Quantifying Grid-Forming Behavior: Bridging Device-Level Dynamics and System-Level Strength

TL;DR

The paper tackles the lack of precise metrics for grid-forming (GFM) behavior and its impact on system stability in converter-dominated grids. It introduces the Forming Index $FI(j\omega)=\bar{\sigma}[S_v(j\omega)]$ to quantify device-level 2D voltage-source behavior and defines system strength $\kappa(j\omega)$ along with grid strength $\alpha(j\omega)$ and bus strength $\kappa_i(\omega)$ to capture multi-bus voltage stiffness, establishing a formal link between device dynamics and system robustness. A small-signal framework proves that GFM converters enhance system strength, and demonstrates how FI and strength metrics can guide both GFM control design (via $\mathcal{H}_\infty$-based optimization) and optimal placement by targeting weak buses. Case studies on the IEEE 39-bus system show that GFM devices improve grid and system strength while GFL devices may degrade them, validating the proposed metrics as practical tools for design, placement, and stability assessment in modern power systems.

Abstract

Grid-forming (GFM) technology is widely regarded as a promising solution for future power systems dominated by power electronics. However, a precise method for quantifying GFM converter behavior and a universally accepted GFM definition remain elusive. Moreover, the impact of GFM on system stability is not precisely quantified, creating a significant disconnect between device and system levels. To address these gaps from a small-signal perspective, at the device level, we introduce a novel metric, the Forming Index (FI) to quantify a converter's response to grid voltage fluctuations. Rather than enumerating various control architectures, the FI provides a metric for the converter's GFM ability by quantifying its sensitivity to grid variations. At the system level, we propose a new quantitative measure of system strength that captures the multi-bus voltage stiffness, which quantifies the voltage and phase angle responses of multiple buses to current or power disturbances. We further extend and define this concept to grid strength and bus strength to identify weak areas within the system. Finally, we bridge the device and system levels by formally proving that GFM converters enhance system strength. Our proposed framework provides a unified benchmark for GFM converter design, optimal placement, and system stability assessment.

Figures (16)

• Figure 1: The diagram of a single converter connected to the grid.
• Figure 2: The common control strategies. (a) PLL-based converter with reactive control (PLL-PQ) or voltage control (PLL-PV). (b) VSG or droop control (droop). (c) PLL-based GFM control (PLL-GFM). (d) dVOC.
• Figure 3: The control diagram of a single converter system.
• Figure 4: The equivalent circuit of a single converter system
• Figure 5: The $FI$s of PLL-PQ. (a) $\omega_{\rm PLL}=10-50$Hz. (b) $L_{\rm g}=0.1-0.5$pu.

Theorems & Definitions (16)

• Definition 2.1: Forming Index at a given frequency
• Lemma 2.2: $FI$ represents the robust stability margin
• proof
• Example 1: PLL-PQ, and PLL-PV in Fig. \ref{['fig2:control']} (a)
• Example 2: VSG, and Droop in Fig. \ref{['fig2:control']} (b), PLL-GFM in Fig. \ref{['fig2:control']} (c), and dVOC in Fig. \ref{['fig2:control']} (d)
• Definition 3.1: System Strength
• remark 1
• Definition 3.2: Grid strength
• Proposition 3.3
• proof