On the Impact of Operating Points on Small-Signal Stability: Decentralized Stability Sets via Scaled Relative Graphs

Abstract

This paper presents a decentralized frequency-domain framework to characterize the influence of the operating point on the small-signal stability of converter-dominated power systems. The approach builds on Scaled Relative Graph (SRG) analysis, extended here to address Linear Parameter-Varying (LPV) systems. By exploiting the affine dependence of converter admittances on their steady-state operating points, the centralized small-signal stability assessment of the grid is decomposed into decentralized, frequency-wise geometric tests. Each converter can independently evaluate its feasible stability region, expressed as a set of linear inequalities in its parameter space. The framework provides closed-form geometric characterizations applicable to both grid-following (GFL) and grid-forming (GFM) converters, and validation results confirm its effectiveness.

Figures (14)

• Figure 1: (a) $\operatorname{SRG}(Y_{\text{grid}}(\textup{j}\omega))$ for all $\omega\in[10^{-2},10^3]$ (b) $\operatorname{SRG}(\overline{Y}_{\text{grid}}(\textup{j}\omega))$ for all $\omega\in[10^{-2},10^3]$.
• Figure 2: Configuration of the multi-converter grid.
• Figure 3: Closed-loop dynamics of a converter-grid system
• Figure 4: (a) $\operatorname{SRG}(Y_{0}(s))$ for $\omega\in[10^{-2},10^3]$ Hz (b) $\operatorname{SRG}({Y_{1}}(s))$ in orange and $\operatorname{SRG}(\overline{Y_{1}}(s))$ in gray for $\omega\in\{0.1,0.21,0.46,1\}$ Hz (c) $\operatorname{SRG}({Y_{2}}(s))$ in orange and $\operatorname{SRG}(\overline{Y_{2}}(s))$ in gray for $\omega\in\{0.1,0.21,0.46,1\}$ Hz.
• Figure 5: (a) Centers of $\operatorname{SRG}(\overline{Y}_{1}(s))$ and $\operatorname{SRG}(\overline{Y}_{2}(s))$ (b) Half-widths of $\operatorname{SRG}(\overline{Y}_{1}(s))$ and $\operatorname{SRG}(\overline{Y}_{2}(s))$, for all $\omega\in[10^{-2},10^3]$.

Theorems & Definitions (16)

• Definition 1: Tight chord approximation
• Definition 2: Disk approximation
• Example 1
• Theorem 1
• Lemma 1
• proof
• Example 2: Application of Lemma \ref{['lemma:SRG_LPV_affine']}
• Example 3: Centers and Half-widths
• Theorem 2
• proof