Dynamic Passivity Multipliers for Plug-and-Play Stability Certificates of Converter-Dominated Grids

TL;DR

This work tackles small-signal stability in converter-dominated grids under uncertain parameters and topology by introducing a dynamic, frequency-dependent multiplier $m(s)$ that renders component admittances effectively passive. The method relies on a decentralised passivity certificate: for all $\omega>0$ and a homotopy parameter $\alpha\in[0,1]$, the Hermitian part $S(j\omega,\alpha)=\operatorname{Her}\big(m(j\omega)Y_{tot}(j\omega,\alpha)\big)\succ 0$, which can be verified locally when a uniform $m(s)$ is used. A linear-homotopy reduction allows endpoint verification, and a parametric, state-space multiplier is synthesized via MATLAB's systune under a PassivityGoal to certify stability for all devices with a single $m(s)$. Case studies on a two-bus system and the IEEE 39-bus system with random inverter placements demonstrate topology-independent stability certificates and an enlarged certified stability region, all without modifying inverter controllers. This approach offers scalable, plug-and-play stability certificates for mixed fleets of grid-forming converters and other grid components, enhancing operational flexibility in modern grids.

Abstract

Ensuring small-signal stability in power systems with a high share of inverter-based resources (IBRs) is hampered by two factors: (i) device and network parameters are often uncertain or completely unknown, and (ii) brute-force enumeration of all topologies is computationally intractable. These challenges motivate plug-and-play (PnP) certificates that verify stability locally yet hold globally. Passivity is an attractive property because it guarantees stability under feedback and network interconnections; however, strict passivity rarely holds for practical controllers such as Grid Forming Inverters (GFMs) employing P-Q droop. This paper extends the passivity condition by constructing a dynamic, frequency-dependent multiplier that enables PnP stability certification of each component based solely on its admittance, without requiring any modification to the controller design. The multiplier is parameterised as a linear filter whose coefficients are tuned under a passivity goal. Numerical results for practical droop gains confirm the PnP rules, substantially enlarging the certified stability region while preserving the decentralised, model-agnostic nature of passivity-based PnP tests.

Figures (8)

• Figure 1: Trajectory of the eigenvalues for two systems each consisting of two GFM inverters during a homotopy. Eigenvalues of one of the system move into the right-half plane (unstable region). Eigenvalues that cross the imaginary axis are marked with “$\circ$".
• Figure 2: Homotopic transformation from a passive reference system ($\alpha=0$) to the actual converter-based system ($\alpha=1$). With passive line admittances as references, only the endpoint at $\alpha=1$ needs verification for stability certification.
• Figure 3: Decentralised stability certification: a single multiplier $m(s)$ is tuned to make all device admittances $Y_k(s)$ and line admittances $Y_{ij}(s)$ passive when combined in series. Each component is certified independently with the same multiplier.
• Figure 4: GFM inverter with LC filter (a) and control architecture (b)
• Figure 5: Eigenvalues of $S_{inv}(j\omega)$ and $S_{line}(j\omega)$ for the tuned multiplier $m(j\omega)$.