---
title: Grid-Forming Mode-Based Allocation (GFM-BA)
url: https://www.emergentmind.com/topics/grid-forming-mode-based-allocation-gfm-ba
type: topic
---

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

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 [2209.10465, 2412.15446, 2307.15623, 2510.26953].

## 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 [2209.10465].

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 [2209.10465].

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 [2510.26953].

## 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 [2510.26953].

## 3. Analytical and Optimization-Based GFM-BA Methods

GFM-BA methodologies fall in two broad categories:

**Closed-form allocation rules:**  
The seminal result [2209.10465] 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 [2209.10465].

**Optimization-based algorithms:**  
Recent frameworks [2412.15446, 2307.15623] embed the GFM/GFL mode decision within network-constrained dispatch and dynamic stability optimization:
- In [2412.15446], 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 [2307.15623], 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) [2307.15623].
- 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 [2307.15623].

## 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 [2209.10465].
- 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 [2412.15446].
- 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 [2510.26953].

## 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) [2209.10465].
- 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 [2307.15623].

## 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 [2510.26953].

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 [2209.10465, 2412.15446, 2307.15623, 2510.26953].

Source: https://www.emergentmind.com/topics/grid-forming-mode-based-allocation-gfm-ba