Virtual Temperature Sensors

Virtual temperature sensors (also referred to as “soft sensors” or “sensorless estimators”) are software-based algorithms that estimate the temperature of battery cells without relying on a direct, physical hardware sensor at the measurement location [2].

Instead of measuring heat directly, these virtual sensors act as real-time state reconstruction algorithms running within the Battery Management System (BMS). They infer the thermal state of a cell by fusing mathematical models with standard operational signals that are already available on the BMS communication bus, such as cell current, terminal voltage, state of charge (SOC), and ambient or coolant temperatures.

Virtual sensors are designed to solve the “instrumentation gap” that occurs when it is too expensive or physically complex to place a physical sensor on every single cell (How Many Temperature Sensors are Required?). They offer several major advantages:

• Accessing the Inaccessible: While physical sensors are generally limited to the battery’s surface, virtual sensors can mathematically reconstruct and estimate the internal core temperatures where heat accumulation is highest and thermal runaway risks originate.
• Monitoring Un-instrumented Cells: By analysing the electrical inputs and outputs of the battery pack, virtual sensors can map out the spatial temperature field across neighbouring cells that have no physical sensors attached to them at all.

To accomplish this, virtual temperature sensors rely on different types of underlying models:

1. Physics-Based Mathematical Models: These utilize thermodynamic equations (such as lumped-parameter thermal circuit models) to track heat transfer. They are frequently paired with closed-loop algorithms, like Kalman filters, which continuously compare estimated temperatures against the few physical sensors that do exist to correct and update their calculations.
2. Data-Driven Machine Learning Models: These treat the battery as a “black box” and employ deep learning architectures—such as Long Short-Term Memory (LSTM) networks or Convolutional Neural Networks (CNNs)—to learn complex, nonlinear thermal patterns directly from historical battery data.
3. Hybrid (Physics-Informed) Models: These integrate physical battery models with machine learning networks. By using the laws of physics to guide the neural network, these virtual sensors achieve highly accurate and robust temperature estimates without requiring massive amounts of experimental training data.

This post has been built based on the support of and sponsorship from: illumion, Custom Power, Quarto Technical Services, Eatron Technologies and About:Energy

How accurate are these Virtual Temperature Sensors?

The accuracy of virtual temperature sensors is typically measured using the Root Mean Square Error (RMSE) or Mean Absolute Error (MAE). While their precision varies depending on the specific cell chemistry, operating conditions, and computational limits, state-of-the-art virtual sensors generally estimate internal and surface temperatures with an RMSE between <0.1 °C and 1.5 °C.

Here is how the accuracy breaks down across the three model types described above:

1. Physics-Based Models (with Observers) Traditional lumped-parameter thermal models that use closed-loop observers, such as the Extended Kalman Filter (EKF), typically achieve an estimation accuracy (RMSE) between 0.38 °C and 1.01 °C.

• Enhancements: Models that explicitly account for “entropic heating” (reversible heat generated by internal chemical reactions) can further reduce this error to a highly accurate 0.1 °C to 0.75 °C range. For instance, one study testing an EKF-based lumped model reported a core temperature RMSE of 0.37 °C and a maximum error of 1.28 °C.
• Limitations: These models can lose accuracy and drift if the battery’s physical parameters (like internal resistance) change due to battery aging or extreme temperature shifts.

2. Data-Driven Machine Learning Models Advanced deep learning models can achieve exceptionally high accuracy by learning complex, nonlinear thermal patterns, provided they are trained on robust datasets.

• Sequential Architectures: Recurrent neural networks excel here because they remember historical thermal states. Bidirectional Long Short-Term Memory (BiLSTM) networks have demonstrated state-of-the-art accuracy, achieving an RMSE as low as 0.018 °C under dynamic loads—outperforming standard LSTM models (0.028 °C).
• Lightweight Architectures: Newer, highly efficient models designed for onboard vehicle computers, such as the Kolmogorov-Arnold Network (KAN), have achieved testing RMSEs of roughly 0.047 °C.
• Limitations: Without proper feature engineering or if operating completely outside of their training data, purely data-driven models can struggle with unseen scenarios, sometimes resulting in higher average errors ranging from 0.9 °C to 3.4 °C.

3. Hybrid and Physics-Informed Models (PINNs) Hybrid models are emerging as the most robust and accurate solution overall. By utilizing physical thermodynamic equations as a baseline, the neural network acts as a “corrector” to compensate for unmodeled dynamics and parameter uncertainties.

• These combined methods routinely achieve accuracies between 0.08 °C and 0.6 °C.
• In one test, integrating a basic physical model with a feedforward neural network (LP+FNN) reduced the average RMSE to 0.3 °C–0.6 °C, down from the 0.8 °C–1.7 °C error rate seen when using the physical model alone.
• A lightweight edge-computing PINN called ThermaPhysLite [4] achieved an RMSE of 0.15 °C to 0.20 °C in offline validation, and maintained an impressive 0.17 °C to 0.24 °C RMSE when quantized and deployed on actual embedded hardware.
• Some advanced hybrid approaches, like Deep Koopman observers fused with an EKF, have achieved errors as low as 0.13 °C [1]. Furthermore, by utilizing transfer learning, some models have successfully driven the RMSE down to <0.1 °C when the physical thermal parameters of the target battery are properly identified. [5]

What are the limitations of virtual temperature sensors?

While virtual temperature sensors provide a solution for monitoring internal battery states without invasive hardware, they face several distinct limitations depending on the underlying methodology used:

Data Dependency and Generalization Issues Purely data-driven machine learning models rely heavily on the quantity and quality of their training data, which is labour-intensive and costly to acquire through experimental sensor embedding. Because they act as “black boxes” without physical interpretability, they struggle to extrapolate outside their specific training domains. If a vehicle encounters unseen ambient temperatures or dynamic loads not present in the training data, these models can easily overfit and generate unreliable predictions that may even violate fundamental thermodynamic laws. This lack of explainability also makes it difficult to certify them for functional safety-critical Battery Management Systems (BMS).

Parameter Drift and Aging Physics-based models require extensive parameter identification for factors like thermal resistance and heat capacity. However, as a battery ages, its physical and electrical parameters—especially internal resistance—change significantly. If a virtual sensor is not equipped with online adaptation mechanisms to account for this degradation, the temperature estimation will experience severe accuracy drift over the battery’s lifespan, with unadjusted models showing errors (RMSE) rising as high as 15 °C in degraded cells.

The “Sim2Real” Gap To avoid the high cost of experimental data collection, engineers often train machine learning estimators using synthetic data generated from physical simulations. However, this introduces a simulation-to-reality (sim2real) gap. Any unmodeled dynamics, simplified assumptions, or mismatched parameters in the simulation will cause the trained virtual sensor to suffer from increased errors when deployed on a real-world physical battery. Addressing this requires complex hybrid training methods that demand high expertise in both deep learning and physical thermodynamics.

Hardware and Measurement Dependencies Virtual sensors are ultimately bound by the accuracy of the physical hardware they rely on. Because they calculate temperature using inputs like terminal voltage, current, and surface temperature, any noise, bias, or failure in those standard BMS sensors will directly degrade the virtual temperature estimation. Furthermore, soft sensors still require benchmark data from physical “hard sensors” (like inserted thermocouples) for their initial offline calibration and validation.

Limitations of Impedance-Based Virtual Sensors For virtual sensors that specifically use Electrochemical Impedance Spectroscopy (EIS) to infer temperature, the algorithms typically only estimate a volume-averaged bulk temperature, which can easily underestimate localized hotspots forming inside the cell. Additionally, because the impedance of a battery tends to converge as it gets hotter, this method loses accuracy at elevated operating temperatures, and it generally struggles to maintain precision when applied to high-power, low-impedance battery cells.

Who has deployed virtual temperature sensors?

Tesla is sometimes said to use virtual temperature sensors across the Model 3 and Y packs, on the basis that each module carries only a couple of physical thermistors for far more cells than that.  That sparse sensor count implies some form of inference is happening. However, we have not found a primary source confirming it, eg: a patent, technical paper, or official documentation. Treat this one as unconfirmed until a proper source turns up (do drop us a line if you have a clear reference).

Mercedes-AMG’s GT 4-Door Coupé is a confirmed case. Mercedes states that, for the first time in one of its production vehicles, the BMS derives temperatures – including within the cell – from a mathematical model based on charging current and power, rather than from a physical sensor at that location. The approach was proven first on the record-breaking CONCEPT AMG GT XX before reaching production [6].

Note: image shows the modular pack from the 2026 Mercedes-AMG GT 4-door Coupe.

Open questions worth a follow-up post

This piece covers the modelling theory reasonably well, but it stops short of being a practitioner-level reference. A few things we’d like to dig into separately:

• Who else is deploying this? Mercedes-AMG has confirmed it; Tesla is rumoured to. Which other OEMs or BMS suppliers are running virtual temperature sensors in production today, and how would we even find out, given most manufacturers don’t publish this level of BMS detail?
• What does this cost in compute? Can an LSTM, PINN, or Koopman observer actually run in real time on production-grade BMS silicon, or does it need dedicated AI accelerator hardware? Where’s the line between what’s been demonstrated offline and what’s deployable on a microcontroller with an automotive BOM cost target?
• How does this clear functional safety? With no physical sensor at the same location to cross-check against, how do virtual sensors get argued through ISO 26262 / ASIL for a safety-relevant measurement? Is there an accepted redundancy pattern, or is this still an open problem for certification?
• What’s the actual business case? Fewer thermocouples and less wiring harness should save cost and weight, but against that sits the compute, validation, and certification burden. Has anyone published a real £/kg comparison, or is this still a qualitative argument?
• How is accuracy maintained in the field? The papers describe online adaptation mechanisms for parameter drift in the lab. What does that look like in a production vehicle over a 10–15 year life – periodic re-calibration against a handful of surviving physical sensors, fleet-data retraining, something else?
• Where does interpolation end and a “virtual sensor” begin? Plenty of existing BMS already infer untested cell temperatures from a sparse set of physical sensors using simple interpolation. Is that meaningfully different from the ML/PINN approaches discussed here, or just a simpler point on the same spectrum?

We have perhaps asked more questions than we have answered, some of these too significant to fully resolve here. We will add to this and if you want to answer any of these for us in the meantime then please do write a post.

References

1. Abdullah, M.H.; Kandil, S.M. Deep Koopman Observer for Lithium-Ion Battery Temperature Estimation. World Electr. Veh. J. 2026, 17, 310. https://doi.org/10.3390/wevj17060310
2. Jinasena  A, Spitthoff  L, Wahl  MS, Lamb  JJ, Shearing  PR, Strømman  AH and Burheim  OS (2022) Online Internal Temperature Sensors in Lithium-Ion Batteries: State-of-the-Art and Future Trends. Front. Chem. Eng. 4:804704. doi: 10.3389/fceng.2022.804704
3. Madani, S.S.; Shabeer, Y.; Fowler, M.; Panchal, S.; Ziebert, C.; Chaoui, H.; Allard, F. Physics-Informed Temperature Prediction of Lithium-Ion Batteries Using Decomposition-Enhanced LSTM and BiLSTM Models. World Electr. Veh. J. 2026, 17, 2. https://doi.org/10.3390/wevj17010002
4. Liu, Z.; Wang, Y.; Gao, P.; Luo, H.; Cai, T.; Su, G.; Wang, Z.; Meng, Y. Online Internal Temperature Estimation Method for Prismatic Li-Ion Battery Using Embedded Physics-Informed Neural Networks. Batteries 2026, 12, 189. https://doi.org/10.3390/batteries12060189
5. Yusheng Zheng, Wenxue Liu, Yunhong Che, Ferdinand Grimm, Jingyuan Zhao, Xiaosong Hu, Simona Onori, Remus Teodorescu, Gregory J. Offer, Merging Physics-Based Synthetic Data and Machine Learning for Thermal Monitoring of Lithium-ion Batteries: The Role of Data Fidelity, Systems and Control, https://doi.org/10.48550/arXiv.2509.10380
6. The new Mercedes-AMG GT 4-Door Coupe: Revolutionary performance. Maximum intensity, Mercedes Media