Virtual Temperature Sensors in Power Transformers Using Neural Ordinary Differential Equations
Authors: Berk Hadzhamolla, Alexander Johannes Stasik, Signe Riemer-Sørensen
Organizations: University of Oslo, Oslo, 0316, Norway · 2,3 SINTEF AS, Department of Mathematics and Cybernetics, Oslo, Norway · Department of Data Science, Norwegian University of Life Sciences, ˚As, Norway
Accurate modeling and forecasting of power transformer thermal behavior are critical for reliability, asset lifetime, and optimized power system operation. Numerical approaches such as finite element methods (FEM) and computational fluid dynamics (CFD) offer high fidelity but are computationally expensive, require complex mesh generation, and are often impractical for real-time or large-scale applications, particularly when transformer geometries are unknown. Lumped-parameter thermal models are more practical but depend on transformer-specific thermal constants and may fail to capture dynamic responses under varying operating and environmental conditions. Purely data-driven machine learning methods, including artificial neural networks, convolutional neural networks, and long short-term memory (LSTM) networks, have shown success in forecasting transformer temperatures but typically require large volumes of high-quality training data and may produce physically inconsistent or uninterpretable results. This paper develops a physics-aware Neural Ordinary Differential Equation (Neural ODE) framework for forecasting transformer thermal behavior from real-world time-series data. Neural ODEs model system dynamics in continuous time, providing smooth trajectory prediction and a natural representation of continuously evolving thermal dynamics. A key contribution is the integration of simplified heat-transfer equations directly into the Neural ODE formulation. The model is evaluated across datasets from fifteen transformers in different regions of Norway with varying designs and cooling mechanisms. The results demonstrate that the developed Neural ODE framework provides a standardized, physics-aware, and robust forecasting approach for heterogeneous transformer units.
As power systems transition toward renewable-rich and inverter-dominated operations, accurate time-domain dynamic analysis becomes increasingly critical. Such analysis supports key operational tasks, including transient stability assessment, dynamic security analysis, contingency screening, and post-fault trajectory evaluation. In practice, these tasks may operate under several challenges, including unknown and time-varying system parameters, privacy constraints on data sharing, and the need for fast online inference. Existing learning-based approaches are typically trained for individual systems and therefore lack generalization across operating conditions and physical parameters. Hence, this paper proposes LArge Scale Small ODE (LASS)-ODE-Power, a learning framework for general-purpose time-domain prediction. The proposed approach leverages large-scale pretraining on more than 40 GB of DAE or ordinary differential-equation (ODE) trajectories to learn transferable representations. The resulting model supports trajectory prediction from short measurement prefixes across diverse dynamic regimes, including electromechanical and inverter-driven systems. Hence, the model can be directly used without data sharing in a zero-shot setting. In addition, the proposed architecture incorporates parallel and linearized computation to achieve fast inference. Moreover, to enhance task-specific performance in power systems, a specialized fine-tuning strategy is developed based on approximately 1 GB of heterogeneous power-system dynamic data. Extensive experiments over diverse power-system simulation scenarios demonstrate that LASS-ODE-Power consistently outperforms existing learning-based models in trajectory prediction accuracy with efficient inference.
Obtaining an accurate short-term forecasting for heat demand is an essential part of operating district heating networks cost-efficient and reliable. Heat consumption time series at the building level are highly dependent on exogenous variables such as outdoor temperature and individual usage patterns, making forecasting in this context a challenging task. Thus, this paper benchmarks novel Transformer-based and xLSTM architectures for short-term heat-demand forecasting. Using hourly data from 25 German buildings (2017-2025), we compare three-hour and 24-hour forecasting horizons relevant for intraday control and day-ahead scheduling. We establish a multi-building benchmark that tests whether models trained on pooled, heterogeneous building data are able to generalize across diverse building stock. The results show that the xLSTM achieves the lowest RMSE (19.88 kWh for three-hour, 21.47 kWh for 24-hour forecasts), while the Temporal Fusion Transformer attains the best MAE (9.16 kWh for three-hour forecasts). As xLSTMs and Transformers require long training times and have a huge number of trainable parameters, their sustainability remains questionable. Therefore, this paper further investigates the trade-off between predictive accuracy and computational resource demand of the evaluated forecasting models. The findings indicate that also low-parameter models like a traditional fully-connected network achieve good predictive results, highlighting that marginal accuracy gains of the novel prediction models come at substantial resource expense for this use case.
Operational weather prediction has long relied on physics-based numerical weather prediction (NWP), whose accuracy comes at the cost of substantial compute and complex simulation workflows. Recent transformer-based forecasters offer efficient data-driven alternatives, however transformers are physics-agnostic models. Additionally, standard transformer encoders evolve representations through discrete layer updates that may be less suited to modeling smooth latent dynamics. In this work, we propose a continuous-depth transformer encoder for weather forecasting that integrates Neural Ordinary Differential Equation (Neural ODE) dynamics within each encoder block. Specifically, we replace discrete residual updates with ODE-based updates solved using adaptive numerical integration. We also introduce a two-branch attention module that combines conventional patch-wise self-attention with an auxiliary branch that applies a derivative operator to attention logits, providing an additional change-sensitive interaction signal. To further align forecasts with governing principles, we propose a customized physics-informed training objective that enforces physical consistency as a soft constraint. We evaluate the proposed method against a standard discrete transformer baseline and an existing continuous-time Neural ODE forecasting variant, demonstrating the importance of PINN-Cast in short term weather forecasting.