Small-Signal Stability Manifolds in Converter-Dominated Power Systems

Abstract: This paper proposes a systematic framework to assess the small-signal stability of power systems with high shares of grid-following inverter-based resources (IBRs) under varying controller parameters and operating conditions. Stability manifolds are introduced to identify controller-parameter regions that ensure stability across multiple scenarios. Full-network linearization and eigenvalue analysis are combined with adaptive sampling based on probabilistic support vector machine classification to approximate stability boundaries efficiently, while surrogate optimization identifies feasible initial controller settings meeting bandwidth and phase-margin constraints. The approach is validated on a modified Cigré European HV network benchmark with 50 operating scenarios and increasing inverter penetration. Results show that stability sensitivity grows with inverter share, interactions among IBRs reshape admissible parameter regions, and simplified equivalent-network models may overlook critical system-level limitations. The framework supports stability-oriented controller design and interconnection studies in converter-dominated systems.

• The paper introduces a multi-scenario stability-manifold framework that combines full-network linearization, eigenvalue analysis, adaptive SVM sampling, and surrogate optimization to identify robust inverter-controller settings.
• The study finds that simultaneous deployment of grid-following inverters can sharply contract admissible PLL-gain regions—especially near bus 11—while Thévenin-equivalent models may overlook these system-level limitations.
• The results identify minimum DC-link proportional gains of approximately 1 pu for voltage control and 0.51 pu for squared-voltage control, while showing little difference between these formulations for PLL and current-control stability.

Motivation and scope

The paper addresses a gap in the small-signal stability assessment of power systems with high shares of grid-following inverter-based resources (IBRs). Most existing analyses rely on Thévenin-equivalent grid representations, which are well suited to single-point interconnection studies but cannot capture interactions among multiple IBRs, between IBRs and synchronous generators (SGs), or the influence of network topology and generation mix. Conversely, full-network linearization approaches are computationally expensive when repeated across many operating points and parameter settings. The authors propose to characterize stability robustness through stability manifolds: regions of the controller-parameter space within which small-signal stability holds simultaneously across multiple operating scenarios. The framework combines full-network linearization with eigenvalue analysis, an adaptive sampling method (ASM) based on probabilistic support vector machine (SVM) classification for efficient boundary identification, and surrogate optimization for initial controller tuning.

Modeling framework

The system model integrates conventional equipment and IBRs into a unified nonlinear differential–algebraic representation. SGs use a sixth-order Sauer–Pai machine model in the rotating $dq$ frame, augmented with a steam turbine-governor (5% droop) and an IEEE Type 1 automatic voltage regulator. Transmission lines, transformers, shunt elements, and loads retain explicit electromagnetic dynamics via $\pi$-equivalents and dynamic equivalent circuits, so that network fast dynamics are preserved at the linearization stage.

Grid-following IBRs are modeled as two-level three-phase voltage source inverters with an LC filter and dc link. The control architecture comprises five stages: $dq$ current control with feedforward decoupling, power-to-current reference generation, $P/\omega$ and $Q/v$ droop laws, PLL synchronization enforcing zero $q$-axis voltage at the point of common coupling, and dc-port voltage regulation implemented either on $v_{dc}$ or on the stored energy $v_{dc}^2$. The tunable parameters are therefore six PI gains (current control, PLL, dc voltage control) plus two droop gains, which are held fixed. A dynamic aggregation procedure extends prior work to map $N$ identical VSIs sharing a busbar onto a single equivalent unit, scaling physical impedances by $1/N$, capacitances and most control gains by $\pi$0, while keeping the PLL gain and voltage references unscaled.

Linearization tool

Each subsystem is linearized individually about a power-flow equilibrium in the global synchronous $\pi$1 frame, with input–output variables assigned per component (e.g., SGs and IBRs take currents as inputs and output voltages; RL branches behave conversely). Subsystems are interconnected through algebraic composition of their state-space matrices, respecting Kirchhoff current constraints, yielding the global state matrix $\pi$2 used for eigenvalue-based stability assessment. The code is released in a public repository, which supports reproducibility of the full pipeline from power flow to modal analysis.

Stability manifolds via adaptive sampling

Stability is defined over a set of $\pi$3 scenarios: a parameter vector $\pi$4 is labeled stable only if all $\pi$5 linearized models have no eigenvalues with nonnegative real parts. Since each label evaluation requires $\pi$6 eigenvalue computations of a large-scale matrix, exhaustive scanning is impractical. The ASM proceeds in four steps: uniform random initial sampling ($\pi$7 points), labeling via the stability function, training of a posterior-calibrated probabilistic SVM, and a refinement round in which $\pi$8 candidate points whose predicted probability is closest to a threshold $\pi$9 are added and the classifier retrained. The method is explicitly framed as a tradeoff between deterministic accuracy and computational cost; the resulting boundary is a probabilistic estimate rather than an exact manifold.

Initial tuning via surrogate optimization

To anchor the analysis, initial gains are obtained by minimizing the worst-case maximum real part of the system eigenvalues, $dq$0, evaluated over $dq$1 candidate IBR connection combinations and $dq$2 scenarios, subject to strict asymptotic stability ($dq$3) and to classical design rules: current-loop bandwidth at least ten times the PLL bandwidth, dc-voltage loop bandwidth below twice the nominal frequency (to reject the $dq$4 ripple), and phase margins above 45° for all loops. Because $dq$5 is non-analytic and non-smooth, gradient-based methods are inapplicable; the authors use MATLAB's derivative-free surrogateopt. Parameter regions satisfying only the classical design rules are termed regions of practical interest (RPIs), and subsequent manifold results are intersected with them, since points outside RPIs would not be adopted in practice regardless of their small-signal stability.

Case study and results

The framework is applied to a modified Cigré European HV benchmark (12 buses, 220/380 kV, 50 Hz) with four SGs at buses 9–12. Modifications include nonzero transformer resistance, transformer ratings matched to generator ratings, G10 derated to 350 MVA, and G9 modeled as a 620 MVA machine rather than an ideal source. Fifty scenarios are constructed from time-shifted, perturbed 24-hour load profiles (25 operating points) under two dispatch conditions: G9 at 620 MVA (Scenarios 1–25) and halved to 310 MVA (Scenarios 26–50), emulating loss of inertia and regulation capability. SGs G10–G12 are progressively replaced by aggregated IBRs built from a 5 MVA commercial converter (70 units for G10, 100 units for G11/G12), giving $dq$6 connection combinations. Sixteen case studies are analyzed across four sets: sets 1–3 vary one focus IBR's gains with others fixed; set 4 varies all installed IBRs' gains simultaneously. ASM parameters are $dq$7, $dq$8, $dq$9.

The principal findings are:

• PLL gains become critical at high penetration. For case study sets 1 and 3 (focus IBR at buses 10 and 12), stability manifolds coincide with or exceed the RPIs, and Thévenin-equivalent analyses are conservative but consistent. For set 2 (focus IBR at bus 11), however, manifolds shrink below the RPIs—dramatically so in case 2 I-I-I with all three IBRs installed—and the Thévenin-based analysis fails to detect this reduction entirely. In set 4, all configurations except 4 I-S-I show reduced manifolds, with the strongest contraction in 4 I-I-I. The sole configuration retaining a full manifold is the one where an SG occupies bus 11, identifying that bus as particularly critical. This constitutes the paper's strongest claim against simplified equivalents: a Thévenin model can substantially overestimate admissible PLL tuning ranges in multi-IBR systems.
• Current control gains yield manifolds close to the RPIs in sets 1–2, slightly smaller in sets 3–4 (implying a higher minimum proportional gain than the classical rules suggest); here the Thévenin approach is overly conservative.
• DC voltage control gains show little variation across case studies, but impose a minimum proportional gain absent from the RPIs: approximately 1 pu for $P/\omega$0 control and 0.51 pu for $P/\omega$1 control.
• Choice of dc-bus control variable ($P/\omega$2 vs. $P/\omega$3) has negligible effect on PLL and current-control manifolds.

Collectively, these results support the paper's central claims that stability sensitivity grows with inverter share, that IBR-to-IBR interactions reshape admissible parameter regions, and that equivalent-network models may overlook system-level limitations—with the important qualification that this evidence derives from a single 12-bus benchmark.

Limitations and open questions

Several limitations are acknowledged or implicit. The validation is confined to one modified Cigré benchmark with 50 synthetically generated scenarios based on assumed load profiles and dispatch rules rather than measured data; generalization to larger networks is untested, and the authors identify scalability (structure-exploiting linearization, eigenvalue screening, advanced sampling) as required future work. The stability boundary is probabilistic, dependent on the SVM calibration and threshold choice, and no formal error bound on the manifold estimate is provided. The study covers grid-following converters only; grid-forming controls, FACTS, HVDC (particularly MMC-based), and Type-3 wind turbines are excluded, as are EMT-level effects such as modulation-induced dynamics, since averaged models are used throughout. Droop gains are fixed, so manifold geometry under joint droop–gain variation remains unexplored. Whether the identified criticality of bus 11 reflects topology-specific resonance or a more general structural property is left open.

Conclusion

The paper introduces stability manifolds as a multi-scenario characterization of small-signal robustness in converter-dominated systems and delivers a practical identification methodology combining full-network linearization, calibrated SVM-based adaptive sampling, and surrogate-optimization tuning. Applied to a modified Cigré HV network, it demonstrates quantitatively that PLL tuning admissibility contracts sharply with simultaneous IBR deployment and that Thévenin-equivalent studies can miss such contractions, while providing concrete minimum-gain requirements for dc-link control. The framework offers transmission system operators and converter designers a systematic tool for controller tuning, interconnection screening, and grid-code parameter specification, contingent on the computational extensions needed for larger, heterogeneous systems.