arXiv subjects

Jakob Niehues

Publications and source records attributed to Jakob Niehues.

7 recordsLinked to original sources

arXiv2026

A graph theoretic view on small signal stability of inverter-based power grids

Dynamic grid stability is traditionally ensured with synchronous generators. Modern grids rely substantially more on inverter-based resources, which require grid-forming control to guarantee adequate system-wide synchronization and stability. Small-signal stability has granted various centralized and decentralized stability certificates - but these have primarily been limited to sufficient criteria only. In this work, we construct a necessary and sufficient small-signal stability criterion for lossless inverter-based power grids with arbitrary topology. We show that asymptotic stability is equivalent to the positive definiteness of a single matrix that combines network topology, operating point, and effective droop gains. We derive graph-theoretic stability criteria based on an augmented cone graph and show that the contribution of graph cycles is typically small, as illustrated for three IEEE test cases. The resulting framework yields decentralized stability criteria, quantifies the conservatism introduced by decentralization, and may support the development of future grid codes.

eess.SY↗

arXiv2026

A Unified Theory of Edge Weights: Stability of General Laplacian Networks from Matrix Phases and Asymmetry Rayleigh Ratios

We study the properties and stability of networks with arbitrary Laplacian coupling. Classic approaches to studying networked systems require unrealistic assumptions, including homogeneous node dynamics, one-dimensional and undirected edges, or constant edge weights. We develop a unified formulation of Laplacian-style couplings that drops these assumptions, providing a unified notion for the edge weights of adaptive, directed, and multi-dimensional edges. We show that the recently developed theory of matrix phases can capture essential stability properties of the network and its edges. We quantify the impact of the asymmetry of the higher-dimensional edge dynamics on the system's phase properties by introducing the Asymmetry Rayleigh Ratio. These theoretical advances allow us to derive new sufficient stability conditions for AC power grids, directed diffusion, and the Kuramoto-Sakaguchi model. The resulting conditions are less conservative than the specific results known for these systems.

nlin.AO↗

arXiv2025

Small-signal stability of power systems with voltage droop

The stability of inverter-dominated power grids remains an active area of research. This paper presents novel sufficient conditions for ensuring small-signal stability in lossless and constant $R/X$ grids with highly heterogeneous mixes of grid-forming inverters that implement an adapted $V$-$q$ droop control. The proposed conditions can be evaluated in the neighborhood of each bus without information on the rest of the grid. Apart from the presence of $V$-$q$ droop, no additional assumptions are made regarding the inverter control strategies, nor is dynamical homogeneity across the system assumed. The analysis is enabled by recasting the node dynamics in terms of complex frequency and power, resulting in transfer functions that directly capture the small-signal frequency and amplitude responses to active and reactive power imbalances. These transfer functions are directly aligned with typical design considerations in grid-forming control. Building on an adapted small-phase theorem and viewing the system as a closed feedback loop between nodes and lines, the derived stability conditions also yield new insights when applied to established inverter control designs. We demonstrate in simulations that our conditions are not overly conservative and can identify individual inverters that are misconfigured and cause instability.

math.OC↗

arXiv2025

Complex Phase Analysis of Power Grid Dynamics

With an increasing share of renewable energy sources, accurate and efficient modeling of grid-forming inverters is becoming crucial for system stability. Linear methods are a powerful tool for understanding dynamics close to an operating point, but usually depend on the reference trajectory. Thus, small deviations can render linear models invalid over time, posing a significant challenge in practice, and complicating theoretical analysis. As a solution, we show that the complex phase offers a robust formulation independent of reference phases and frequencies, thus preserving invariance properties under linearization. This enables robust system identification during realistic conditions and opens the road to powerful stability analysis of inverter-based grids.

eess.SY↗

arXiv2024

Phase and gain stability for adaptive dynamical networks

In adaptive dynamical networks, the dynamics of the nodes and the edges influence each other. We show that we can treat such systems as a closed feedback loop between edge and node dynamics. Using recent advances on the stability of feedback systems from control theory, we derive local, sufficient conditions for steady states of such systems to be linearly stable. These conditions are local in the sense that they are written entirely in terms of the (linearized) behavior of the edges and nodes. We apply these conditions to the Kuramoto model with inertia written in adaptive form, and the adaptive Kuramoto model. For the former we recover a classic result, for the latter we show that our sufficient conditions match necessary conditions where the latter are available, thus completely settling the question of linear stability in this setting. The method we introduce can be readily applied to a vast class of systems. It enables straightforward evaluation of stability in highly heterogeneous systems.

nlin.AO↗

arXiv2024

Resonant Solitary States in Complex Networks

Partially synchronized solitary states occur frequently when a synchronized system of networked oscillators is perturbed locally. Several asymptotic states of different frequencies can coexist at the same node. Here, we reveal the mechanism behind this multistability: additional solitary frequencies arise from the coupling between network modes and the solitary oscillator's frequency, leading to significant energy transfer. This can cause the solitary node's frequency to resonate with a Laplacian eigenvalue. We analyze which network structures enable this resonance and explain longstanding numerical observations. Another solitary state is characterized by the effective decoupling of the synchronized network and the solitary node at the natural frequency. Our framework unifies the description of solitary states near and far from resonance, allowing to predict the behavior of complex networks.

nlin.AO↗

arXiv2024

Self-organized quantization and oscillations on continuous fixed-energy sandpiles

Atmospheric self-organization and activator-inhibitor dynamics in biology provide examples of checkerboard-like spatio-temporal organization. We study a simple model for local activation-inhibition processes. Our model, first introduced in the context of atmospheric moisture dynamics, is a continuous-energy and non-Abelian version of the fixed-energy sandpile model. Each lattice site is populated by a non-negative real number, its energy. Upon each timestep all sites with energy exceeding a unit threshold re-distribute their energy at equal parts to their nearest neighbors. The limit cycle dynamics gives rise to a complex phase diagram in dependence on the mean energy $μ$: For low $μ$, all dynamics ceases after few re-distribution events. For large $μ$, the dynamics is well-described as a diffusion process, where the order parameter, spatial variance $σ$, is removed. States at intermediate $μ$ are dominated by checkerboard-like period-two phases which are however interspersed by much more complex phases of far longer periods. Phases are separated by discontinuous jumps in $σ$ or $\partial_μσ$ - akin to first and higher-order phase transitions. Overall, the energy landscape is dominated by few energy levels which occur as sharp spikes in the single-site density of states and are robust to noise.

cond-mat.stat-mech↗