Spatial Dynamic Thermal Discretization for Enhanced Transformer Condition Assessment

Introduction

Reliable insulation assessment in liquid-filled transformers requires accurate representation of internal thermal behaviour, which is inherently non-uniform due to oil circulation, winding losses, and liquid inertia. Conventional models used in standards such as IEC 60076‑7 [1] and IEEE guides [2] rely on lumped assumptions and fixed time constants, limiting their ability to capture axial temperature gradients and transient cooling dynamics. To address this, spatial dynamic transformer thermal modelling (SDTTM) resolves thermal behaviour along the winding height by combining standard measurements with dual-exponential liquid dynamics, enabling reconstruction of the internal temperature field with higher fidelity [3-5]. This spatial-temporal approach provides a more realistic prediction of hot-spot evolution, thermal stress, and cooling performance, and when coupled with moisture dynamics, supports advanced applications such as insulation condition assessment, dynamic rating, and digital-twin-based asset management.

Limitations of Conventional Thermal Models

Traditional transformer thermal models rely on simplified assumptions:

• Uniform oil temperature throughout the tank
• Instantaneous liquid response to load variations
• Fixed winding and oil thermal time constants
• Dependence on top‑oil and hot‑spot estimation

These simplifications lead to several practical limitations:

• Inability to capture axial temperature gradients
• Overshoot errors in transient conditions
• Underestimation of localized overheating
• Poor representation of cooling curves
• Reduced accuracy in insulation aging predictions

Most importantly, these models cannot describe how thermal energy propagates spatially through the transformer during dynamic loading. Since insulation aging and moisture migration are highly temperature-dependent, this limitation directly impacts condition assessment accuracy.

Concept of Spatial Thermal Discretization

The proposed approach transforms the transformer from a lumped system into a spatially resolved thermal domain. The winding height is discretized into multiple axial segments, each characterized by its own thermal response. Physically, the temperature increases from bottom oil to top oil due to natural convection. The top region responds faster to load changes, while the bottom region exhibits stronger thermal inertia [3]. This spatial heterogeneity forms the basis of the discretized model. The key innovation lies in representing local thermal dynamics at each axial position, rather than using a single global temperature.

Capturing Liquid Inertia Using Dual-Exponential Dynamics

Transformer cooling curves reveal that oil does not follow a single time constant. Instead, its thermal response consists of two distinct phases:

• A fast transient component, associated with liquid inertia
• A slow component, representing bulk oil cooling

This behavior can be accurately captured using a dual-exponential formulation [3]:                            

This dual representation significantly improves the fidelity of thermal transient modeling compared to standard single-exponential models. Figures 1-3 collectively illustrate how liquid properties govern transformer thermal behaviour under dynamic loading conditions. Figure 1 shows that natural ester exhibits slower heating and cooling compared to mineral oil, primarily due to its higher viscosity and reduced heat transfer capability. Building on this, Figure 2 demonstrates that increased viscosity leads to larger thermal time constants, resulting in a slower transient response and highlighting the direct link between fluid properties and thermal inertia. Figure 3 further shows that viscosity remains higher for natural ester across the operating range and varies with temperature, directly affecting heat transfer, oil circulation, and overall cooling effectiveness. Together, these observations emphasize the critical role of viscosity in controlling thermal response and cooling performance in transformers. 

Deriving Spatial Dynamics from Measurable Quantities

One of the strengths of this approach is that it relies on standard heat-run test measurements, specifically:

• Top-oil (TO) temperature
• Bottom-oil (BO) temperature

These two signals naturally bound the thermal behavior of the liquid:

• TO exhibits fast dynamics (low inertia)
• BO exhibits slow dynamics (high inertia)

By fitting dual-exponential curves to both TO and BO cooling responses, the model extracts:

• Fast and slow time constants
• Relative contributions of inertia and bulk cooling

These parameters are then interpolated along the winding height to construct a continuous axial model.

Axial Distribution of Thermal Time Constants

To generate spatial resolution, the extracted TO and BO parameters are mapped along the winding height Z:

This interpolation yields:

• Higher inertia (larger time constants) near the bottom
• Faster dynamics near the top

Each axial segment is then governed by its own local dynamics:

This formulation enables reconstruction of the full temperature field inside the transformer, including:

• Transient thermal gradients
• Hot-spot evolution and migration
• Interaction between winding and liquid temperatures

Figure 4 presents the axial contour of liquid temperature over time, revealing a distinctly non-uniform thermal response along the winding. The top region exhibits a rapid cooling response, whereas the bottom region shows a delayed cooling behavior. This difference is primarily driven by temperature-dependent viscosity: hot oil at the top has lower viscosity, enabling faster flow and more efficient heat transport, while the colder oil at the bottom has higher viscosity, resulting in slower flow and delayed heat removal. Figure 5 shows the corresponding winding temperature contour, where the winding exhibits a quasi-lumped thermal response characterized by a single dominant time constant. However, spatial variations are still evident due to the non-uniform cooling provided by the surrounding liquid. This indicates that the winding temperature is strongly influenced by the local fluid field, leading to time-dependent axial temperature gradients. These evolving gradients are critical, as they directly impact hot-spot temperature prediction and cannot be accurately captured using constant gradient assumptions.

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Reconstruction of Thermal Fields and Cooling Behaviour

Unlike conventional measurements, which provide only a few temperature points, the discretized model reconstructs the entire internal thermal profile.

The reconstructed cooling curves reveal that:

• Windings behave as a relatively lumped system with a single time constant
• Liquid cooling is highly non-uniform and spatially dependent
• Top, middle, and bottom oil regions exhibit distinct dynamics

This granular insight allows accurate prediction of:

• Peak temperature location
• Time-dependent thermal stress
• Cooling efficiency under varying loads

Figure 6 illustrates the cooling curves at different axial positions, highlighting a clear variation in cooling rates along the winding height as well as differences in effective thermal inertia. The results demonstrate that different sections of the winding respond at different speeds due to the non-uniform thermal interaction with the surrounding fluid. This behavior confirms that the cooling process cannot be accurately represented by a single time constant, as thermal gradients evolve dynamically over time and space.

Integration with Moisture Dynamics

Thermal behavior and moisture distribution are strongly coupled in transformer insulation systems, as moisture migration depends on local temperature and equilibrium conditions. While the present work focuses on spatial dynamic thermal modelling (SDTTM), the reconstructed temperature fields provide a strong foundation for integrating moisture behaviour.

The spatial thermal model enables:

• Accurate local temperature inputs along the winding
• Realistic transient thermal gradients
• Consistent boundary conditions for moisture exchange

When coupled with a dynamic transformer moisture model (DTMM), this framework can support:

• Prediction of moisture migration under dynamic loading
• Assessment of localized insulation risk
• Improved evaluation of bubble formation conditions and other hydrolysis aging factors. 

Engineering and Asset Management Applications

The spatially discretized framework offers significant benefits across multiple domains:

1. Design Optimization

• Improved cooling duct and oil flow design
• Better understanding of thermal bottlenecks
• Optimization of winding geometry

2. Operational Planning

• Accurate dynamic thermal rating
• Realistic overload capability estimation
• Reduced risk of thermal overstressing

3. Condition Monitoring

• Enhanced hot-spot prediction
• Improved interpretation of heat-run tests
• Support for digital twin development

4. Insulation Life Assessment

• More accurate aging estimation
• Coupled thermal-moisture modeling
• Better prediction of failure risk mechanisms

Toward Digital Twins and Predictive Diagnostics

The spatial thermal discretization framework represents a foundational step toward transformer digital twins. By combining real-time measurements with physics-based models, it becomes possible to:

• Continuously estimate internal thermal states
• Predict insulation degradation
• Enable condition-based maintenance strategies

This evolution moves transformer monitoring from reactive diagnostics to predictive and risk-informed operation.

Conclusion

The transition from lumped thermal models to spatially resolved dynamic frameworks represents a significant advancement in transformer thermal analysis. By capturing axial temperature gradients, liquid inertia, and transient cooling behavior, the proposed spatial dynamic transformer thermal modeling (SDTTM) approach provides a more realistic representation of internal transformer conditions. Most importantly, it demonstrates that temperature gradients and hot‑spot formation are inherently dynamic and evolve both in space and time, rather than remaining fixed as assumed in conventional models. As transformer operation becomes increasingly dynamic, driven by variable loading and evolving grid demands, such physically consistent and spatially resolved modeling approaches are essential for improving thermal assessment, enhancing reliability, and enabling advanced applications such as dynamic rating and digital-twin-based asset management.

References

• IEC 60076-7:2018, Power Transformers – Part 7: Loading Guide for Mineral-Oil-Immersed Power Transformers, IEC.
• IEEE Std C57.91-2011, IEEE Guide for Loading Mineral-Oil-Immersed Transformers and Step-Voltage Regulators, IEEE.
• A. Al-Abadi, A. Gamil, P. Klys, R. Daszkiewicz, “New Concepts on Dynamic Thermal Behavior of Ester Liquid-Filled Distribution Transformers,” CIGRE SC A2 & 6th International Colloquium Transformer Research and Asset Management, Split, Croatia, Nov. 29-Dec. 2, 2023.
• A. Al-Abadi, A. Gamil, “Steady State and Dynamic Thermal Performance of Liquid-Filled Distribution Transformers,” Cigre Paris session, Ref A2-11027-2024, 2024.
• A. Al-Abadi, A. Gamil and A. Sbravati, “Determination of Moisture Content during Dynamic Loading of Liquid-Filled Distribution Transformers,” 2024 IEEE Electrical Insulation Conference (EIC), Minneapolis, MN, USA, pp. 134-138, 2024.