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

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.

💡 Research Summary

The paper addresses the growing challenge of guaranteeing small‑signal stability in power systems that are increasingly dominated by inverter‑based resources (IBRs), especially grid‑forming inverters (GFMs) employing P‑Q droop control. Traditional passivity‑based plug‑and‑play (PnP) stability certificates require each component to be strictly passive, a condition that is rarely satisfied by practical droop‑controlled converters, particularly at low frequencies where the droop introduces negative damping. To overcome this limitation, the authors introduce a dynamic, frequency‑dependent multiplier m(ω) that, when premultiplied to the admittance of every component, renders the Hermitian part of the product positive‑definite across the entire frequency range.

The theoretical foundation builds on a homotopy argument: the total admittance Y_tot(s)=Y_net(s)+Y_D(s) of the network plus devices is continuously transformed from a fictitious, provably stable reference (α=0) to the actual system (α=1). If a uniform multiplier M(ω)=blkdiag{m(ω),…,m(ω)} is independent of the homotopy parameter α and each device admittance varies linearly with α, then the positive‑definiteness condition needs to be verified only at the two endpoints. Since the reference network composed solely of passive lines is inherently stable, the certification reduces to checking Her{m(jω)Y_k(jω)}≻0 for every device k and Her{m(jω)Y_ℓ(jω)}≻0 for every line ℓ, for all ω>0.

The key contribution lies in the systematic synthesis of the multiplier. Rather than using a simple piecewise‑constant matrix that rotates the low‑frequency complex plane by 90°, the authors parametrize m(ω) as a fixed‑order linear state‑space filter: m(s)=C_m(sI−A_m)^{-1}B_m+D_m. The filter coefficients (A_m, B_m, C_m, D_m) are tuned using MATLAB’s systune toolbox with a PassivityGoal. For each device admittance Y_k(s), the condition I+mY_k must be minimum‑phase; passivity is then equivalent to the H∞ norm of the scattering matrix R_k(s)=(I−mY_k)(I+mY_k)^{-1} being ≤ 1 over all frequencies. The optimization seeks to minimize the worst‑case H∞ norm across all devices simultaneously, thereby ensuring a common multiplier works for the entire system without altering any inverter firmware.

Numerical validation is performed on a detailed IEEE 39‑bus test system populated with realistic GFM models that include multiple cascaded control loops (inner current, outer voltage, and droop). Various droop‑gain settings are examined. The dynamic multiplier significantly enlarges the certified stability region compared to the earlier piecewise‑constant approach, allowing many droop configurations that would otherwise violate strict passivity to be certified as stable. The method also accommodates the full 2×2 dq‑frame MIMO admittance representation, demonstrating applicability to complex multi‑input‑multi‑output converter dynamics.

In summary, the paper delivers three major advances: (1) a rigorous, frequency‑dependent multiplier framework that transforms non‑passive converter admittances into passive equivalents; (2) a plug‑and‑play stability certificate that requires only local admittance data and a common multiplier, eliminating the need for global system models or exhaustive topology enumeration; and (3) a practical synthesis algorithm that leverages modern optimization tools to produce low‑order filters, enabling scalable stability assessment for large, converter‑dominated grids. This work paves the way for decentralized, model‑agnostic stability verification in future renewable‑rich power systems.