Towards Digital Twin Modeling and Applications for Permanent Magnet Synchronous Motors

Abstract

This paper explores the potential of Digital Twin (DT) technology for Permanent Magnet Synchronous Motors (PMSMs) and establishes a foundation for its modeling and applications. While DTs have been widely applied in complex systems and simulation software, their use in electric motors, especially PMSMs, remains limited. This study examines physics-based, data-driven, and hybrid modeling approaches and evaluates their feasibility for real-time simulation, fault detection, and predictive maintenance. It also identifies key challenges such as computational demands, data integration, and the lack of standardized frameworks. By assessing current developments and outlining future directions, this work provides insights into how DTs can be implemented for PMSMs and drive advancements in industrial applications.

1. Introduction

Permanent Magnet Synchronous Motors (PMSMs) have become a cornerstone in various applications, ranging from electric vehicles (EVs) to industrial machinery, due to their high efficiency, power density, and robust performance characteristics [1,2]. With the rapid advancement of digital technologies, the Digital Twin (DT) has the potential to enhance the design, monitoring, and optimization of systems, though its application in electric motors remains largely unexplored. A DT creates a real-time, digital counterpart of a physical motor, allowing for continuous interaction between the physical and virtual models, which is useful for predictive maintenance, performance optimization, and fault detection.

The application of DT technology in electric motors, especially PMSMs, offers significant advantages. It allows for the detailed analysis of motor behavior under various operating conditions without the risks associated with physical testing. By integrating real-time data with simulation models, DTs can provide information about factors such as electromagnetic performance, thermal effects, and mechanical stresses, which are difficult to assess through conventional methods [3,4]. This virtual prototyping is essential for optimizing the motor’s design and operational parameters, thus extending its lifespan and reducing maintenance costs [5].

Recent studies have explored various methods for implementing DTs in PMSMs, utilizing modeling techniques such as finite element analysis (FEA), machine learning, and real-time data integration. Some approaches use neural networks to predict thermal behavior, ensuring motor efficiency and preventing overheating, while others integrate advanced control algorithms to optimize performance in real time [6]. Moreover, DT technology plays a crucial role in fault diagnosis and health management by continuously monitoring the motor’s condition and comparing real-time data with its digital counterpart. This enables early issue detection and preventive actions, which are particularly valuable in critical applications such as coal mine belt conveyors, where motor reliability is essential [7].

The concept of the DT has gained tremendous traction in recent years, evidenced by a sharp rise in related publications. As shown in Figure 1, a search on Google Scholar reveals an exponential increase, with references to DTs growing from 135 in 2004 to 39,800 in 2024. This growth is driven by technological advancements, particularly in areas that facilitate real-time data exchange, such as sensor systems, cloud computing, and AI. Among these, the IoT plays a critical role in enabling DT implementations by providing the necessary data infrastructure for real-time monitoring and interaction. While IoT and DT are distinct fields, their interconnection is crucial: IoT supplies the data whereas the DT uses these data for simulation, optimization, and predictive analysis. The figure also shows that while publications on Industry 4.0 peaked in 2021, both IoT and DT research continue to grow. Notably, the number of DT-related publications in 2023 is almost double that of IoT, which indicates a faster growth trajectory.

The history of DTs can be divided into three stages: formation, incubation, and growth [8,9]. The formation stage began with the concept’s introduction by Grieves in 2003, followed by a period of limited publication due to technological constraints. The incubation stage, from 2011 to 2014, saw foundational developments such as NASA’s formal definition of DTs in 2012 and early applications in aerospace [10]. In 2014, DTs entered the growth stage, marked by broader recognition of their potential in diverse fields and the publication of the first white paper on DT applications. By 2017, Gartner identified DTs as one of the top ten technological trends, reflecting their rising prominence and practical feasibility [11].

This growth aligns with advancements in Industry 4.0 and the IoT, as illustrated in Figure 1, where publications on these technologies also surged. While the IoT provides the connectivity and real-time data acquisition capabilities necessary for DT implementation, the DT extends beyond data collection by enabling real-time decision-making, predictive analytics, and operational optimization. The integration of cloud computing and AI further strengthens DT applications by enhancing computational efficiency and the ability to model complex systems. This convergence underscores the importance of continued research into DTs as a key driver of industrial innovation.

DT technology has already seen widespread adoption in other fields, notably in the automotive industry. In automotive applications, DTs are used to simulate and optimize vehicle dynamics, predict maintenance needs, and enhance the design process through real-time data integration. The ability to create a detailed virtual model of an entire vehicle or its subsystems allows manufacturers to conduct extensive testing and validation without the need for physical prototypes, significantly reducing development time and costs. This success in the automotive sector highlights the potential for DT technology to bring similar benefits to the realm of PMSMs and other electrical machinery. In particular, DT-enabled predictive maintenance has been shown to reduce unplanned downtime in manufacturing systems, production lines, and heavy machinery by up to 45% while lowering maintenance costs by 30% [12]. For motors, DTs continuously monitor performance and analyze sensor data on temperature, vibration, and electrical currents to detect early signs of faults, such as overheating, misalignment, or bearing wear.

Despite its promising potential, Digital Twin (DT) technology is still relatively new in the field of electric motors. Companies such as ANSYS and Siemens have integrated DT capabilities into their analysis software, TwinBuilder and Simcenter Amesim, respectively. However, while DT technology has seen significant advancements in other industries, its application to electric motors—particularly PMSMs—remains in its early stages, with limited published literature on its development. Although ongoing research is exploring the full capabilities of DT technology, its implementation is currently confined to commercial software, with no dedicated studies in the literature on how to build DTs for electric motors or how they can be effectively utilized. The novelty of the DT in this domain means that many methodologies and frameworks are still under development, and there is considerable scope for innovation and refinement.

Additionally, there is ambiguity in the definition of DT. Often, the DT model is confused with the digital shadow (DS) and digital model (DM). Reviews in [13,14] explain this issue in detail. The distinction lies in the data flow—whether the “twin” receives feedback from or sends it to the physical object and whether this feedback is managed automatically or manually. Researchers in [15] have proposed the term “Digital Twin Framework” (DTF) to encapsulate all relevant work and avoid confusion. Figure 2 below illustrates the differences between these three categories.

In this paper, the aim is to explore the implementation of DT technology in PMSMs and its effects. This study does not intend to compile all literature on DT modeling and applications for PMSMs, as the available data are very limited and DT technology is considered relatively new. As seen in the References section, most of the literature was published recently, with many contributions coming from the year 2024. The term DT is often confused with other terminologies, making it challenging to review previous literature with a focus on the “correct” DT, which involves automatic data flow to and from the physical counterparts. Therefore, the literature included here specifically contains the term "Digital Twin", without categorizing it further, yet the discussion strives to remain faithful to its proper terminology. By reviewing existing studies, this paper seeks to provide a guideline on how DTs can be potentially used in PMSMs and to highlight potential directions for future work. This should serve as a foundational reference for researchers and practitioners looking to explore and expand the applications of DT technology in electric motors. Section 2 delves into the fundamentals of DT, elaborating on what can be considered DT. Section 3 focuses on modeling techniques that can be used to build DTs for PMSMs. Section 4 discusses DT applications in the electric motor field, especially PMSMs. Section 5 explores some challenges and possible future research directions.

2. Fundamentals of Digital Twin Technology

Despite the ambiguity in the definition of DT technology and differing opinions regarding its components, there is no consensus on the properties of DTs and their corresponding component architecture [17,18]. However, as illustrated in Figure 2, it can be concluded that a DT must consist of at least three core components: the physical model, the digital model, and the communication protocol between them. DT technology represents a significant advancement in the digitization of physical assets and processes, enabling real-time monitoring, simulation, and optimization. This technology functions by creating a virtual replica of a physical system that is continuously updated through data streams, allowing for a synchronized view of the asset’s condition and behavior. Initially developed for aerospace applications, DTs have since been widely adopted across various industries, including manufacturing, healthcare, and notably in electric motors such as PMSMs.

The core components of DTs include the physical model, the virtual model, data integration, and the communication and feedback mechanisms that connect the physical and digital worlds. The physical model refers to the real-world asset or system, such as a PMSM, which is equipped with various sensors that provide continuous data on operational parameters like temperature, torque, current, and voltage. These sensors form the foundation of the DT system, supplying the raw data necessary to update the virtual model. The accuracy and granularity of these data are critical for ensuring the fidelity of the DT. For example, PMSMs, which are extensively used in applications ranging from electric vehicles to industrial machinery, rely on sensors to monitor their electromagnetic, thermal, and mechanical states. Without high-quality sensor data, a DT cannot accurately reflect the physical model’s current or future behavior [19,20].

DTs rely on real-time data acquisition technologies, including IoT-enabled sensor networks, to facilitate continuous synchronization between physical and digital models. As illustrated in Figure 3, a general DT framework for electric motors should consist of a physical system (the actual motor), sensors for real-time data collection, a data processing unit, cloud-based digital storage, and a digital model for analysis and decision-making. Sensors capture real-time data from the motor, measuring parameters such as temperature, current, and torque, which are then processed and stored on a central server. These data update the digital model, which performs simulations to predict potential faults, optimize control strategies, and improve motor efficiency. The insights generated by the digital model are then communicated back to the motor’s controller, enabling real-time adjustments to its operation. This continuous feedback loop enhances predictive maintenance, reduces downtime, and improves overall system performance, making DTs highly beneficial for electric motor applications.

The virtual model is the digital counterpart of the physical entity, constructed using a combination of physics-based simulations and data-driven methods. Physics-based models rely on the fundamental principles of mechanics, electromagnetism, and thermodynamics to replicate the behavior of the physical system. These models are particularly effective for systems like PMSMs, where the physical laws governing motor operation are well understood. For example, electromagnetic and thermal modeling techniques are employed to predict motor performance under various operating conditions. However, these models can be computationally expensive, especially when simulating complex scenarios in real time [21].

To complement physics-based models, data-driven approaches such as machine learning are increasingly employed. These techniques utilize historical and real-time data to predict outcomes, detect anomalies, and optimize performance. Data-driven models excel at handling large datasets and adapting to new data trends more effectively than purely physics-based models. For instance, a neural network might be trained to predict motor faults based on historical failure data and real-time sensor readings. The use of hybrid models, which combine physics-based and data-driven approaches, is gaining popularity as it offers both the accuracy of physical laws and the flexibility of data-driven predictions [22]. More details about modeling are available in Section 3.

Data integration is another crucial component of a DT, ensuring continuous synchronization between physical and digital models. To facilitate efficient and scalable communication, the study in [23] utilizes MQTT and OPC UA as core data exchange protocols. MQTT, a lightweight publish–subscribe messaging protocol, enables low-latency and bandwidth-efficient data transmission, making it ideal for real-time updates from IoT-enabled devices. Meanwhile, OPC UA provides a standardized and secure framework for interoperability between industrial systems, allowing structured data exchange between PLCs, sensors, and cloud platforms. Together, these protocols enable seamless data flow in smart manufacturing, where MQTT efficiently transmits sensor data, while OPC UA ensures compatibility across diverse industrial devices. In PMSM applications, this combination allows the DT to collect and process real-time metrics such as speed, voltage, and current, ensuring accurate synchronization for predictive maintenance and optimized motor performance.

Effective communication between the physical and virtual models ensures that any changes in the real-world system are immediately reflected in the digital representation. This two-way communication also enables control feedback, allowing adjustments in the physical system based on insights gained from the virtual model. For example, if a DT of a PMSM detects an anomaly that indicates an impending failure, the system can automatically adjust operational parameters, such as reducing motor speed or increasing cooling to mitigate the issue [24]. This feedback loop is a defining feature of DT technology, providing not only real-time monitoring but also the ability to dynamically optimize and control physical assets.

Lastly, evaluation metrics are required to assess the accuracy and reliability of a Digital Twin. The studies in [25,26] demonstrate the importance of using root mean square error (RMSE) and mean absolute error (MAE) for evaluating model performance. In [25], RMSE and MAE assess the precision of machine learning-based vector control in PMSMs. Meanwhile, Ref. [26] applies these metrics to validate the effectiveness of a surrogate model that optimizes the design of an interior IPMSM for electric vehicles. Similarly, these evaluation methods help Digital Twin models reflect the physical system’s behavior accurately, which enables effective optimization and predictive analysis.

The continuous feedback loop created between the physical and virtual models leads to the final core component of a DT: its ability to learn and evolve over time. As a DT receives more data from the physical model, it becomes more accurate and capable of making better predictions and recommendations. This self-improving aspect of DTs is particularly important in industries where systems frequently experience changes in operating conditions, such as manufacturing, energy, and transportation. In the context of PMSMs, DTs can and should adapt to changes in load, temperature, and mechanical stress, continuously optimizing performance to reduce energy consumption and prevent failures.

3. Modeling Techniques for Digital Twins in PMSMs

Accurate and efficient modeling is fundamental to the success of DTs in PMSMs. Two main approaches dominate the modeling landscape: physics-based models and data-driven models. Physics-based models rely on established physical laws—such as electromagnetic and thermal dynamics—to simulate the real-world behavior of motors. Although these models are highly accurate, they can be computationally expensive. On the other hand, data-driven models, which include machine learning algorithms, utilize historical and real-time data to predict motor behavior and diagnose potential issues. Hybrid approaches that combine the strengths of both physics-based and data-driven models are increasingly used to balance accuracy with computational efficiency.

Figure 4 illustrates the key components of a PMSM and their characteristics. Temperature plays a significant role in PMSM modeling, as it affects magnet strength, winding resistance, and overall efficiency. In addition to temperature, vibration, torque, current, flux, and their harmonics are also crucial parameters that must be considered. Comprehensive modeling of a PMSM must account for all these characteristics to accurately mimic the actual motor. This section explores the modeling techniques that can be used to generate a DT model, discusses their general applications, and also compares them.

3.1. Physics-Based Models

Physics-based models offer a robust foundation for DTs in PMSMs by capturing the essential physical dynamics of the motor. Modeling techniques like the Finite Element Method (FEM) can be incorporated into a DT model by continuously synchronizing historical and model parameters with measurements, sensor data, and historical data of the product instance [27]. FEM softwares can be used to simulate electromagnetic, thermal, and mechanical interactions within the motor, providing a high-fidelity representation of motor behavior under various operating conditions. Currently, there is a wide range of FEM software options available that even offer multiphysics solutions, such as ANSYS, Simcenter, COMSOL, and many others. Figure 5 illustrates the workflow of physics-based motor modeling. In addition to FEM, the Maxwell stress tensor and lumped parameters, which are often used for thermal modeling, are also commonly utilized.

In the case of PMSMs, high-fidelity models simulate electromagnetic force and torque, which are critical for accurately predicting the motor’s dynamic responses, including torque fluctuations and vibrations. The study in [28] highlights how FEA-based models can be developed to represent the interactions between electromagnetic fields and structural forces. The model includes a two-dimensional electromagnetic module for torque and vibration simulations and a rolling element bearing (REB) model to capture contact forces accurately under fault conditions. Another study in [29] also emphasizes the role of high-frequency models in addressing EMI concerns, showing how FEA can incorporate parameters like parasitic capacitance and eddy currents, enhancing the model’s ability to analyze EMI effects at frequencies of up to 30 MHz.

A significant advantage of physics-based modeling lies in its capability to offer precise insights into specific fault types, such as bearing faults, which contribute substantially to motor failures. Integrating electromagnetic and structural forces enables a comprehensive analysis of the motor’s response to varying load conditions and faults. By coupling electromagnetic simulations with structural models, such as those using Maxwell’s stress tensor method, it becomes feasible to observe the motor’s dynamic response in real time, ensuring high accuracy in vibration predictions, which is a crucial factor for fault detection and condition monitoring. The study in [30] further exemplifies this advantage by utilizing back-EMF estimators based on the Finite Element Method to detect inter-turn faults accurately. This approach combines thermal and electromagnetic modeling, proving effective even under complex conditions such as harmonic loads and variable speeds.

Another benefit of physics-based models is their ability to encompass thermal effects, which is an important aspect when designing PMSMs. Temperature fluctuations impact motor performance, influencing factors like magnetic flux density and electrical resistance. The study in [31] demonstrates the integration of thermal, electromagnetic, and mechanical domains in a unified optimization framework. By employing a dynamic phase-variable physics-based model combined with FEA, it effectively links motor geometry to drive circuits, accounting for heat dissipation and thermal constraints during operation. This ensures that performance metrics, such as torque ripple and harmonic distortion, remain optimized even under varying thermal loads. The study in [32] demonstrates how physics-based models can integrate multi-domain analysis, including electromagnetic, thermal, and mechanical analyses, to optimize generator design, minimizing material usage and costs while maintaining efficiency and fault tolerance.

As these models require detailed parameter inputs and complex simulations, they come at a significant computational cost. Accuracy tends to be proportional to the time required for simulations. While incorporating more factors into the model increases its complexity and setup time, detailed physical modeling with extensive data provides the advantage of enabling the analysis of various aspects [33]. However, the demand for substantial computational resources poses a challenge for real-time applications. The study in [34] presents strategies for reducing computational demands in DTs by employing machine learning algorithms and efficient simulation techniques, which facilitate faster data processing while maintaining accuracy in modeling complex systems. These methods are particularly effective in applications requiring real-time monitoring and optimization. Nonetheless, further advancements in computational efficiency are necessary to enhance their applicability in DTs.

3.2. Data-Driven Models

DTs facilitate the optimization of PMSM designs by enabling virtual prototyping and testing. Engineers can simulate various design parameters and operating conditions to identify the optimal configuration. This reduces the need for physical prototypes, saving time and resources. Data-driven models are increasingly utilized in DTs for PMSMs, as they too offer significant advantages for optimization and fault detection. These models incorporate historical and real-time data to create predictive analytics and diagnostic tools that enhance the motor’s operational efficiency without precise physical parameters.

In the context of data-driven models, different approaches—often categorized as black box, white box, and gray box models—are used based on the level of understanding and transparency of the model’s internal workings. Black box models, such as many deep learning algorithms, are highly effective at pattern recognition but provide little insight into the underlying mechanisms, making them useful where the complexity is high and the exact phenomena are not fully understood. White box models are based on known laws and equations and offer complete transparency, making them ideal where understanding and trust in the model’s predictions are required. Gray box models combine elements of both, using empirical data to fine-tune theoretical models, increasing both performance and interpretability.

By employing machine learning and deep learning algorithms, data-driven models can reveal patterns in motor behavior, thus facilitating the rapid identification of faults and potential optimizations for PMSM configurations. Figure 6 illustrates this method. Initially, data selection is performed to choose those viable for training before beginning the machine learning process.

For instance, research in [35] describes a data-driven approach for diagnosing incipient inter-turn short-circuit faults, which are among the most common issues in PMSMs. This approach employs a nonlinear auto-regressive model with exogenous inputs (NARX) to detect minor deviations in three-phase current residuals, providing early warning signs of short circuits even in systems with significant harmonic distortions. This model demonstrates high sensitivity and fault characteristic detection under challenging conditions, improving the motor’s reliability and maintenance planning.

In addition, research in [36] presents the importance of optimal sensor placement in data-driven DTs, emphasizing that data accuracy directly impacts fault detection performance. Their study employs a genetic algorithm to determine optimal sensor locations within PMSMs, improving the detectability of short-winding faults. By optimizing sensor placement, the DT achieves high fault classification accuracy with minimal sensors, thereby enhancing efficiency and reducing system complexity.

Furthermore, as illustrated in [37], data-driven models in PMSM DTs contribute to the broader goals of Industry 4.0 by enabling real-time monitoring and predictive maintenance. These models support virtual prototyping, allowing engineers to test and optimize motor designs under various simulated conditions, thereby reducing dependency on physical prototypes. This application of DT technology allows for significant resource savings while facilitating continuous improvement in the motor’s operational parameters.

Incorporating advanced data-driven methods like deep transfer learning, as demonstrated in [38], enhances the accuracy and practicality of fault diagnosis in real-world conditions. By combining self-sensing signal visualization with transfer learning, their model achieves a fault diagnosis accuracy of over 99%, even under variable operational conditions. Such high precision makes data-driven DTs an invaluable tool for predicting system health and extending the motor’s service life.

Despite the significant advantages of data-driven models in enhancing fault detection, optimization, and predictive maintenance, these methods have limitations. A primary challenge is the dependency on large volumes of high-quality data, which may not always be available or accurately representative of all operational conditions. Data-driven models also require extensive computational resources for training, especially when utilizing complex machine learning or deep learning algorithms. Additionally, while these models excel at pattern recognition and prediction, they often lack the interpretability of physics-based approaches, making it difficult to understand the underlying physical mechanisms of faults or optimizations. Studies have also shown that the performance of data-driven models can degrade under conditions that differ significantly from the training dataset, such as unanticipated faults or extreme operational scenarios [39,40]. Moreover, as discussed in [41], the risk of overfitting, the lack of generalizability to unseen conditions, and the challenges of integrating these models into larger systems further complicate their widespread industrial adoption, emphasizing the need for hybrid approaches that combine data-driven and physics-based methods for more robust and interpretable solutions.

3.3. Integrating Physics-Based and Data-Driven Models

The integration of physics-based and data-driven models, often referred to as hybrid modeling, has emerged recently as a promising approach to overcome the limitations inherent in each method when used independently. This method, often referred to as physics-informed, physics-assisted, or physics-based machine learning, integrates physical laws and principles into data-driven frameworks to enhance model accuracy, interpretability, and consistency with real-world system dynamics. By combining the interpretability and reliability of physics-based models with the adaptability and predictive power of data-driven techniques, as shown in Figure 7, hybrid approaches combine the strengths of both methodologies. These methods address challenges such as uncertainties in parameter estimation and unmodeled dynamics by coupling analytical models with data-driven techniques, such as neural networks, to enhance system accuracy [42].

Recent advancements, such as physics-informed neural networks (PINNs), demonstrate the potential of this integration by embedding physical laws into neural network training. By leveraging partial differential equations, PINNs constrain model predictions to ensure consistency with underlying physics. For example, in electric motors, PINNs have been applied to estimate electromagnetic responses with accuracy comparable to FEA while significantly reducing computational time [43]. The proposed PINN is trained on electromagnetic response datasets from both experimental measurements and high-fidelity FEA. The training process incorporates three key governing equations: (1) the magnetic vector potential equation, (2) Ampère’s circuit law, and (3) the residual equation for electric and magnetic field consistency. Additionally, a domain decomposition strategy with an interface loss and an adaptive learning rate annealing method is implemented to enhance convergence and accuracy. The effectiveness of the model is validated through comparisons with FEA and experimental data, demonstrating its capability to estimate electromagnetic responses with high accuracy and computational efficiency.

A hybrid approach in [44] models iron loss in PMSMs by integrating electromagnetic coupling, mechanical stress, and thermal effects. It combines pre-trained neural networks with physics-based insights to enhance iron loss predictions across varying conditions. Similarly, hybrid methods have been applied to stator insulation monitoring in inverter-fed machines [45]. By embedding time–frequency features from high-frequency switching oscillations into convolutional neural networks, these methods achieve high sensitivity and robustness in fault detection.

In optimization contexts, physics-informed Bayesian optimization (PIBO) methods have shown improvements in computational efficiency and design accuracy for electrical machines [46]. These approaches utilize Gaussian processes and FEA to optimize complex design variables such as slot fill factors, offering significant performance gains with reduced development time.

Overall, the integration of physics-based and data-driven methods enhances system modeling by combining theoretical rigor with the adaptability of data-driven approaches. However, this approach is still relatively novel and holds significant potential for further development. In the age of Artificial Intelligence (AI), future advancements in hybrid modeling are anticipated to refine these techniques further, expanding their applicability to increasingly complex and dynamic systems. A comparison between physics-based, data-driven, and hybrid models is presented in

Table 1. Comparison between modeling methods for Digital Twins.

	Physics-Based	Data-Driven	Hybrid
Based on	Physical behaviors	Historical and/or real-time data	Both physical laws and data
Common examples	FEA Maxwell’s stress tensor	Machine learning Neural networks Genetic algorithms	Physics-informed neural networks (PINNs)
Accuracy	High	Varies (high potential with optimal data and algorithms)	High (if well integrated)
Interpretability	High	Low (without domain knowledge)	High
Computational demand	High	Variable, often high during training phases	Potentially reduced through efficient integration methods
Best to use in	Detailed fault diagnostic Design optimization	Rapid fault identification Predictive maintenance	Fault prediction and prognostic System/control optimization

, while

Table 2. Summary of References on modeling techniques (Section 3.1, Section 3.2 and Section 3.3).

Ref.	Focus Area	Application	Methodology	Result
[27]	Physics-based modeling of mechatronic systems	DT modeling for a flexible manipulator	FEM Model order reduction H-infinity loop shaping control	High-fidelity modeling and control workflow. DT models to enhance real-time monitoring and control.
[28]	High-fidelity multiphysics modeling of PMSMs	Fault data generation for PMSM diagnosis	FEM Rolling element bearing model (REB) Structural dynamic modeling	A multiphysics model integrating electromagnetic, REB, and structural characteristics. High-fidelity vibration data for fault diagnosis, particularly for bearing spall faults.
[29]	High-frequency modeling of PMSMs	EMI studies in variable-speed drive systems	Behavioral modeling with impedance measurements Physics-based modeling using 3D FEM Analytical calculations for parameter extraction	Improved high-frequency PMSM model for EMI analysis. Combined behavioral and physics-based modeling for better mid-frequency accuracy.
[30]	Physics-based back-EMF modeling	Inter-turn fault detection in PMSMs	Physics-based back-EMF estimator FEM coupled with thermal analysis	Physics-based back-EMF estimation method for inter-turn fault detection. Real-time fault detection across a wide speed range.
[31]	Physics-based multiphysics modeling for motor-drive optimization	Multi-objective optimization of PMSMs for reduced torque ripple and improved efficiency	FEM Physics-based phase variable modeling Hybrid Genetic Algorithm–Particle Swarm Optimization (GA-PSO)	Optimization framework integrating PMSM design with drive dynamics. Reduced copper area by 18.5% and magnet area by 1.75%. Reduced torque variance by 29.7% and speed ripple by 75%. Decreased total harmonic distortion (THD) by 41.6%.
[32]	Physics-based multiphysics modeling	Cost and efficiency optimization of a 55 kW PMSG for wind energy conversion	FEM Multiphysics modeling (electrical, mechanical, thermal, magnetic, and economic) Sequential Quadratic Programming (SQP) optimization	Multidisciplinary design optimization framework integrating PMSG and power converter design. Reduced PMSG cost by optimizing PM volume and copper usage. 88.6% total system efficiency.
[33]	Physics-based data-driven modeling	Building energy consumption prediction	Physics-based simulation using IDA-ICE Data-driven modeling with Long Short-Term Memory (LSTM) networks Hybrid model integrating simulation results as input data	A hybrid model combining physics-based and data-driven methods. Improved prediction accuracy compared to conventional data-driven models.
[34]	Computational modeling for real-time DT (Review)	Reducing computational demands in Digital Twin applications	Literature review of real-time modeling methods Analysis of machine learning and physics-based acceleration techniques	A structured review of methods for reducing Digital Twin computational costs. Identified gaps in real-time analysis techniques. Proposed a roadmap for future research on Digital Twin efficiency.
[35]	Data-driven DT modeling	Early detection of inter-turn short-circuit faults in PMSMs	NARX DT model trained on healthy motor data Current residual analysis for fault detection	Data-driven Digital Twin model using NARX networks. Early fault detection without requiring prior fault data. High sensitivity to incipient inter-turn short-circuit faults.
[36]	Data-driven DT modeling	Optimal sensor placement for PMSM condition monitoring	FEM Data-driven reliability-based design optimization Genetic algorithm for optimal sensor selection	Digital twin-assisted framework for optimizing sensor placement. Identified the optimal number and placement of Hall-effect sensors for fault detection. High fault classification accuracy with minimal sensor usage.
[37]	Multiphysics-based DT modeling	Design and optimization of PMSMs and drive systems	Multiphysics modeling (electromagnetic, thermal, and mechanical) System-level optimization integrating control strategies	DT framework for PMSM design optimization. Improved efficiency and reliability in PMSM drive systems.
[38]	Data-driven fault diagnosis modeling	Fault diagnosis in PMSM drive systems using self-sensing signals	Short-time Fourier transform for time–frequency analysis Image fusion of three-phase current spectrograms Deep transfer learning using SqueezeNet	Data-driven fault diagnosis method based on current signal spectrograms. Achieved 99.03% fault detection accuracy without requiring external test instruments. Enabled real-time, system-level diagnosis for PMSM drives.
[39]	Nonlinear residual-based fault diagnosis modeling	Fault isolation and estimation in nonlinear systems	Multivariate Adaptive Regression Splines (MARS) for nonlinear redundancy modeling Directional residual-based fault diagnosis using local linearization Comparison with machine learning methods	A nonlinear residual-based fault diagnosis approach using MARS. Improved fault isolation accuracy over linear models. Demonstrated better fault estimation accuracy than machine learning in unseen conditions.
[40]	Data-driven resilience modeling	Transmission defense planning against extreme weather events	Historical weather data for component survivability estimation Probabilistic modeling of system failure states Failure state sparsification for computational efficiency	Data-driven model to optimize line hardening and construction. Eliminated the need for predefined uncertainty budgets by estimating failure probabilities. Proposed a failure state sparsification method to improve computational efficiency.
[41]	Data-driven computational modeling (Review)	Machine learning applications in chemical and industrial processes	Review of supervised, unsupervised, and reinforcement learning techniques Computational analysis of challenges in industrial-scale data-driven models	Reviewed the state-of-the-art applications of data-driven modeling in process systems. Identified major challenges such as data availability, scalability, and security. Outlined future research directions for integrating ML in industrial-scale applications.
[42]	Hybrid multi-domain analytical and data-driven modeling	Tracking error prediction for ball screw feed systems in CNC machine tools driven by PMSM	Multi-domain analytical modeling using energy flow principles Data-driven Back Propagation Neural Network (BPNN) Hybrid modeling by coupling analytical and data-driven models	Hybrid model integrating analytical and data-driven approaches. Higher accuracy in tracking error prediction than pure analytical models.
[43]	Physics-informed neural network (PINN)	Estimating the electromagnetic response of a PMSM	Governing equations with rotational coordinate transformation Domain decomposition with separate rotor and stator networks Learning rate annealing for adaptive weight optimization	Novel PINN architecture for PMSM electromagnetic modeling. Achieved similar accuracy to FEA but with 10× faster inference speed. Applicability for Digital Twins and predictive maintenance.
[44]	Hybrid mechanism-data-driven iron loss modeling	Iron loss estimation in PMSMs considering multiphysics coupling effects	Physics-based iron loss analytical model (accounting for mechanical stress, temperature, harmonics, and load currents) Convolutional Neural Network (CNN) for feature extraction and pattern recognition Hybrid training: pre-training on simulation data and fine-tuning with experimental data	Hybrid iron loss model integrating physics-based and data-driven approaches. Higher prediction accuracy than traditional analytical and CNN-only models.
[45]	Hybrid physics-based and data-driven modeling	Monitoring stator insulation degradation in inverter-fed PMSMs	Physics-based analysis of high-frequency common-mode switching oscillations Continuous Wavelet Transform (CWT) for time–frequency feature extraction CNN for fault classification	A hybrid model integrating physics-based analysis with CNN-based classification. Achieved 94.53% accuracy in identifying groundwall and turn insulation degradation.
[46]	Physics-informed Bayesian optimization	Rapid optimization of slot fill factor in traction motors	Physics-informed Bayesian optimization (PIBO) with Maximum Entropy Sampling Algorithm (MESA) Gaussian Process (GP)-based surrogate modeling Coupled FEM for electromagnetic validation	PIBO-MESA optimization framework for improving slot fill factor. Achieved a 20% increase in slot fill factor, leading to higher electromagnetic performance. Reduced computation time by 45% compared to NSGA-II, improving design efficiency.

provides a summary of the References discussed in Section 3.1, Section 3.2 and Section 3.3.

3.4. Reduced Order Modeling (ROM)

Reduced Order Modeling (ROM) is a mathematical technique used to simplify complex systems by reducing their number of variables while maintaining essential dynamic characteristics. It is particularly useful in scenarios where full-scale simulations are computationally expensive or impractical, such as real-time monitoring and control. ROM techniques achieve this by reducing the number of equations or state variables, making simulations faster without significantly compromising accuracy. ROM plays a vital role in enabling efficient real-time simulations in DTs of PMSMs, especially where high-fidelity and computationally demanding models are impractical. ROM techniques reduce model complexity while preserving essential dynamic characteristics, thus balancing computational efficiency with accuracy, and even enabling real-time simulations. Furthermore, techniques such as Krylov subspace methods and Proper Orthogonal Decomposition (POD) have been widely adopted to accelerate ROM simulations [47].

ROM techniques have been successfully implemented to enhance the efficiency and scalability of DTs by reducing computational demands. Recent advancements focus on Projection-Based Reduced Order Models (PROMs), which compress high-fidelity models into lower-dimensional representations while maintaining essential dynamics [48]. This study highlights the integration of PROMs with high-performance computing (HPC) workflows to address computational bottlenecks during the training and deployment phases. By employing POD through parallel singular value decomposition (SVD), PROMs facilitate real-time applications such as motor thermal monitoring and predictive maintenance, enabling rapid model updates across edge and cloud environments.

Another promising approach integrates component-based ROM with machine learning for adaptive model selection [49]. Decomposing complex systems into smaller substructures, enables flexible model updates and scalability, even in parameter-rich systems. This strategy supports the creation of a library of reduced-order models that dynamically adapt to real-time data, ensuring responsiveness to changing operational conditions. Adaptive ROM frameworks also enhance the ability of DTs to manage diverse conditions and failure modes efficiently. By incorporating interpretable machine learning, these models align the DT with the motor’s current state based on sensor data.

A study in [50] employs Proper Generalized Decomposition (PGD)-based methods for their ROM. This approach enables the computation of high-dimensional parametric solutions in electric motors with remarkable efficiency. By using a fully separated representation, PGD techniques decompose complex geometries, such as those found in PMSMs, into simpler subdomains, facilitating the accurate modeling of electromagnetic fields under varying operating conditions. This method supports the creation of virtual charts for real-time evaluation of key parameters, such as magnetic vector potential, based on factors like rotor position and material properties. The combination of spatial and parametric decomposition significantly reduces computational costs, achieving solutions equivalent to high-resolution finite element models within minutes on standard hardware.

Hybrid ROMs, which combine physics-based models with data-driven enhancements, significantly improve computational efficiency and precision. A study in [51] introduced a hybrid twin for magnetic bearings that uses machine learning, such as Long Short-Term Memory (LSTM) networks, to correct ROM prediction errors, achieving near real-time computation with high accuracy. By applying spectral decomposition and reduced basis methods, these frameworks enable adaptive updates based on sensor inputs, ensuring fidelity under varying operational conditions. Such methods are particularly effective for complex, nonlinear systems like PMSMs, where standard ROM approaches may be insufficient.

ROM techniques in DTs for PMSMs have shown significant promise for enhancing predictive diagnostics, real-time monitoring, and efficient system optimization. By minimizing the computational demands of full-order models, ROM enables faster simulations, supporting both condition-based monitoring and proactive fault detection across a range of operating conditions.

4. Possible Applications of Digital Twins in PMSMs

PMSMs are widely utilized across a variety of sectors. Figure 8 shows some actual PMSMs in real-life applications. Applying DT technology to PMSMs enhances the efficiency and intelligence of systems such as vehicles and appliances, driven by extensive applications that leverage continuous data collection and analysis to unlock significant potential. This section will explore a few key applications of DT technology for PMSMs.

4.1. Real-Time Monitoring, Predictive Maintenance, and Fault Detection

It has been established that DTs have emerged as transformative tools for real-time monitoring and predictive maintenance. By integrating real-time data acquisition, high-fidelity modeling, and advanced analytics, DTs enable the continuous monitoring of key parameters such as temperature, vibration, and electrical currents and this allows predictive maintenance, ensuring that PMSMs operate efficiently while minimizing downtime and maintenance costs.

A prominent example of real-time monitoring involves the integration of IoT sensors and thermo-magnetic FEM in DTs for induction motors [52]. This approach monitors parameters such as temperature, current, and resistive losses, providing insights into motor health and enabling preemptive maintenance actions. Moreover, the approach emphasizes the use of real-time data synchronization, which enables live updates of motor conditions.

As illustrated in Figure 9, the proposed monitoring framework in [52] consists of IoT-enabled sensors for real-time data collection, a data processing system, and a Digital Twin model for fault detection and predictive maintenance. The dotted line in the figure represents a possible feedback route from the DT model to the physical twin, although the referenced study does not specify this connection. In this setup, data collected from the physical motor is continuously fed into the DT model, where advanced algorithms analyze trends and detect potential issues before they escalate. By incorporating real-time monitoring and predictive analytics, the framework enables proactive maintenance, reducing unexpected failures and improving motor longevity. Further research integrating DTs with feedback mechanisms could enhance temperature regulation by adjusting control inputs dynamically. This concept can be extended to PMSMs, where real-time DT models can play a crucial role in ensuring operational reliability by identifying early signs of wear and detecting potential faults before they cause significant performance degradation.

The predictive maintenance capabilities of DTs are further illustrated in [53], which presents a system that uses a DT powered by intelligent predictive maintenance tools. By incorporating data-driven models and multiphysics simulations, this framework identifies motor degradation and predicts remaining useful life (RUL). The lightweight computational design ensures that these tools can be applied to PMSMs without imposing excessive processing demands, making real-time fault detection and diagnostics feasible even under dynamic operational conditions.

Furthermore, advancements in visualization and remote diagnostics, such as those detailed in [54], underscore the importance of integrating cloud-based platforms with DTs. These platforms allow maintenance teams to visualize real-time data and address emerging issues proactively. The visualization tools offer clear representations of motor health, making it easier to implement predictive maintenance strategies aligned with Industry 4.0 objectives.

Anomaly detection frameworks, as described in [55], demonstrate another valuable application of DTs. These frameworks use model-based techniques to analyze vibrations and frequency domain characteristics, enabling precise identification of faults such as misalignments or imbalances. The adaptation of such frameworks for PMSMs could significantly enhance their fault detection capabilities, especially in high-stress environments.

The predictive capabilities of DTs are further enhanced by incorporating machine learning algorithms to analyze operational patterns, identify anomalies, and estimate the RUL of critical components. This approach reduces unplanned downtime, minimizes maintenance costs, and improves motor reliability. Real-time monitoring systems can also synchronize with physical systems at near-zero latency, providing instant feedback and enabling precise adjustments during operation, as demonstrated in CNC machinery applications [56].

Furthermore, the integration of DTs with cloud-based platforms facilitates seamless data visualization and remote diagnostics, making it possible for maintenance teams to proactively address emerging issues. By creating a dynamic virtual representation of PMSMs, DTs bridge the gap between predictive maintenance strategies and the demands of Industry 4.0, ensuring that motors operate at peak efficiency while extending their operational lifespan.

4.2. Power Management

DT technology offers transformative potential in power management for PMSMs by enabling precise energy monitoring, optimization, and real-time control. Through the integration of advanced modeling and IoT-based systems, DTs provide a virtual platform to simulate, analyze, and optimize power consumption under various operating conditions. One study highlights the use of DTs in light electric vehicles to compare different motor types, including PMSMs. It demonstrates how real-time DT systems can optimize motor performance by analyzing power losses under varying load conditions, thereby reducing energy waste and extending battery life [57]. The motor selection process using a DT simulates acceleration and brake control according to the drive cycle, as shown in Figure 10. This capability is particularly valuable in electric vehicle applications, where energy efficiency directly impacts range and operational costs.

Another study explores the role of DTs in integrating renewable energy systems into motor-driven applications. By simulating the interaction between motors and variable energy inputs, DTs enable adaptive power management strategies that minimize dependency on traditional power sources while ensuring consistent performance [58]. This aligns with Industry 4.0’s emphasis on sustainability and efficiency.

The study in [59] reviewed DT platforms specifically designed for electric vehicles. These platforms utilize real-time sensor data to predict power demands and optimize motor control strategies, achieving a balance between performance and energy efficiency. In addition, machine learning algorithms can be integrated into these platforms to predict energy consumption trends, enabling precise allocation of energy resources and proactive adjustments during fluctuating loads.

A notable application in induction motors shows how energy-efficiency models embedded in digital shadows can evolve into fully functional DTs [60]. These models simulate electromagnetic and thermal behaviors, optimizing operational parameters to reduce power consumption and enhance reliability. The success of these systems with induction motors highlights their potential for PMSM applications, where maintaining high efficiency under dynamic conditions is essential. PMSMs are widely used not only in industrial and automotive settings but also in home appliances, and the application of DT technology in this context could enable the development of smarter, more energy-efficient home appliances.

4.3. Integrated Designs

Another possible application, though less discovered, is using DTs in the design process. A study in [61] highlights how DTs unify the traditionally siloed processes of design, manufacturing, and maintenance. It provides a cohesive platform for evaluating designs, simulating performance, and incorporating real-world operational data into iterative improvements. For PMSMs, this integration allows designers to evaluate motor configurations early, ensuring compatibility with manufacturing techniques and anticipating future maintenance needs. Similarly, a study in [62,63] reports that using DTs in the design process can reduce manufacturing risks and create smart designs that predict operational performance under various conditions, incorporating adaptive features for improved efficiency with machine learning and IoT platforms. For PMSMs, this translates to better modeling of electromagnetic and thermal behaviors, minimizing design iterations and accelerating the prototyping phase.

The concept of nominal Digital Twins (NDTs), which is a framework for aggregating data from multiple instances of a product to optimize future designs, is introduced in [64]. This approach is particularly advantageous for PMSMs, as it enables the synthesis of operational data from existing motors to refine subsequent designs. It uses the history and database of previous motors. By creating a feedback loop, NDTs improve performance and reliability across generations of PMSMs, making each iteration more efficient and reliable.

Using DTs along with their adaptive features can allow for virtual testing of designs, and in this process, it can also minimize material waste. Additionally, operational data collected from in-service motors enable predictive adjustments in future designs, improving reliability and energy efficiency [65]. This comprehensive integration aligns with the goals of Industry 4.0, enabling future electric motors to be not only high-performing but also sustainable and adaptive to evolving technological needs.

5. Discussions: Challenges and Limitations

The integration of DT technology in PMSMs offers enhanced accuracy, cost savings, and improved reliability and efficiency. By ensuring real-time data integration, a DT accurately reflects the current state of the PMSM, enhancing the precision of simulations and predictions, which leads to better decision-making. Additionally, the technology reduces the need for physical prototypes and enables predictive maintenance, thus saving costs and reducing the time required for product development and maintenance. Often, new products use the same motor type across different products with similar ratings. Therefore, developing DTs for electric motors, particularly PMSMs, becomes crucial for evaluating new control algorithms and system engineering. Virtual testing accelerates the design process and minimizes the risk of costly failures, while real-time monitoring and predictive analytics improve the reliability and efficiency of PMSMs by maintaining optimal performance and preventing unexpected breakdowns.

Despite the growing number of publications and technological advancements in DT technology, practical applications beyond research have been limited. Most current research focuses on system-level applications, such as power plants, buildings, and vehicles, often overlooking individual units like motors. However, effective system-wide DT implementation necessitates modeling key components, including motors. The limited research on DTs at the unit level restricts available references for developing robust DTs for motors. While there is some research on induction motors, the broader application of PMSMs justifies a more focused exploration of their DTs.

Several technical and operational challenges hinder the widespread adoption of DT technology across industries, including PMSM applications. One significant issue is the lack of standardized frameworks and definitions for DTs, leading to inconsistent implementations and difficulties in scaling solutions [66]. Without universally accepted standards, integrating DT technology into complex systems like PMSMs becomes a bespoke and resource-intensive process.

Data-related challenges also present substantial limitations. High-fidelity DT models rely on large volumes of accurate, real-time data collected through sensors and monitoring devices, often facilitated by IoT systems. However, ensuring data quality, security, and interoperability remains a persistent problem [67]. In PMSM applications, inconsistent or incomplete data can result in less reliable models, undermining the predictive and diagnostic capabilities of DTs. Additionally, concerns over data ownership and privacy complicate collaborations, particularly in industries requiring compliance with stringent regulations.

Another key challenge lies in the computational demands of DT technology. High-fidelity simulations and real-time analytics require significant processing power and storage, which are often unavailable in resource-constrained environments. This creates barriers, particularly for small and medium-sized enterprises. Moreover, DT implementations demand interdisciplinary expertise, integrating knowledge from domains such as physics, data science, and engineering. The lack of skilled personnel capable of designing, maintaining, and interpreting DT systems is a significant bottleneck, which is particularly acute for PMSMs [68].

Finally, the economic viability of DT technology is a concern. The initial costs of developing and deploying DT solutions are often prohibitive, particularly for industries with tight margins. The added cost of implementing DT systems may not be immediately justifiable for PMSM manufacturers without clear and measurable benefits. Additionally, the complexity of DT modeling can vary depending on PMSM structures, as differences in rotor topology and winding configurations may require tailored modeling approaches, further influencing development costs.

Despite these challenges, the future of DTs in PMSMs is promising, with opportunities for advancements in Artificial Intelligence, real-time processing, and broader industrial applications. Future research should focus on integrating advanced data analytics and machine learning to enhance predictive maintenance, fault detection, and operational optimization for PMSMs. While the IoT remains essential for real-time data collection, the primary advancements in DT technology for PMSMs will come from improved simulation accuracy, adaptive control strategies, and real-time decision-making capabilities. These developments will enable PMSMs to operate more efficiently, reducing energy consumption and extending their lifespan. As DT technology continues to evolve, its applications in PMSMs are expected to expand, driving innovation in motor design, monitoring, and industrial automation.

6. Conclusions

The integration of DT technology with PMSMs presents a promising avenue for advancing motor efficiency and functionality. Instead of focusing solely on the system, it is crucial to consider the components at a unit level, which underscores the importance of DTs for motor applications. The benefits of creating a DT for PMSMs include enhanced predictive maintenance, optimized power management, and improved design processes. This approach is particularly valuable in product development, where the same motor may be used across several product generations. Despite challenges related to data quality, computational demands, and the need for standardized frameworks, the potential for DTs to revolutionize PMSM applications is evident. Future work should focus on overcoming these barriers and exploring advanced data analytics to fully harness the capabilities of DTs. As technology progresses, the application of DTs in PMSMs is expected to become more prevalent, driving significant advancements in motor design, monitoring, and optimization, thus contributing to smarter manufacturing practices and the evolution of Industry 4.0.