Virtual Temperature Sensors in Power Transformers Using Neural Ordinary Differential Equations

Accurate modeling and forecasting of power transformer thermal behavior are essential for grid reliability, condition monitoring, and extending asset lifetime. Traditional numerical approaches like finite element methods and computational fluid dynamics offer high fidelity but are computationally prohibitive and require detailed geometry. Meanwhile, lumped-parameter thermal models depend on transformer-specific constants that are frequently unknown, and purely data-driven machine learning methods—such as LSTMs—lack physical interpretability and can drift during long-horizon rollouts. This paper introduces a physics-aware Neural Ordinary Differential Equation (Neural ODE) framework designed to combine the flexibility of deep learning with the continuous-time constraints of heat-transfer physics.

The thermal dynamics of power transformers are governed by energy balance equations involving windings, cooling oil, and ambient conditions. The proposed framework respects this physics by decomposing inputs into a state vector of internal temperatures and a control vector of external loads and ambient temperatures. Instead of predicting future temperatures directly, the Neural ODE learns a continuous-time vector field representing the time derivatives of the thermal states. The model uses an adaptive numerical ODE solver to integrate these derivatives over time. Continuous access to external drivers between discrete sensor observations is maintained via piecewise-linear interpolation. Training utilizes a composite loss function combining mean squared error and a physical smoothness penalty to prevent unrealistic high-frequency oscillations, alongside a sliding-window curriculum learning strategy.

The framework is evaluated on multivariate time-series data from fifteen distinct power transformers located across Norway, capturing diverse designs and cooling mechanisms over multiple months or years. The results reveal three distinct performance regimes based on the amount of available training data. With very short training sets of one week or one month, the model suffers from instability and catastrophic trajectory divergence. However, when trained on three months or more of data, the Neural ODE enters a stable and useful regime, achieving median absolute errors of roughly two degrees Celsius over extended forecasting horizons. This confirms that continuous-time physics-aware neural networks can robustly generalize across heterogeneous grid assets.