Hinrikus Wolf

Luis Böttcher, Hinrikus Wolf, Bastian Jung et al.

In this paper, we propose a graph neural network architecture to solve the AC power flow problem under realistic constraints. To ensure a safe and resilient operation of distribution grids, AC power flow calculations are the means of choice to determine grid operating limits or analyze grid asset utilization in planning procedures. In our approach, we demonstrate the development of a framework that uses graph neural networks to learn the physical constraints of the power flow. We present our model architecture on which we perform unsupervised training to learn a general solution of the AC power flow formulation independent of the specific topologies and supply tasks used for training. Finally, we demonstrate, validate and discuss our results on medium voltage benchmark grids. In our approach, we focus on the physical and topological properties of distribution grids to provide scalable solutions for real grid topologies. Therefore, we take a data-driven approach, using large and diverse data sets consisting of realistic grid topologies, for the unsupervised training of the AC power flow graph neural network architecture and compare the results to a prior neural architecture and the Newton-Raphson method. Our approach shows a high increase in computation time and good accuracy compared to state-of-the-art solvers. It also out-performs that neural solver for power flow in terms of accuracy.

Hinrikus Wolf, Luca Oeljeklaus, Pascal Kühner et al.

Graph homomorphism counts, first explored by Lovász in 1967, have recently garnered interest as a powerful tool in graph-based machine learning. Grohe (PODS 2020) proposed the theoretical foundations for using homomorphism counts in machine learning on graph level as well as node level tasks. By their very nature, these capture local structural information, which enables the creation of robust structural embeddings. While a first approach for graph level tasks has been made by Nguyen and Maehara (ICML 2020), we experimentally show the effectiveness of homomorphism count based node embeddings. Enriched with node labels, node weights, and edge weights, these offer an interpretable representation of graph data, allowing for enhanced explainability of machine learning models. We propose a theoretical framework for isomorphism-invariant homomorphism count based embeddings which lend themselves to a wide variety of downstream tasks. Our approach capitalises on the efficient computability of graph homomorphism counts for bounded treewidth graph classes, rendering it a practical solution for real-world applications. We demonstrate their expressivity through experiments on benchmark datasets. Although our results do not match the accuracy of state-of-the-art neural architectures, they are comparable to other advanced graph learning models. Remarkably, our approach demarcates itself by ensuring explainability for each individual feature. By integrating interpretable machine learning algorithms like SVMs or Random Forests, we establish a seamless, end-to-end explainable pipeline. Our study contributes to the advancement of graph-based techniques that offer both performance and interpretability.

Hinrikus Wolf, Luis Böttcher, Sarra Bouchkati et al.

In the course of the energy transition, the expansion of generation and consumption will change, and many of these technologies, such as PV systems, electric cars and heat pumps, will influence the power flow, especially in the distribution grids. Scalable methods that can make decisions for each grid connection are needed to enable congestion-free grid operation in the distribution grids. This paper presents a novel end-to-end approach to resolving congestion in distribution grids with deep reinforcement learning. Our architecture learns to curtail power and set appropriate reactive power to determine a non-congested and, thus, feasible grid state. State-of-the-art methods such as the optimal power flow (OPF) demand high computational costs and detailed measurements of every bus in a grid. In contrast, the presented method enables decisions under sparse information with just some buses observable in the grid. Distribution grids are generally not yet fully digitized and observable, so this method can be used for decision-making on the majority of low-voltage grids. On a real low-voltage grid the approach resolves 100\% of violations in the voltage band and 98.8\% of asset overloads. The results show that decisions can also be made on real grids that guarantee sufficient quality for congestion-free grid operation.

Jan Tönshoff, Martin Ritzert, Hinrikus Wolf et al.

We propose CRaWl, a novel neural network architecture for graph learning. Like graph neural networks, CRaWl layers update node features on a graph and thus can freely be combined or interleaved with GNN layers. Yet CRaWl operates fundamentally different from message passing graph neural networks. CRaWl layers extract and aggregate information on subgraphs appearing along random walks through a graph using 1D Convolutions. Thereby it detects long range interactions and computes non-local features. As the theoretical basis for our approach, we prove a theorem stating that the expressiveness of CRaWl is incomparable with that of the Weisfeiler Leman algorithm and hence with graph neural networks. That is, there are functions expressible by CRaWl, but not by GNNs and vice versa. This result extends to higher levels of the Weisfeiler Leman hierarchy and thus to higher-order GNNs. Empirically, we show that CRaWl matches state-of-the-art GNN architectures across a multitude of benchmark datasets for classification and regression on graphs.

Tobias Schumacher, Hinrikus Wolf, Martin Ritzert et al.

We systematically evaluate the (in-)stability of state-of-the-art node embedding algorithms due to randomness, i.e., the random variation of their outcomes given identical algorithms and graphs. We apply five node embeddings algorithms---HOPE, LINE, node2vec, SDNE, and GraphSAGE---to synthetic and empirical graphs and assess their stability under randomness with respect to (i) the geometry of embedding spaces as well as (ii) their performance in downstream tasks. We find significant instabilities in the geometry of embedding spaces independent of the centrality of a node. In the evaluation of downstream tasks, we find that the accuracy of node classification seems to be unaffected by random seeding while the actual classification of nodes can vary significantly. This suggests that instability effects need to be taken into account when working with node embeddings. Our work is relevant for researchers and engineers interested in the effectiveness, reliability, and reproducibility of node embedding approaches.

Jan Toenshoff, Martin Ritzert, Hinrikus Wolf et al.

Many combinatorial optimization problems can be phrased in the language of constraint satisfaction problems. We introduce a graph neural network architecture for solving such optimization problems. The architecture is generic; it works for all binary constraint satisfaction problems. Training is unsupervised, and it is sufficient to train on relatively small instances; the resulting networks perform well on much larger instances (at least 10-times larger). We experimentally evaluate our approach for a variety of problems, including Maximum Cut and Maximum Independent Set. Despite being generic, we show that our approach matches or surpasses most greedy and semi-definite programming based algorithms and sometimes even outperforms state-of-the-art heuristics for the specific problems.