Set a desired stability margin, then determine minimum GFM fraction accordingly.
• Deploy or retune inverters to meet or exceed the calculated GFM share at targeted weak nodes.
• Validate the final allocation with full-order dynamic simulation due to inherent model reduction limitations (e.g., gSCR neglects resistive elements, high-frequency resonances, or hardware saturation) (Xin et al., 2022).
• Software-defined control platforms may execute GFM-BA at sub-hourly intervals, but care is required to avoid rapid large-scale switching between modes and to maintain cyber-security (Cui et al., 2023).

7. Limitations, Extensions, and Open Directions

GFM-BA frameworks as currently formulated:

• Are restricted primarily to small-signal stability (device models assume infinite DC-side storage and do not directly address large-signal or transient instability).
• Often use surrogate or linearized indices to enforce grid-strength or frequency-constrained dispatch; their accuracy under high-penetration, rapidly reconfiguring IBR fleets may require periodic coefficient retraining and model validation.
• Extensions include robust or distributed optimization, uncertainty-aware design, and coordination with protection and grid-code requirements.
• The use of forming index (FI) and singular-value–based strength metrics are being further developed for H∞-based control design, multi-objective placement, and real-time monitoring (Zhuang et al., 30 Oct 2025).

GFM-BA thus enables explicit, system-optimal allocation of GFM capability, quantitatively linking device-level control, network strength, scheduling, and economic operation. It supports planners, operators, and automation in constructing stable, low-inertia, converter-dominated grids (Xin et al., 2022, Chatterjee et al., 2024, Cui et al., 2023, Zhuang et al., 30 Oct 2025).