🤖 machine learning

Virtual Temperature Sensors in Power Transformers Using Neural Ordinary Differential Equations

This paper proposes a physics-aware Neural Ordinary Differential Equation (Neural ODE) framework that integrates simplified heat-transfer equations to accurately and robustly forecast the thermal behavior of diverse power transformers using real-world time-series data, overcoming the computational limitations of traditional numerical methods and the physical inconsistency of purely data-driven models.

Original authors: Berk Hadzhamolla, Alexander Johannes Stasik, Signe Riemer-Sørensen

Published 2026-08-14

📖 9 min read🧠 Deep dive

CC BY 4.0

Original authors: Berk Hadzhamolla, Alexander Johannes Stasik, Signe Riemer-Sørensen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the weather inside a giant, glowing metal box that never sleeps. This box is a power transformer, the heavy-duty heart of the electrical grid that steps voltage up or down so your lights stay on. But like any hardworking machine, it gets hot. If it gets too hot, the insulation inside ages faster, and the whole thing might break, causing blackouts. To keep it safe, engineers need to know the temperature of the hottest spots inside, but those spots are buried deep in oil and copper, making them impossible to stick a thermometer into. So, they have to guess the temperature based on what's happening on the outside, like how much electricity is flowing and how hot the air is.

For a long time, scientists tried to solve this puzzle in two main ways. The first was like building a perfect, miniature 3D model of the transformer in a computer and simulating every drop of oil and every wire. It was incredibly accurate but took so much computing power that it was useless for real-time decisions. The second way was to use simple math formulas based on average rules, but these often failed when the weather changed suddenly or the load got weird. Recently, people started using "Artificial Intelligence" (AI) to learn the patterns from past data. Think of it like teaching a parrot to squawk the right answer after hearing thousands of examples. But these AI parrots can be tricky; if you ask them about a situation they've never seen before, they might just make up a wild guess that sounds plausible but is physically impossible.

This paper introduces a new kind of "smart guesser" called a Neural Ordinary Differential Equation (Neural ODE) to act as a virtual temperature sensor. Instead of just memorizing past data like a parrot, or building a heavy 3D simulation, this new method learns the rules of physics while it learns from the data. Imagine teaching a student not just the answers to a math test, but also the actual laws of how heat moves. The researchers tested this on real transformers across Norway, using data from fifteen different units with different designs. They found that while the new method needs at least three months of data to get its bearings, once it learns, it becomes a much more reliable predictor than the standard AI models, especially for looking far into the future. It doesn't just guess; it understands that heat takes time to travel, making it a sturdy tool for keeping the grid safe.

The Problem: The Invisible Heat

Power transformers are the unsung heroes of our electrical grid. They take high-voltage electricity and convert it to a level safe for our homes and businesses. But this process generates a lot of heat. The hottest part, called the "hotspot," is usually deep inside the winding coils, hidden under layers of oil. If this hotspot gets too hot, the paper insulation inside ages rapidly, shortening the transformer's life and risking a catastrophic failure.

Ideally, we would just stick a sensor right on the hotspot. But in many transformers, that's impossible. So, engineers use "virtual sensors"—software that estimates the hidden temperature based on things we can measure, like the outside air temperature and how much power is flowing through the wires.

The Old Ways: Too Heavy, Too Simple, or Too Guessy

Before this new approach, there were three main ways to build these virtual sensors, and each had a big flaw:

The Super-Computer Simulation: Scientists used complex physics models (like Finite Element Methods) to simulate every drop of oil and wire. These were very accurate but required so much computing power that they couldn't run in real-time. They were like trying to predict the weather by simulating every single molecule of air in the atmosphere—too slow for a quick forecast.
The Simple Math Models: These used simplified equations with fixed numbers (constants) for each transformer. They were fast but rigid. If the transformer was old, or the weather was weird, or the cooling system worked differently, these models often got it wrong because they couldn't adapt.
The Pure AI Models: Recently, people tried using standard AI (like LSTM networks) to learn from data. These are great at spotting patterns but act like a "black box." They don't know the laws of physics. If you ask them to predict the temperature for a day they've never seen, they might drift into impossible territory, like predicting the oil gets colder while the load increases. They need massive amounts of data to work well and can be unstable over long periods.

The New Solution: The Physics-Aware AI

The authors of this paper proposed a hybrid approach: Neural Ordinary Differential Equations (Neural ODEs).

Think of a standard AI as a student who memorizes the answers to a practice test. If the test changes slightly, the student panics. A Neural ODE is like a student who memorizes the formulas and the logic behind the answers.

In this specific framework, the researchers didn't just let the AI guess the next temperature. Instead, they forced the AI to guess the rate of change (how fast the temperature is rising or falling) based on the current state and the physics of heat transfer.

The Analogy: Imagine you are driving a car. A standard AI tries to guess where you will be in 10 minutes by looking at your past speed. A Neural ODE understands that you are currently pressing the gas pedal (load) and that the car has weight (thermal mass). It calculates your acceleration and then integrates that to find your future position. This ensures the prediction follows the laws of physics, even if the data is messy or irregular.

They built this model to handle real-world data, which is often messy. Sensors might miss a reading here or there, or the data might come in at irregular intervals. Because Neural ODEs work in "continuous time" (like a smooth movie) rather than "discrete time" (like a flipbook), they handle these gaps much better than older AI models.

The Experiment: Testing on Real Transformers

The team didn't just run this on a computer simulation; they tested it on 15 real power transformers located in different parts of Norway. These transformers had different designs, cooling systems, and operating histories.

They fed the model data recorded every 900 seconds (15 minutes). The data included:

The active power load (how much electricity was flowing).
The ambient temperature (how hot the air was outside).
The actual oil and winding temperatures (to train the model).

They compared their new Neural ODE model against a standard AI model (LSTM) to see who could predict the future temperature better. They tested different amounts of training data, from just one week to two years, and different forecast lengths, from one day to a full year.

The Findings: Patience Pays Off

The results were clear, but they came with a catch.

1. The "Three-Month" Rule:
The new model needs time to learn. If you only give it one week of data, it fails miserably. The predictions become chaotic and physically impossible. The authors found that the model needs at least three months of data to stabilize. This makes sense because three months covers a full seasonal quarter in Norway, allowing the model to see how the transformer reacts to the full range of temperature changes from winter to summer.

2. Better than the Competition:
Once the model had at least three months of data, it outperformed the standard AI (LSTM) significantly.

Accuracy: In the "stable" regime (3 months of training), the Neural ODE had a median error (MAE) of 2.04 °C, while the LSTM had 2.45 °C.
Reliability: Over long forecast periods (like 30 to 365 days), the Neural ODE stayed much closer to the real temperature. The standard AI tended to drift away, producing "physically inconsistent" results, while the Neural ODE's predictions remained smooth and grounded in reality.
Consistency: Across all 15 transformers, the Neural ODE was more accurate in 64.8% of the cases and had a lower error in 64.8% of the cases compared to the LSTM. It performed better on 14 out of the 15 individual transformers.

3. The Long-Term View:
The model is particularly good at "medium-to-long horizon" forecasting (30 days to a year). While standard AI struggles to predict a year ahead without drifting off, the Neural ODE maintained a reasonable accuracy, with a median error staying around 2 °C even for a 365-day forecast. This is crucial for asset managers who need to plan maintenance schedules far in advance.

Why This Matters

This isn't just a theoretical win; it's a practical tool for keeping the lights on.

Virtual Sensing: It allows grid operators to "see" the hidden hotspots without installing expensive, fragile physical sensors inside every transformer.
Cost Savings: It reduces the need for dense sensor networks, lowering maintenance costs.
Safety: By providing more reliable long-term forecasts, it helps prevent overheating and extends the life of the transformers.

The authors suggest that while the model needs a "warm-up" period of about three months, it is a robust, physics-aware tool that works well across different types of transformers. It bridges the gap between the slow, heavy physics simulations and the unreliable, data-hungry pure AI, offering a "sweet spot" that is both smart and scientifically grounded.

In short, this new method teaches the computer the rules of heat, not just the answers, making it a much better weather forecaster for the invisible heat inside our power grid.

Technical Summary: Virtual Temperature Sensors in Power Transformers Using Neural Ordinary Differential Equations

Problem Statement
Accurate estimation and forecasting of power transformer internal temperatures (specifically top-oil, winding, and hotspot temperatures) are critical for condition monitoring, overload control, and predictive maintenance. Existing approaches suffer from distinct limitations:

High-fidelity numerical methods (e.g., Finite Element Methods, Computational Fluid Dynamics) offer detailed physical representations but are computationally prohibitive for real-time applications and require complete knowledge of internal geometries and parameters.
Lumped-parameter thermal models are computationally efficient but rely on transformer-specific thermal constants that are often unknown and struggle to capture dynamic responses under varying operating and environmental conditions.
Purely data-driven machine learning models (e.g., ANNs, LSTMs, CNNs) can learn nonlinear mappings without explicit physical formulations but typically require large volumes of high-quality training data. They often lack physical interpretability and risk producing physically inconsistent results or diverging during long-horizon forecasting.

Methodology
The paper proposes a Physics-Aware Neural Ordinary Differential Equation (Neural ODE) framework to forecast transformer thermal behavior using real-world time-series data.

Thermal Dynamics Formulation: The authors model the transformer thermal system using coupled heat transfer equations between windings, cooling oil, and the ambient environment. These equations define a first-order Markovian state-space structure where the rate of change of thermal states depends on current internal states and external drivers (active power load and ambient temperature).
Neural ODE Architecture: Instead of predicting the next temperature step directly (as in discrete-time models like LSTMs), the framework parameterizes the continuous-time derivative of the state vector, $dy/dt$, using a neural network ( $NN_\theta$ ). The temperature trajectory is obtained by numerically integrating this learned vector field:
$y(t) = y(t_0) + \int_{t_0}^{t} NN_\theta(y(t'), x(t')) dt'$
The architecture explicitly preserves the physical decomposition between internal thermal states ( $y$ ) and external exogenous inputs ( $x$ ).
Physics-Aware Integration: The model incorporates simplified heat-transfer equations directly into the learning process. The network outputs the rate of change ($dy/dt$) rather than absolute temperatures, enforcing a continuous-time heat-transfer structure. Exogenous inputs are handled via piecewise-linear interpolation to provide continuous-time access during integration.
Training Strategy: The model is trained using a composite loss function comprising Mean Squared Error (MSE) and a smoothness penalty ( $L_{smooth}$ ) to discourage high-frequency oscillations inconsistent with thermal dynamics. A curriculum learning strategy is employed, gradually increasing the prediction horizon from 1.25 hours to 2 days to accelerate convergence.
Dataset: The framework was evaluated on operational data from 15 distinct transformers across Norway, characterized by varying designs and cooling mechanisms (ONAN, ONAF, OFAF, ODAF). Data was recorded at 15-minute intervals.

Key Contributions

Physics-Aware Formulation: The integration of simplified heat-transfer equations directly into the Neural ODE architecture, enabling physically consistent solutions while reducing the data requirements compared to purely data-driven methods.
Continuous-Time Modeling: Utilizing Neural ODEs to model system dynamics in continuous time, providing robustness to irregular sampling and improved extrapolation capabilities compared to discrete-time models like LSTMs.
Standardized Framework: Development of a standardized forecasting tool capable of generalizing across heterogeneous transformer units without requiring transformer-specific thermal constants.

Results
The performance of the Neural ODE was compared against a stacked LSTM baseline across 823 combinations of transformer units, training horizons, and forecast horizons.

Data Requirements: The Neural ODE requires a minimum of approximately three months of training data to enter a stable and useful regime. With less than one month of data, the model exhibits unstable trajectories and catastrophic divergence.
Performance Metrics: In the stable regime (training $\ge$ $\geq$ 3 months, forecast $\ge$ $\geq$ 30 days):
- The Neural ODE achieved a median $R^2$ of 0.591 and a median Mean Absolute Error (MAE) of 2.04 °C.
- The LSTM baseline achieved a median $R^2$ of 0.493 and a median MAE of 2.45 °C.
- The Neural ODE outperformed the LSTM in 65.2% of cases regarding $R^2$ and 70.1% of cases regarding MAE in this operational regime.
Long-Horizon Stability: The Neural ODE demonstrated superior stability in long-horizon forecasting (up to 365 days). While both models showed performance degradation over time due to error accumulation in open-loop integration, the Neural ODE's physically bounded trajectories resulted in slower degradation compared to the LSTM.
Per-Unit Consistency: The Neural ODE achieved higher median $R^2$ on 14 of the 15 units and lower median MAE on all 15 units, indicating consistent performance across diverse transformer designs.

Significance and Claims
The paper claims that the proposed Physics-Aware Neural ODE framework successfully bridges the gap between computationally expensive numerical methods and brittle purely data-driven models. Its primary significance lies in its ability to serve as a robust, physics-aware virtual sensor for heterogeneous transformer fleets.

Key practical advantages highlighted include:

Data Efficiency: The model produces useful forecasts with as little as three months of training data, making it suitable for new or recently instrumented units where long historical records are unavailable.
Operational Reliability: By providing reliable virtual temperature estimates (with an average error of ~2 °C over a year), the framework can reduce the need for dense physical sensor deployment, lowering monitoring costs while supporting early detection of abnormal thermal behavior.
Generalization: The framework generalizes effectively across different transformer designs and cooling mechanisms without requiring unit-specific parameter tuning.

The authors note that the main limitation remains the minimum data requirement (approx. 3 months) for stability, and future work will investigate warmup strategies and transfer learning to lower this threshold.