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Data-Driven Permanent Magnet Temperature Estimation in Synchronous Motors with Supervised Machine Learning
Wilhelm Kirchgässner, Oliver Wallscheid, Joachim Böcker
TL;DR
PMSM magnet temperature is difficult to monitor directly, despite its importance for avoiding overheating and motor deterioration. This paper empirically evaluates supervised ML models on test-bench data for high-dynamic temperature estimation. The reported results indicate that data-driven models can match classical thermal-model performance, with OLS offering a strong accuracy–size balance among ML approaches.
Problem
Direct rotor-temperature sensing is not economically and technically feasible for industrial series production, although precise magnet-temperature estimates matter because overheating can severely deteriorate a PMSM.
Source: §I · p. 1Method
Results
Classical supervised learning achieves state-of-the-art accuracy on high-dynamic drive cycles, while OLS combines one of the best accuracies with by far the fewest parameters among ML methods.
Source: §V · p. 8Takeaways & Limitations
For this application, OLS is the first-choice ML approach after optimal moving-average factors are found during feature engineering; the authors expect deep neural networks to gain prevalence as data and computing capabilities grow.
Source: §V · p. 8
Abstract
from arXiv · showhide
Monitoring the magnet temperature in permanent magnet synchronous motors (PMSMs) for automotive applications is a challenging task for several decades now, as signal injection or sensor-based methods still prove unfeasible in a commercial context. Overheating results in severe motor deterioration and is thus of high concern for the machine's control strategy and its design. Lack of precise temperature estimations leads to lesser device utilization and higher material cost. In this work, several machine learning (ML) models are empirically evaluated on their estimation accuracy for the task of predicting latent high-dynamic magnet temperature profiles. The range of selected algorithms covers as diverse approaches as possible with ordinary and weighted least squares, support vector regression, $k$-nearest neighbors, randomized trees and neural networks. Having test bench data available, it is shown that ML approaches relying merely on collected data meet the estimation performance of classical thermal models built on thermodynamic theory, yet not all kinds of models render efficient use of large datasets or sufficient modeling capacities. Especially linear regression and simple feed-forward neural networks with optimized hyperparameters mark strong predictive quality at low to moderate model sizes.
I. INTRODUCTION
PMSM magnet temperature matters for safe, high-utilization operation, but direct rotor-temperature sensing is not yet economically and technically feasible for industrial series production. This work evaluates data-driven ML estimation as an alternative to physically motivated methods, with potential application beyond magnet temperature.
I. INTRODUCTION
- Precise runtime temperature information matters because overheating can severely deteriorate the motor, while the thermal state informs control and power derating.The paper links better-informed control to watching for critical operation and applying power derating.Source: §I · pp. 1–2
- Rotor magnets are failure-prone, and overheating can irreversibly demagnetize them; direct rotor monitoring remains outside usual economic and technical boundaries for industrial series production.The paper identifies infrared thermography and shaft-mounted thermocouples as examples of direct techniques that fall short of series production.Source: §I · p. 1
- CFD and heat-equation FEA offer rigorous modeling but demand too much computation for real-time monitoring; LPTNs are lightweight but require expert choices of structure and parameters.LPTNs reduce degrees of freedom to meet real-time requirements, forfeiting some structural and parameter interpretability.Source: §I · p. 1
- The paper empirically evaluates ML models using observational data without a priori heat-theory knowledge, with runtime demand scaling less drastically with model complexity.The authors argue this makes domain knowledge less relevant and allows insights to transfer to neighboring fields such as power electronics and batteries.Source: §I · p. 2
- Although magnet temperature is the sole target here, the considered ML models are also presented as applicable to torque, stator temperatures, and additional spatial temperature regions.The paper describes adding spatially targeted regions as a virtually minor design overhead.Source: §I · p. 2
Related Work
Prior PMSM temperature-estimation work includes recurrent and temporal-convolutional networks as well as preprocessed linear regression, while this paper broadens the comparison to intermediate-complexity regression methods. It also introduces the regression-algorithm families and the closed-form basis of OLS.
Related Work
- Earlier studies evaluated recurrent networks on low-dynamic profiles, temporal convolutional networks on high-dynamic data, and linear regression after low-pass filtering.Source: Related Work · p. 2
- The paper targets the gap between fast, simple least-squares methods and computation-heavy deep learning by systematically evaluating intermediate-complexity regression models.Experiments use the dataset from,.Source: Related Work · p. 2
II. REGRESSION ALGORITHMS
- Regression approaches span linear models, feed-forward networks, decision trees, support vector machines, k-nearest neighbors, and ensembles, differing in capacity, assumptions, parameters, update rules, and runtime.Source: §II · p. 2
A. Ordinary Least Squares
- OLS assumes a linear relationship between input vectors and real-valued outputs, and its coefficients can be inferred through a closed-form solution minimizing residual sum of squares.Ridge and LASSO add ℓ2 and ℓ1 penalties, respectively, but are excluded here because they were shown inefficient for this dataset.Source: A. Ordinary Least Squares · pp. 2–3
B. Weighted Least Squares
The section presents weighted least squares as an observation-weighted OLS variant motivated by the greater importance of estimating high temperatures accurately. It then describes epsilon-support vector regression and its tolerance-band and slack-variable design.
B. Weighted Least Squares
- WLS modifies OLS by applying an observation weight matrix W when estimating the coefficients.Source: B. Weighted Least Squares · p. 3
- WLS weights observations to improve robustness and, in this application, to prioritize accurate estimates of high temperatures because overheating avoidance matters more than lower thermal states.The authors note that penalizing underestimates more than overestimates through gradient descent is not covered.Source: B. Weighted Least Squares · p. 3
C. Epsilon-Support Vector Regression
- Epsilon-SVR uses a regularized linear model in a kernel-transformed feature space, accumulating cost only for prediction errors that exceed the ε threshold.Nonnegative slack variables allow support-vector deviations beyond the ε-band; C balances coefficient regularization against ε-band alleviation.Source: C. Epsilon-Support Vector Regression · p. 3
D. K-Nearest Neighbors
K-nearest neighbors estimates new samples from nearby stored training observations, while randomized-tree ensembles reduce variance by averaging diverse trees. The methods differ in where computation occurs and in how tree splits are selected.
D. K-Nearest Neighbors
- k-NN stores training observations and estimates a new sample by averaging nearby points, with neighborhoods determined by Euclidean distance and optionally weighted by distance.Its lazy-learning design defers computation to inference, which counteracts real-time capability.Source: D. K-Nearest Neighbors · p. 3
E. Randomized Trees
- Random forests and extremely randomized trees ensemble trees trained on random observation and feature subsets, with averaging reducing variance at the cost of additional bias.Source: E. Randomized Trees · p. 3
- ETs add randomness by drawing split thresholds, whereas RFs search for the most discriminative thresholds; this further amplifies ET variance reduction and bias increase.Source: E. Randomized Trees · p. 3
F. Neural Network Architectures
The paper describes feed-forward neural networks that transform inputs through layers into predictions, with trainable weights and activation functions. It also includes recurrent and temporal-convolutional architectures for comparison and uses measured motor signals and magnet-surface temperature in the modeling task.
F. Neural Network Architectures
- MLPs transform regressors through layers into predictions, with each layer applying trainable weights and an activation function.The paper notes that MLPs are end-to-end differentiable and can be optimized with stochastic gradient descent and its variants.Source: F. Neural Network Architectures · p. 3
- Batch normalization scales each training batch after every layer, while self-normalizing networks use scaled ELUs and a dropout variant to normalize activations implicitly.The paper also identifies weight decay and dropout as regularization schemes.Source: F. Neural Network Architectures · pp. 3–4
- LSTM and TCN architectures use neighboring-observation dependencies without explicit feature engineering, but have more parameters than MLPs.Their performance is reported for comparison with benchmarks, following extensive optimization in prior work.Source: F. Neural Network Architectures · p. 4
III. BLACK-BOX THERMAL MODELING
- The models use commonly accessible traction-drive signals to estimate permanent-magnet surface temperature, recorded wirelessly from a 50 kW PMSM test bench.The dataset contains 139 hours of recordings sampled at 2 Hz.Source: §III · p. 4
A. Data Preprocessing and Feature Engineering
The preprocessing adds exponentially weighted moving averages and standard deviations to standardized input signals, supplying trend information for models that assume independent and identically distributed data. This feature engineering is motivated by the similarity between EWMA filtering and the low-pass RC behavior of lumped-parameter thermal networks.
A. Data Preprocessing and Feature Engineering
- EWMA and EWMS features are computed for each input parameter at every timestep and supplied to the models.Input representations are standardized using the training-set sample mean and unit variance.Source: A. Data Preprocessing and Feature… · p. 4
- The EWMA span controls the weighting of preceding observations, with those within span s accounting for about 86.5% of total weight.A computationally efficient recursive form is available, and a corresponding efficient EWMS form exists.Source: A. Data Preprocessing and Feature… · p. 4
- Adding trend features addresses the mismatch between the models’ independent-and-identically-distributed assumption and the motor’s dynamic thermal behavior.The paper describes the additional features as supporting sufficiently accurate approximating functions.Source: A. Data Preprocessing and Feature… · p. 4
B. Analogy to LPTN RC Circuits
- LPTNs behave like low-pass filters, motivating EWMA filtering of sensor time series to create regressors with similar patterns.The paper reports that this analogy was empirically confirmed in.Source: B. Analogy to LPTN RC Circuits · p. 4
- The PCA plot shows no clear high- versus low-temperature trend, particularly in the middle region.It projects measurements onto the first two principal components, with recording sessions ordered by increasingly random excitation from left to right.Source: Figure 2 · p. 5
C. Cross-Validation
Generalization is evaluated on an unseen seven-hour test set, with a separate training-data validation split for MLP early stopping. Prediction quality is measured using error metrics and maximum test-set deviation.
C. Cross-Validation
- Generalization error is evaluated on seven hours of data withheld from training.For MLPs, 10% of the training set is also withheld as a validation set for early stopping.Source: C. Cross-Validation · p. 5
- The reported scores include MSE, MAE, R², and the maximum test-set deviation measured by the ℓ∞ norm.The paper identifies the maximum deviation as an important indicator of temperature-estimation quality.Source: C. Cross-Validation · p. 5
D. Cross-Validation for Hyperparameter Tuning
Hyperparameters are tuned with Bayesian optimization using stratified group-three-fold cross-validation, keeping measurement sessions separate across folds and excluding the final test set from tuning. The model search uses four optimized span values and model-specific hyperparameters, with constrained intervals and training settings for the neural networks.
D. Cross-Validation for Hyperparameter Tuning
- Bayesian optimization uses a Gaussian-process surrogate to map hyperparameters to test-set error and balance exploration with exploitation.Its objective is average MSE over stratified group-three-fold cross-validation; folds distribute sessions with similar maximum temperatures evenly and keep each measurement session in one fold.Source: D. Cross-Validation for Hyperparameter… · p. 5
- The final test set is excluded from hyperparameter cross-validation, and normalization is fitted separately within each fold using its training observations.Keeping sessions intact across folds is intended to mitigate overfitting.Source: D. Cross-Validation for Hyperparameter… · p. 5
E. Hyperparameters and Intervals
- Each model optimizes four span values; the count is fixed to preserve comparability with prior work,.The cited prior work found four spans to balance modeling accuracy and computational demand, while interval bounds are manually set to constrain runtime.Source: E. Hyperparameters and Intervals · p. 5
- Random forests and extremely randomized trees tune ensemble size, maximum depth, and split and leaf sample thresholds to control growth and overfitting.The two tree ensembles use the same hyperparameters.Source: E. Hyperparameters and Intervals · p. 5
- MLP search bounds limit layer and unit counts to constrain model parameters and flexibility.The paper notes that smaller neural networks can still reach satisfactory performance.Source: E. Hyperparameters and Intervals · p. 5
- MLP training considers six optimizers and reduces learning rate after 10 epochs without training-loss improvement.Batch size increases from 32 to 64 and then 128 after successive 33-epoch intervals, for 99 epochs absent early stopping.Source: E. Hyperparameters and Intervals · p. 6
IV. EXPERIMENTAL RESULTS
Optimized models span a clear accuracy–size trade-off: CNNs lead in MSE, LPTNs in maximum error and compactness, while OLS offers strong accuracy with few parameters. OLS, ET, and MLP predictions are shown as potential intermediate approaches for automotive temperature estimation.
A. Model Performance
- CNNs have the best MSE, LPTNs the lowest ℓ∞ norm with 46 parameters, and OLS combines strong MSE, low maximum deviation, and 109 parameters.The benchmark table reports test-set values, while the prose gives these comparisons.Source: Table III · p. 6, A. Model Performance · p. 6
- OLS, ET, and MLP predictions are presented as intermediate approaches until automotive hardware can support deep learning.Their predictions are shown for the first test-set profile in Fig. 3.Source: A. Model Performance · p. 6
- k-NN, RF, and SVR did not find a sufficient function approximation despite optimization that pushed model size toward its allowed upper bounds.Source: A. Model Performance · p. 6
B. Learn Curves
OLS, MLP, and ET reach high performance with about 70 hours of training data, while SVR performance appears to diminish as data increases. For high-temperature estimation, WLS subtly shifts residual behavior, but further load profiles are needed to assess its advantage over OLS.
B. Learn Curves
- All models plateau at about half the training-set size except SVR, whose performance appears to diminish as training data increases.The authors suggest limited modeling capacity may explain SVR’s difficulty mapping the observed operating points.Source: B. Learn Curves · p. 6
- OLS, MLP, and ET achieve high performance with around 70 hours of training data, with no significant gains from more data.Source: B. Learn Curves · p. 6
C. Error Residuals
- Because avoiding overheating is the goal, accurate and robust estimates at high temperatures matter more industrially than estimates at low temperatures.The paper identifies subsampling toward high temperatures and cost-function adjustment as possible approaches.Source: C. Error Residuals · p. 6
- WLS subtly shifts residuals: outlying predictions often fall at lower temperatures, while predictions at higher temperatures show less variance.Observations are weighted from 0.33 to 1 according to closeness to the dataset’s minimum and maximum magnet temperatures.Source: C. Error Residuals · p. 7
- Assessing WLS’s advantage over OLS requires additional load profiles with more temperature variation.Source: C. Error Residuals · p. 7
V. CONCLUSION AND OUTLOOK
Classical supervised learning algorithms achieve state-of-the-art estimation accuracy during high-dynamic drive cycles, and OLS combines strong accuracy with a very small model. The authors identify cross-motor generalization as unresolved and expect deep neural networks to gain prevalence as data and computing become more available.
V. CONCLUSION AND OUTLOOK
- With rich test-bench or production datasets, certain classical supervised learning algorithms can monitor PMSM component temperatures in real time without motor-sheet specifications or special domain expertise.Source: §V · p. 7
- Fig. 5 presents OLS and WLS error residuals on the test set.Source: Figure 5 · p. 8
- Classical supervised learning algorithms achieve state-of-the-art estimation accuracy during high-dynamic drive cycles.Source: §V · p. 8
- OLS has one of the best accuracies and by far the fewest model parameters among ML methods, making it the first choice after moving-average factors are optimized during feature engineering.Source: §V · p. 8
- Deep neural networks are expected to become prevalent as their scalability, measurement-data availability, and computing capabilities increase.Source: §V · p. 8
- Generalization across motors from one or multiple manufacturers remains an open question pending a suitably diverse dataset.Source: §V · p. 8