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Frank Hellmann

Publications and source records attributed to Frank Hellmann.

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Graph Machine Learning: An Opportunity for Power Systems

Modern power systems face growing operational complexity driven by the integration of renewable energy sources, decentralization, and the need for real-time decision-making across a wide range of timescales. Addressing these challenges traditionally relies on model-based methods that, while accurate, can be too slow for operational demands. Machine learning (ML) has therefore emerged as a faster, data-driven alternative. As grid topology plays a central role in power system operation, graph machine learning (GML) methods offer a natural framework for incorporating topological dependencies as an inductive bias. We survey nearly 800 papers at the intersection of GML and power systems, covering forecasting, state estimation, optimization, control, fault diagnosis, and cybersecurity. Power systems constitute an unusually rich benchmark setting for GML, as they combine hard physical constraints, multi-scale dynamics, safety-critical requirements, and scarce labeled data within a single, well-defined domain. Conversely, power systems can benefit from utilizing GML to complement classical solvers, as GML provide scalable, topology-aware approximations with promising generalization and computational efficiency. We identify open challenges, including limited real-world deployment and the need for interpretable models in safety-critical settings. Despite the rapidly growing number of publications, standardized benchmarks and open datasets remain scarce, leaving many results difficult to reproduce and undermining the long-term scientific credibility of the field. We further derive a structured requirements catalog for ML-ready power grid benchmarks, intended to guide future dataset development and improve reproducibility across studies. We call on the community to prioritize dedicated benchmark studies and the release of open datasets and models.

cs.LG

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

Nodal Braess's Paradox and Inertia Destabilization with Dynamic Node and Line Failures in Power Grids

Large-scale power outages are typically caused by cascading failures. These unfold dynamically through complex interactions between network dynamics and individual component failures. In contrast, the study of cascading failures in physics has focused on analyzing line overloads in the quasi-static regime. We introduce a new model that integrates the dynamics of node and line failures with a paradigmatic oscillator model for power grid synchronization. This enables us to investigate the collective cascading behavior of coupled failures for the first time. We study the impact of nodal robustness, the ability of nodes to tolerate transient disturbances, and inertia, the ability of nodes to resist frequency deviations, on cascade sizes. We discover two novel mechanisms driving system fragility: i) While low inertia is widely considered a major challenge for power grids, we find that high inertia can amplify cascade sizes unless accompanied by appropriate adjustments of other dynamical properties. ii) Further, we find that an increase in the robustness of individual nodes can paradoxically lead to larger cascades. This latter phenomenon constitutes a novel type of Braess's paradox. Understanding such counterintuitive collective effects may become central for achieving resilient future power grids.

nlin.AO

Learning Dynamic Stability Landscapes in Synchronization Networks

The robustness of synchronization is typically characterized by scalar, per-node stability indices whose dependence on topology is studied via network science or graph neural networks (GNNs). We propose a novel upstream task, learning stability landscapes, which provide deeper insights into synchronization behavior and from which many such scalar indices can be derived. Crucially, we pioneer a graph-to-image prediction paradigm: learning image-like landscapes as per-node targets directly from graph topology, a formulation we are not aware of having been established elsewhere in the literature. To support this task, we release two datasets of 10,000 graphs each at 20 and 100 nodes with per-node landscape labels, based on a conceptual oscillator model, capturing power grid synchronization behavior. A GNN encodes topology and a CNN decoder renders per-node images, learned end-to-end with good in-distribution accuracy, generalizing across graph sizes and to realistic power grid topologies. This demonstrates that stability landscapes, while beyond the reach of conventional network science, are learnable from topology and open new avenues for moving beyond scalar stability indices in biology, neuroscience, and power grids.

cs.LG

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

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

Probabilistic Behavioral Aggregation: A Case Study on the Nordic Power Grid

This study applies the Probabilistic Behavioral Tuning (ProBeTune) framework to transient power grid simulations to address challenges posed by increasing grid complexity. ProBeTune offers a probabilistic approach to model aggregation, using a behavioral distance measure to quantify and minimize discrepancies between a full-scale system and a simplified model. We demonstrate the effectiveness of ProBeTune on the Nordic5 (N5) test case, a model representing the Nordic power grid with complex nodal dynamics and a high share of RESs. We substantially reduce the complexity of the dynamics by tuning the system to align with a reduced swing-equation model. We confirm the validity of the swing equation with tailored controllers and parameter distributions for capturing the essential dynamics of the Nordic region. This reduction could allow interconnected systems like the Central European power grid to treat the Nordic grid as a single dynamic actor, facilitating more manageable stability assessments. The findings lay the groundwork for future research on applying ProBeTune to microgrids and other complex sub-systems, aiming to enhance scalability and accuracy in power grid modeling amidst rising complexity.

eess.SY

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

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

Projected Neural Differential Equations for Learning Constrained Dynamics

Neural differential equations offer a powerful approach for learning dynamical systems from data. However, they do not inherently respect known constraints, such as conservation laws, that should be obeyed by the learned dynamics. It is well known that enforcing constraints in data-driven models can enhance their generalizability and numerical stability. In this paper, we introduce projected neural differential equations (PNDEs), a method for constraining neural differential equations based on projection of the predicted velocities onto the tangent space of the manifold that fulfills the constraint. In tests on several examples from different fields, including chaotic dynamical systems and power grid models, PNDEs outperform existing methods for constraining learned dynamics, require fewer hyperparameters, and are computationally more efficient. The proposed approach demonstrates potential for enhancing the modeling of constrained dynamical systems, particularly in domains where accuracy and reliability are essential.

cs.LG

Complex-Phase, Data-Driven Identification of Grid-Forming Inverter Dynamics

The increasing integration of renewable energy sources (RESs) into power systems requires the deployment of grid-forming inverters to ensure a stable operation. Accurate modeling of these devices is necessary. In this paper, a system identification approach to obtain low-dimensional models of grid-forming inverters is presented. The proposed approach is based on a Hammerstein-Wiener parametrization of the normal-form model. The normal-form is a gray-box model that utilizes complex frequency and phase to capture non-linear inverter dynamics. The model is validated on two well-known control strategies: droop-control and dispatchable virtual oscillators. Simulations and hardware-in-the-loop experiments demonstrate that the normal-form accurately models inverter dynamics across various operating conditions. The approach shows great potential for enhancing the modeling of RES-dominated power systems, especially when component models are unavailable or computationally expensive.

eess.SY

Dirac--Bianconi Graph Neural Networks -- Enabling Non-Diffusive Long-Range Graph Predictions

The geometry of a graph is encoded in dynamical processes on the graph. Many graph neural network (GNN) architectures are inspired by such dynamical systems, typically based on the graph Laplacian. Here, we introduce Dirac--Bianconi GNNs (DBGNNs), which are based on the topological Dirac equation recently proposed by Bianconi. Based on the graph Laplacian, we demonstrate that DBGNNs explore the geometry of the graph in a fundamentally different way than conventional message passing neural networks (MPNNs). While regular MPNNs propagate features diffusively, analogous to the heat equation, DBGNNs allow for coherent long-range propagation. Experimental results showcase the superior performance of DBGNNs over existing conventional MPNNs for long-range predictions of power grid stability and peptide properties. This study highlights the effectiveness of DBGNNs in capturing intricate graph dynamics, providing notable advancements in GNN architectures.

cs.LG

Predicting Fault-Ride-Through Probability of Inverter-Dominated Power Grids using Machine Learning

Due to the increasing share of renewables, the analysis of the dynamical behavior of power grids gains importance. Effective risk assessments necessitate the analysis of large number of fault scenarios. The computational costs inherent in dynamic simulations impose constraints on the number of configurations that can be analyzed. Machine Learning (ML) has proven to efficiently predict complex power grid properties. Hence, we analyze the potential of ML for predicting dynamic stability of future power grids with large shares of inverters. For this purpose, we generate a new dataset consisting of synthetic power grid models and perform dynamical simulations. As targets for the ML training, we calculate the fault-ride-through probability, which we define as the probability of staying within a ride-through curve after a fault at a bus has been cleared. Importantly, we demonstrate that ML models accurately predict the fault-ride-through probability of synthetic power grids. Finally, we also show that the ML models generalize to an IEEE-96 Test System, which emphasizes the potential of deploying ML methods to study probabilistic stability of power grids.

eess.SY

Instability in Complex Oscillator Networks: Limitations and Potentials of Network Measures and Machine Learning

A central question of network science is how functional properties of systems emerge from their structure. For networked dynamical systems, structure is typically captured through network measures. We investigate the relationship between these measures and stability metrics across non-linear and linear oscillators, as well as real-world power grid topologies and dynamics. We find that this relationship is highly sensitive to the underlying ensemble: minor changes in the networks considered, such as going from mean degree 6 to mean degree 8, can invert the correlation between a network measure and stability. We also investigate network measures as inputs for machine learning, as well as Graph Neural Networks (GNNs) as predictors of stability. Both GNNs and the non-linear combination of many network measures can accurately predict stability within a given ensemble, yet both can fail when the ensemble changes. We conclude that neither approach reliably identifies the underlying structural causes of instability.

nlin.AO

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

Complex Couplings -- A universal, adaptive and bilinear formulation of power grid dynamics

The paper is now published in PRX Enegry. Please refer to the PRX version from now on. Anna B\"uttner and Frank Hellmann. "Complex Couplings-A Universal, Adaptive, and Bilinear Formulation of Power Grid Dynamics." The energy transition introduces new classes of dynamical actors into the power grid. There is a growing need for so-called grid-forming inverters (GFIs) that can contribute to dynamic grid stability as the share of synchronous generators decreases. Understanding the collective behavior and stability of future grids, featuring a heterogeneous mix of dynamics, remains an urgent and challenging task. Two recent advances in describing such modern power grid dynamics have made this problem more tractable: First, the normal form for grid-forming actors provides a uniform, technology-neutral description of plausible grid dynamics, including grid-forming inverters and synchronous machines. Secondly, the notion of the complex frequency has been introduced to effortlessly describe how the nodal dynamics influence the power flows in the grid. The major contribution of this paper is to show how the normal form approach and the complex frequency dynamics of power grids combine, and how they relate naturally to adaptive dynamical networks and control affine systems. Using normal form and complex frequency, we derive a remarkably elementary and universal equation for the collective grid dynamics. Notably, we obtain an elegant equation entirely in terms of a matrix of complex couplings, in which the network topology does not explicitly appear. These complex couplings give rise to new adaptive network formulations of future power grid dynamics. We give a new formulation of the Kuramoto model with inertia as a special case.

nlin.AO

A Framework for Synthetic Power System Dynamics

The paper is published in Chaos. Please refer to the Chaos version from now on. Anna B\"uttner, Anton Plietzsch, Mehrnaz Anvari, Frank Hellmann; A framework for synthetic power system dynamics. Chaos 1 August 2023; 33 (8): 083120. https://doi.org/10.1063/5.0155971 Information on power grids is confidential and thus real data is often inaccessible. This necessitates the use of synthetic power grid models in research. So far the models used, for example, in machine learning had to be very simple and homogeneous to produce large ensembles of robust grids. We present a modular framework to generate synthetic power grids that considers the heterogeneity of real power grid dynamics but remains simple and tractable. This enables the generation of large sets of synthetic grids for a wide range of applications. We also include the major drivers of fluctuations on short-time scales. The synthetic grids generated are robust and show good synchronization under all evaluated scenarios, as should be expected for realistic power grids. This opens the door to future research that studies grids under severe stress due to extreme events which could lead to destabilization and black-outs. A software package that includes an efficient Julia implementation of the framework is released as a companion to the paper.

nlin.AO

Protecting the Texas power grid from tropical cyclones: Increasing resilience by protecting critical lines

The Texan electric network in the Gulf Coast of the United States is frequently hit by Tropical Cyclones (TC) causing widespread power outages, a risk that is expected to substantially increase under global warming. Here, we introduce a new approach of combining a probabilistic line fragility model with a network model of the Texas grid to simulate the temporal evolution of wind-induced failures of transmission lines and the resulting cascading power outages from seven major historical hurricanes. The approach allows reproducing observed supply failures. In addition, compared to a static approach, it provides a significant advantage in identifying critical lines whose failure can trigger large supply shortages. We show that protecting only 1% of total lines can reduce the likelihood of the most destructive type of outages by a factor of between 5 and 20. The proposed modelling approach could represent a tool so far missing to effectively strengthen the power grids against future hurricane risks even under limited knowledge.

physics.soc-ph