A Brief Review of Multi-Physics Coupling Research on Hydroelectric Generators

Abstract

Hydropower, with its high degree of flexibility, plays an important role in the transformation of the global energy mix. Generators are the core component of the hydropower units; their performance directly affects the efficiency and reliability of the hydroelectric units. The dynamic characteristics of a generator during operation are usually the result of the coupling and interaction of multiple physical fields. Therefore, the interactions among electromagnetic, thermal, structural, and fluid fields inside hydroelectric generators have become of great concern. This paper briefly reviews the hydroelectric generator multi-physics coupling investigations, which include research conducted through field measurements, theoretical analysis, and numerical simulations. The review covers electromagnetic vibrations of generators under the influence of electromagnetic and structural fields, heat generation of generators under the influence of electromagnetic and thermal fields, ventilation and heat dissipation of generators under the influence of flow and thermal fields, and physical field changes of generators under the influence of electromechanical signals. The review also highlights unresolved issues in the field of hydropower that could benefit from fundamental research using a multi-physics coupling approach.

1. Introduction

In recent years, global climate change has emerged as a pressing concern, prompting a heightened focus on sustainable and clean energy solutions. Hydropower, a cornerstone of renewable energy, is poised to assume an increasingly pivotal role within the global energy landscape. Characterized by its low carbon footprint across its entire lifecycle, hydropower stands as a primary source of low-carbon electricity, accounting for nearly 60% of global renewable energy generation, as illustrated in Figure 1 [1]. The rapid development of wind and solar power, driven by the ongoing transformation of the energy structure, introduces new dynamics to the energy mix. However, the inherent instability of these sources escalates the demand for adjustable low-carbon power generation resources. Hydropower, with its flexible regulation capabilities, emerges as a key player in balancing the fluctuating output of renewable energy sources, fortifying the stability and flexibility of the new energy system, and accelerating the low-carbon transition of the global energy structure.

At the heart of hydropower systems lies the hydroelectric generator, which converts mechanical energy into electrical energy. This energy is subsequently transmitted to the power grid via the transmission system, ensuring a stable and reliable power supply to consumers. Consequently, the stability of hydroelectric generator design and operation is directly linked to the reliability and efficiency of hydropower generation.

Despite their critical importance, hydroelectric generators are not immune to operational challenges. One such challenge is vibration, which stems from the interplay of hydraulic, mechanical, and electromagnetic factors. Electromagnetic vibration, particularly in the stator and rotor core, is a root cause of numerous hydroelectric generator accidents and severe damage [2,3,4]. This vibration not only generates noise but also significantly impacts the safe and stable operation of the unit [5]. Electromagnetic vibration can be categorized into low-frequency rotational frequency vibration and high-frequency extreme frequency vibration. Rotational frequency vibration is primarily attributed to factors such as rotor out-of-roundness, rotor inter-turn short circuits, rotor dynamic imbalance, shaft bending, and misalignment of the stator–rotor magnetic field axis. On the other hand, extreme frequency vibration often results from stator fractional slot harmonic magnetic potential, negative sequence current, poor seat seams, and loose stator cores.

Another significant concern is temperature rise, which refers to the internal temperature distribution within the generator. An uneven temperature distribution can lead to a multitude of issues. Excessive temperature rise in the stator and rotor windings can compromise insulation integrity, thereby affecting the service life and operational reliability of the generator [6]. During the start-up and shutdown processes, abrupt temperature changes induce significant thermal stress, making the generator susceptible to failures [7,8]. Additionally, power electronics, voltage, and current waveforms, switching frequency, and other motor control components also influence motor temperature [9,10]. Temperature rise can also cause thermal expansion and deformation between the stator and rotor, disrupting the air gap distribution [11].

To address these challenges, numerical calculations and multi-physics analysis offer promising solutions. These techniques enable the examination of physical field distributions within the generator, taking into account the coupling relationships between different physical fields. By exploring the root causes of heat generation and vibration, they provide a theoretical foundation for fault prevention and resolution. The current research in this domain has encompassed magnetic-thermal coupling, magnetic-structure coupling, and thermo-fluid coupling, among others, addressing a limited array of problems. However, a more comprehensive and accurate analysis necessitates the consideration of additional physical field coupling relationships.

In this review, we delve into the significant advancements achieved through multi-physics analysis techniques in the context of hydroelectric generator analysis. This coupling between the multi-physics is illustrated in Figure 2. We also explore the benefits and challenges presented by these techniques. Beginning with the common issues faced by hydroelectric generators, we examine how multi-physics analysis techniques have been employed to study these problems.

For instance, the electromagnetic vibration problems of hydroelectric generators are investigated using coupled electromagnetic and structural field analysis. This approach allows for a deeper understanding of the interactions between electromagnetic forces and the mechanical structure of the generator, helping to identify and mitigate vibration sources. Similarly, coupled electromagnetic and temperature field analysis is utilized to study temperature rise and heat generation problems in hydroelectric generators. This analysis helps to elucidate the complex interplay between electromagnetic losses, thermal conduction, and convection, enabling the optimization of cooling systems and the enhancement of generator efficiency and reliability. Furthermore, coupled fluid field and temperature field analysis is applied to investigate the ventilation and heat dissipation problem of hydroelectric generators. By simulating the flow of cooling air or water through the generator, this analysis helps to design more effective cooling strategies, ensuring that the generator operates within safe temperature limits. In addition to these applications, we also discuss the study of the impact of electromechanical signals, such as hydrodynamic forces, grid fluctuations, and control signals, on the internal physical fields of the generator. Understanding these interactions is crucial for the development of robust control strategies and the enhancement of generator performance under varying operating conditions.

Finally, we provide an overview of current outstanding issues in this field, highlighting potential research directions that may guide future efforts. As the energy landscape continues to evolve, the role of hydropower and the importance of reliable and efficient hydroelectric generators will only grow. By leveraging the power of multi-physics coupled analysis techniques, we can address the challenges faced by these generators, ensuring their continued contribution to a sustainable and low-carbon future.

Compared with existing reviews in the industry, the novelty of this review lies mainly in the following aspects:

• This review is focused specifically on the physics of hydroelectric generators rather than discussing general generators in a broad sense. This targeted approach enables the review to dive into the unique problems and challenges of hydroelectric generators, providing the readers with more precise and professional information. We concentrate on the practical issues encountered by hydroelectric generators during operation, such as vibration, temperature increase, and ventilation and heat dissipation, ensuring that the content discussed is closely related to the actual application of hydroelectric generators.
• This review provides a comprehensive discussion on the multi-physics calculation of hydroelectric generators. We not only cover common issues such as vibration and temperature rise in hydroelectric generators, but also systematically review the application of multi-physics coupling analysis techniques in the study of these problems. Specifically, we elaborate on the significance of the coupling analysis of electromagnetic and structural fields in the research of electromagnetic vibration problems, revealing the interaction mechanism between electromagnetic forces and structural deformation. Meanwhile, we also explore the application of the coupling analysis of electromagnetic and temperature fields in the study of temperature rise and heating problems, analyzing the impact of electromagnetic losses, heat conduction, and thermal stress on the performance of generators. Additionally, we systematically introduce the role of the coupling analysis of fluid and temperature fields in the research of ventilation and heat dissipation problems, revealing the influence law of fluid flow on heat dissipation effects.
• The discussion content of this review is cutting-edge. We particularly focused on the research of the influence of electromechanical signals (such as grid fluctuations, control signals, etc.) on the internal physical fields of motors. This field is a current hot topic in the research of hydro-generator technology, which is of great significance for in-depth understanding of the operation mechanism of hydro-generators and optimizing their design. We deeply analyzed how these electromechanical signals interact with the electromagnetic field, temperature field, structural field, etc., inside the motor, jointly affecting the operation performance and stability of hydro-generators, providing readers with the latest research trends and ideas.

2. Electromagnetic Vibration

Under normal operating circumstances, the air gap within a hydroelectric generator undergoes periodic alterations in the electromagnetic field. This dynamic environment is a result of the interplay between the stator and rotor, which, influenced by electromagnetic force waves, generate electromagnetic vibration. The intricacies of this phenomenon extend beyond the immediate operational parameters, encompassing factors related to the installation, manufacturing processes, and the unit’s lifespan, including natural wear and tear and thermal expansion. These elements can lead to an uneven distribution of the air gap between the stator and rotor, giving rise to a variable electromagnetic force that acts upon the rotor, thereby inducing its vibration. This vibration is subsequently transmitted through the main shaft to the turbine system, potentially leading to intense vibrations under specific resonance conditions, which can result in substantial economic losses.

2.1. The Influence of Generator Structure on Electromagnetic Vibration

Electromagnetic vibration is inherently tied to the design of the generator structure itself [12]. The generator’s architecture, including the method of magnet mounting, the distribution pattern of the number of generator gears and slots, and the configuration of the stator end windings, all play crucial roles. In the nascent stages of electromagnetic vibration analysis, the distribution of the field within the generator structure was often overlooked. Researchers primarily relied on methods such as the Maxwell tensor method [13] or the virtual displacement method [14] to calculate and analyze the electromagnetic force and electromagnetic moment. However, with the advent and progression of multi-physics analysis techniques, a new era of research was ushered in. Led by pioneers like Kim et al. [15], researchers began to delve into the complex relationships between the electromagnetic field, electromagnetic force, and structural vibration that impact the electromagnetic vibration of motors. By coupling and comparing the fields generated by these physical quantities, and by integrating mechanical vibration with the magnetic field, they were able to derive the vibration response of the rotor. This magnetic-structure coupled analysis technique has enabled scholars to conduct more in-depth mechanistic analyses of electric motors, continuously refining and developing this approach in subsequent research. For instance, Yu et al. employed the magnetic-structure coupling calculation method to perform comprehensive calculations and analyses on electromagnetic forces and vibrations. Building on this foundation, they designed a novel magnet connection structure [16] that aimed to mitigate electromagnetic vibration. Similarly, Ren et al. proposed a method of overall electromagnetic–local structure coupling simulation to analyze the mechanical vibration performance of the stator end windings of turbo generators [17]. They compared the vibration performance under different reinforcement strategies, shedding light on potential improvements in generator design. Bang et al., leveraging the magnetic-structure coupling finite element calculation method, conducted a comparative analysis of the noise, vibration, and harshness (NVH) characteristics generated by various fractional pole/slot combinations in permanent magnet synchronous motors [18]. Their findings contributed to a deeper understanding of how these combinations influence the overall performance of the generator, particularly in terms of vibration and noise reduction.

2.2. The Influence of Unbalanced Magnetic Pull on Electromagnetic Vibration

Beyond the generator’s structural design, electromagnetic vibration is also significantly influenced by unbalanced magnetic tension. As early as 1900, Behrend formulated a linear expression for the magnetic pulling force, assuming that the magnetic density in the air gap is inversely proportional to the air gap length [19]. This pioneering work laid the foundation for subsequent research in this area. In the 1960s, Belmans [20] and Richard [21] further advanced our understanding by deducing an analytical expression for the unbalanced magnetic pulling force. Their contributions provided a more comprehensive framework for analyzing the effects of unbalanced magnetic tension on generator performance. In 1993, DeBortoli [22] was the first to compute the unbalanced magnetic pulling force using the transient finite element method, specifically analyzing its effect when the rotor is eccentric. This marked a significant milestone in the evolution of electromagnetic vibration analysis, as it enabled more accurate and detailed simulations of generator behavior under real-world conditions. Bai [23] initially proposed the linear unbalanced magnetic tension formula in 1982, which was later experimentally validated by Zhang [24]. Zhang demonstrated that the unbalanced magnetic tension is approximately proportional to the eccentricity, further solidifying the theoretical foundations established by Bai. In 1997, Qu et al. [25] summarized the nonlinear relationship between eccentricity and unbalanced magnetic tension, highlighting the complexity of this phenomenon. This work underscored the need for more sophisticated models to accurately capture the behavior of motors under varying conditions. Subsequently, Li and others compared and summarized the differences in computational accuracy between linear and nonlinear models based on previous studies [26]. Their findings provided valuable insights into the strengths and limitations of each approach, guiding future research in this area.

2.3. Research Progress of Magnetic-Structure Coupling Analysis Techniques

In recent years, with the significant enhancement of computer calculation speed and accuracy, the calculation of unbalanced electromagnetic force has been transformed [5,27,28,29,30,31,32,33,34,35,36]. Researchers have shifted from relying on empirical formulas to utilizing finite element analysis, and from single electromagnetic field analysis to coupled electromagnetic and structural field analysis. This evolution has enabled more accurate and comprehensive simulations of generator behavior, leading to improved designs and reduced vibration. For example, Li utilized the finite element method for hydroelectric generators to determine the radial unbalanced electromagnetic force of the generator and incorporated it into the structural field to calculate the vibration endured by the stator [37]. That work demonstrated the feasibility and effectiveness of this integrated approach in predicting and mitigating electromagnetic vibration. He et al. focused on the impact of inclined static air-gap eccentricity (ISAGE) on electromagnetic characteristics and mechanical properties, including vibration, deformation, strain, and stress [38]. Through finite element analysis of both the electromagnetic field and the structural field, they compared the effects of various fault types on the stator radial vibration, as shown in Figure 3. Their findings provided valuable insights into the complex interplay between electromagnetic forces and structural dynamics in motors, further advancing our understanding of this phenomenon. Considering the inhomogeneity of the electromagnetic force extended in the axial direction, Zhang et al. [39,40,41] started to study the full three-dimensional electromagnetic vibration analysis of hydroelectric generators. Literature [40] adopts the full three-dimensional numerical calculation method to analyze the distribution of motor electromagnetic force and electromagnetic moment in the case of hydroelectric generator radial eccentricity, axial offset, and shaft deflection, as shown in Figure 4 and Figure 5. They concluded that within a small deviation range, the linear variation laws of force and moment with respect to the deviation were obtained, which is important for practical engineering applications. In conclusion, electromagnetic vibration in hydroelectric generators is a multifaceted phenomenon influenced by a variety of factors, including the generator’s structural design, unbalanced magnetic tension, and the interplay between electromagnetic and structural fields. Over the years, researchers have made significant progress in understanding and mitigating this phenomenon, leveraging advanced analysis techniques and computational models to improve generator performance and reduce vibration. As technology continues to evolve, we will likely see even more sophisticated and effective solutions to this long-standing challenge in the field of electric generator design and operation.

In the operation of power stations, there exists the issue of electromagnetic vibration. If it is not resolved promptly and effectively, it may pose a threat to the safe operation of the engineering project. However, multi-field coupling analysis technology can identify the causes of electromagnetic vibration and provide solutions, thereby ensuring the safe operation of the project. Taking the analysis of the unstable vibration problem of the No. 3 hydro-turbine generator unit at the Tianshengqiao Hydropower Station by Yao et al. [28] as an example, they used multi-field coupling analysis technology to discover that the main causes of the unit’s vibration were the weak connection stiffness of the upper bracket, rotor mass imbalance, and generator electromagnetic imbalance. In response to these issues, they proposed corresponding solutions and implemented comprehensive treatment measures, ultimately successfully eliminating the unit’s vibration. In addition, multi-field coupling analysis technology can also provide theoretical references for the design of large-capacity hydro-turbine generators [29].

A comprehensive review of the literature reveals that there is a paucity of finite element studies on the three-dimensional electromagnetic force of motors or generators, and the objects of most existing studies are small. The electromagnetic vibration analysis of large hydroelectric generators mostly remains at the two-dimensional level, relying either on empirical formulas or finite element calculations. The radial electromagnetic force is typically treated as an excitation applied to the three-dimensional structure field to analyze vibration. Throughout the calculation, the distribution of the electromagnetic force in the axial direction is assumed to be constant. This approach fails to account for the non-uniformity of the electromagnetic force distribution along the axial direction in the actual turbine generator system and neglects the analysis of the impact of the electromagnetic force distribution on the structural field under three-dimensional air gap eccentricity. Future research should focus on filling these gaps to enhance our understanding and control of electromagnetic vibration in large hydroelectric generators, thereby improving their operational stability and reliability. This could involve more detailed three-dimensional finite element simulations, taking into account the actual operating conditions and structural characteristics of large hydroelectric generators, as well as experimental validations to ensure the accuracy and practicality of the analysis results.

3. Ventilation and Heat Dissipation Characteristics

Heat generation and cooling have always been challenging issues that the hydroelectric generator industry must pay close attention to and thoroughly study. These aspects are not solely dependent on the magnitude of the generator’s electromagnetic load, line load, and various types of losses; they are also intimately linked to the selection of materials and the design of the ventilation and cooling system. As the capacity of hydroelectric generators continues to increase, both the unit losses and the radial length of the generator grow accordingly. Especially during short-term overload operations, phase-in processes, or in situations where the generator’s internal ventilation system malfunctions, insulation fails, or the machine network encounters issues, the temperature of the generator components rises significantly. Consequently, the temperature difference and thermal stress escalate, potentially causing the temperature rise of an air-cooled generator to exceed permissible limits. This directly impacts the safe and stable operation of the generator. Therefore, the investigation of large-scale ventilation and cooling systems is not only concerned with electromagnetic load and line load but also closely tied to material selection and generator design.

Ensuring the safe and stable operation of generators necessitates an in-depth study of the ventilation structure within large air-cooled turbine generators, particularly focusing on the internal flow distribution and temperature rise control. Continuously improving turbine generator air cooling technology to reduce the generator’s hot spot temperature and optimize the average internal temperature distribution has become crucial. This has emerged as a pivotal consideration in the design of contemporary large-scale hydroelectric generators.

3.1. Research History of Ventilation and Cooling Systems

The exploration of ventilation and cooling mechanisms in hydroelectric generators dates back several decades. As early as the 1980s, American scientist Tandon et al. [42] pioneered the use of the finite element method to simulate and calculate the gas flow state inside the rotor of the motor. This groundbreaking work resulted in the delineation of the flow field distribution and isothermal curves, providing valuable insights into the internal thermodynamics of the generator. On this basis, Shanel and Lampard, after extensive research, further published their findings in the book Application of Computational Fluid Dynamics to The Cooling of Salient Electrical Machines in 2000 [43]. They revealed that the flow field distribution and isothermal curves of the rotor in an electrical machine mirror those of an electric motor, highlighting the universality of these principles. Furthermore, their work elucidated the impact of internal fluid distribution on heat dissipation within electric motors. In 2005, German engineers Ujiie et al. presented their methodology for utilizing computational fluid dynamics (CFD) techniques in the ventilation and cooling analysis and structural optimization of hydroelectric generators at the international conference on large power grids [44]. They discussed the synergy between CFD methodology and traditional statistical methods, underscoring the advantages of CFD in capturing complex flow dynamics and thermal behaviors within generators.

Earlier research methodologies primarily relied on single physical field calculations, which gradually exposed their limitations in addressing the intricate ventilation and cooling mechanisms of generators. In reality, the internal structure of a motor is exceedingly complex, with the electromagnetic field, temperature field, and flow field intertwined and mutually influential, forming a highly nonlinear and tightly coupled physical system. During generator operation, changes in the electromagnetic field induce Joule heating in the core and windings. The accumulation and dissipation of this heat directly determine the temperature field distribution. An increase in temperature leads to alterations in gas density, viscosity, and other physical parameters, profoundly affecting the flow characteristics of the gas, i.e., the state of the flow field. In turn, changes in the flow field influence the heat transfer pathways and efficiency, once again impacting the temperature field. This cycle of mutual constraints and influences repeats continuously.

Single physical field calculations sever the intrinsic coupling relationships that exist among the physical fields, leading to significant deviations between the calculation results and the actual physical processes within the generator. Consequently, such calculations cannot accurately reflect the real situation of ventilation and heat dissipation. To overcome these limitations, the multi-physics finite element calculation method has been developed with advancements in computer technology.

The multi-physics finite element method integrates the complex interactions between the electromagnetic field, temperature field, and flow field within the generator into a unified computational framework. A common approach to coupling these fields involves an iterative solution strategy. Initially, independent calculations of the electromagnetic field, temperature field, and flow field are performed to obtain preliminary results for each. Subsequently, the heat source derived from the electromagnetic field calculation is input into the temperature field calculation. The temperature distribution obtained from the temperature field calculation is then used to update the physical properties of the fluid in the flow field calculation. The results of the flow field calculation, such as flow velocity and pressure, in turn, affect the boundary conditions and parameters like heat transfer coefficients in the electromagnetic field and temperature field calculations. Through numerous iterations, the calculation results for each field are continuously refined until specific convergence criteria are met. Ultimately, this process yields accurate distributions of the electromagnetic field, temperature field, and flow field within the motor, accounting for the three-field coupling.

3.2. Advances in Magnetic-Thermal-Fluid Coupling Analysis Techniques

Based on the insights gained from multi-physics finite element calculations, engineers can obtain comprehensive and in-depth information to precisely optimize the generator’s ventilation and cooling system [45,46,47,48,49,50,51,52,53,54,55,56]. For instance, Jamshidi et al. [57] discovered that incorporating a deflector plate into a particular generator design enhanced flow characteristics, resulting in a more uniform flow field within the stator channel, as shown in Figure 6. This improvement led to reduced motor losses and enhanced heat dissipation. Similarly, Verkhovtsev et al. [58] evaluated a self-ventilating system for the rotor windings of a gas-cooled turbine generator. They computationally analyzed the flow and temperature fields of the rotor windings for various sub-slot configurations. The results indicated that the temperature distribution inhomogeneity in the slot portion of the rotor winding was reduced, and the average temperature of the winding remained constant when using a sub-slot duct with a variable cross-section compared to one with a constant cross-section. Furthermore, Li et al. [59] established the differential equations governing fluid motion in the radial ventilated groove of the stator. They compared the flow field distributions within the radial ventilation trench for different inlet wind speeds. Concurrently, based on the fluid field calculations, they computed the heat dissipation coefficients for each wind wall in the radial ventilation channel. Using these data, they constructed mathematical and physical models of the three-dimensional temperature field of the stator and subsequently calculated this temperature field. The computed results were then compared with measured values. The findings revealed that the cooling effect on the stator temperature field is closely related to the speed and temperature of the cooling medium.

3.3. Significance and Application of Multi-Physics Field Analysis Techniques

The application of multi-physics techniques has significantly advanced our understanding of the complex interactions within hydroelectric generators. By accurately modeling the electromagnetic, thermal, and fluid dynamic processes, engineers can design more efficient and reliable ventilation and cooling systems. This, in turn, enhances the overall performance and lifespan of the generators. Moreover, the ability to predict and mitigate potential hot spots and temperature gradients reduces the risk of thermal stress and insulation failure, ensuring the safe and continuous operation of these critical infrastructure components.

In addition to the theoretical advancements, practical implementations of these techniques have also demonstrated their efficacy. For example, the incorporation of deflector plates or variable cross-section sub-slot ducts in generator designs has led to measurable improvements in flow uniformity and heat dissipation. These enhancements not only improve the operational efficiency of the generators but also contribute to energy savings and reduced environmental impact.

As the demand for renewable energy sources continues to grow, the importance of optimizing hydroelectric generator designs cannot be overstated. By leveraging multi-physics finite element calculations, engineers can push the boundaries of current technology, developing more efficient, reliable, and sustainable generator systems. This ongoing research and development will play a crucial role in meeting the global energy demands while minimizing environmental footprint and promoting sustainable development.

In conclusion, the study of ventilation and cooling systems in hydroelectric generators is a multifaceted and complex endeavor that requires a deep understanding of the intertwined physical fields within the generator. The evolution from single physical field calculations to multi-physics finite element methods has marked a significant leap forward in our ability to accurately model and optimize these systems. By continuing to refine and apply these techniques, we can ensure the safe, efficient, and sustainable operation of hydroelectric generators, contributing to the global transition towards a greener and more resilient energy future.

4. Thermal Stress Characteristics

With the relentless and continuous growth of modern power demand, the capacity of individual generators has been on an upward trajectory. This increase in single-machine capacity is a direct response to the escalating energy requirements of various sectors, including industrial, residential, and commercial. In tandem with this expansion, the line load and air-gap magnetic flux density of generators have also witnessed a corresponding surge. This uptick in these parameters directly leads to a persistent rise in the electromagnetic load and thermal load within the motor. As a consequence, the heating phenomenon within generators has become increasingly pronounced, posing significant challenges to their operational efficiency and longevity. Particularly during the critical phases of start up and shut down, generators experience drastic changes in operating conditions. These transitions are characterized by sharp fluctuations in temperature, which can be quite severe. Such frequent and intense temperature variations subject multiple components of the generator to immense thermal stresses. These stresses can lead to structural deformations, material fatigue, and even catastrophic failures if not managed properly. Consequently, they pose a serious threat to the safe and reliable operation of the unit. Therefore, conducting in-depth research on the temperature distribution and thermal stress distribution of hydro-generators under diverse operating conditions holds immense significance.

From a design standpoint, the exploration of temperature and thermal stress distribution is indispensable. It empowers designers to foresee potential damage to the structure caused by the thermal stress exerted by large-scale hydroelectric generators. This foresight is crucial, as it provides powerful data support and a scientific basis for optimizing design schemes. By understanding how thermal stresses manifest within the generator, designers can make informed decisions to enhance the structural integrity and durability of the unit. This proactive approach not only improves the reliability of the generator but also reduces maintenance costs and downtime, ultimately contributing to the overall efficiency and profitability of the power plant. Moreover, from the perspective of fault handling, the insights gained from research on temperature and thermal stress distribution are invaluable. The results of such studies can offer a solid theoretical foundation for the detection and treatment of generator faults. By understanding the thermal behavior of the generator under various conditions, engineers can implement effective preventive measures before faults occur. This proactive strategy helps to mitigate the risk of unexpected failures, which can be costly and disruptive. In the event of a fault, the research findings can guide precise repair measures, ensuring that the generator is restored to optimal working condition swiftly and efficiently.

4.1. The Development of Thermal Stress Calculation of a Generator

The development of motor thermal stress calculation is intrinsically linked to the evolution of temperature field calculation. Tracing back to the pre-1970s era, the thermodynamic calculation of motors primarily relied on analytical methods. During this period, the theory of motor temperature calculation matured significantly. Researchers developed various models and formulas to predict the temperature distribution within motors, laying the groundwork for subsequent advancements. It is worth highlighting that Armor et al. [60] were pioneers in introducing the finite element method into the temperature field calculation of motors. This groundbreaking innovation revolutionized the approach to solving complex thermal problems and provided important references and ideas for future calculations using the finite element method. As we ventured into the 1980s, the field of motor temperature rise calculation witnessed substantial progress, particularly in the realm of numerical methods. Field methods, such as the finite element method, gained prominence due to their ability to handle complex geometries and boundary conditions. Several notable figures emerged during this period, contributing significantly to the advancement of motor temperature rise calculation. Doi et al. from Japan [61] are among the prominent researchers whose work has had a lasting impact. Their research findings not only improved the accuracy of temperature predictions but also paved the way for more sophisticated models and simulation techniques.

The dawn of the 21st century marked a new era in the field of motor thermal stress calculation. This period saw a transformation from single temperature field calculation to multi-physical field coupling calculation. Researchers began to recognize the interconnectedness of different physical phenomena within motors and generators. Consequently, they started combining electromagnetic fields and structural fields to calculate the thermal stress characteristics of these devices. This holistic approach allowed for a more comprehensive understanding of the thermal behavior and stress distribution within motors.

4.2. Magnetic-Thermal-Structure Coupling Analysis of Thermal Stress

In this context, numerous scholars have made significant contributions to various aspects of motor thermal stress calculation [62,63,64,65,66,67,68,69,70]. Australian scholar Preis et al. [71] was among the first to conduct in-depth research on the coupling calculation of electric fields, thermal fields, and stress fields. His work focused on simulating the operation of thermoelectric generators based on the Seebeck effect. By optimizing their structures, he aimed to maximize the operating efficiency of these generators, demonstrating the potential of multi-physical field coupling in enhancing device performance. Jiang [72] took a practical approach by examining a real-life case study: the SF-K50-30/6400 hydroelectric generator stator core silicon steel sheet outward displacement accident at a hydroelectric power station. Utilizing ANSYS finite element software, he calculated the three-dimensional temperature field and stress field of the stator. Through a qualitative analysis of the calculation results, he was able to pinpoint the root cause of the motor failure, highlighting the practical applications of thermal stress calculation in fault diagnosis and prevention. Sun [73] focused research on the Y100L-2 motor, constructing a three-dimensional thermal conductivity model of the motor rotor using the finite element method. Their simulation study on the temperature field and thermal stress field of the rotor revealed that the temperature rise of the motor rotor directly influences the magnitude of the thermal stress. This finding underscores the importance of monitoring and controlling temperature rises to prevent thermal stress-related issues. Kou et al. [70] conducted a comprehensive study on the electromagnetic analysis, temperature, and thermal stress-related aspects of large hydroelectric generators. They began by analyzing the transient electromagnetic field using an electromagnetic finite element model. This analysis allowed them to calculate the copper and iron losses of the hydroelectric generator stator system under no-load and rated load conditions. Subsequently, they established a three-dimensional structure model of the hydroelectric generator stator system and imported the electromagnetic loss density into it for coupled electromagnetic-steady state temperature field analysis. This approach enabled them to obtain the temperature distribution of the hydroelectric generator stator system under different load conditions. Finally, by combining the temperature field analysis with structural thermodynamic analysis, they determined the thermal stress distribution within the hydroelectric generator. This study not only provides a theoretical basis for reducing temperature rises during operation but also offers insights into preventing structural thermal stress deformation. Notably, it was found that, under load conditions, the maximum value of the thermal stress in the entire rotor occurs at the connection between the guide bar and the end ring, indicating that this area is most susceptible to fracture failures. This groundbreaking study represents the first instance of coupled calculation of magnetic, thermal, and structural physical fields in the context of hydroelectric generators.

In addition to these pioneering studies, numerous other related studies have enriched the field of thermal stress calculations for electric motors. For instance, research on the temperature and thermal stress analysis of the abort ring during unit braking [74] has shed light on the specific challenges faced by these components during operational transitions. Similarly, comprehensive stress analyses during the startup of a subcritical turbine [75,76] have provided insights into the dynamic changes in stress distributions during this critical phase. Studies on the effect of thermal stresses on the fatigue life of a unit [71,77] have highlighted the importance of considering thermal stresses in long-term operational planning. Furthermore, analyses of the thermal stresses of the magnetic poles under the influence of a short-time fault [78] have underscored the need for robust fault detection and management strategies to mitigate the impact of thermal stresses on motor components, as shown in Figure 7. Hengmin [79] calculated the influence of the physical parameters of the stator of the steam turbine generator on the thermal deformation. Figure 8 shows the influence of the elastic modulus ($E_x,E_y$) on the generator structure. It can be found from the figure that the reduction of the elastic modulus will lead to the decrease of the material stiffness, and the deformation will increase under the same force.

During the operation of power stations, there exists the issue of thermal stress. If not analyzed and controlled promptly and effectively, thermal stress may pose severe threats to the structural safety and operational stability of hydro-turbine generators. Multi-field coupling analysis technology, however, can deeply delve into the mechanisms underlying thermal stress and offer effective solutions, enabling the avoidance of unfavorable operating conditions, thereby ensuring the safe and stable operation of hydro-turbine generators. For example, Li et al. calculated the thermal stress distribution of the Three Gorges hydro-generator under short-circuit faults using multi-field coupling analysis technology [62], which served as a reference for diagnosing inter-turn short-circuit faults in the generator. Furthermore, multi-field coupling analysis technology can also provide theoretical guidance for the design of large-capacity hydro-generators, assisting engineers in designing more efficient and reliable generators.

Collectively, these studies have contributed to a deeper understanding of the complex interplay between temperature, thermal stress, and the operational performance of electric motors. They have enriched the results in the field of thermal stress calculation of electric machines from diverse perspectives and propelled the continuous development of this vital area of research. As power demand continues to grow and generator capacities increase, the insights gained from these studies will be instrumental in ensuring the safe, reliable, and efficient operation of electric motors in various applications.

5. The Influence of Control Strategies on the Multi-Physical Fields

In the operation process of the hydroelectric generator, the control part plays a vital role in its operation status. As the core device for controlling the rotational speed of the hydroelectric generator, the governor can ensure the stable operation of the motor under different working conditions. In the running process of the hydraulic turbine, the governor regulates the flow rate, rotational speed, and other parameters of the hydraulic turbine so that the output active power and frequency of the generator are kept stable. The excitation controller is an important device for controlling the magnetic field strength of the generator, which changes the output voltage of the generator by adjusting the excitation current. When the generator starts, runs, and the load changes, the excitation controller ensures the magnetic field of the generator is stable to ensure the output reactive power and voltage of the generator is stable.

5.1. Progress of Control Technology of the Hydro-generator

The traditional hydraulic turbine governor adopts a mechanical-hydraulic type governor or electrical-hydraulic type governor, which can only realize fixed proportional-integral (PI) or proportional-integral-differential (PID) control law. However, the operating conditions of hydropower units are complex and variable, and they need to meet the requirements of different tasks, such as frequency tracking and power regulation. Therefore, the microcomputer governor that can adapt to different working conditions comes into being. It can realize more advanced hydraulic turbine control strategies, such as adaptive PID control [80], fuzzy PID chaotic control [81], fractional PID contro [82], robust control [83], sliding mode control [84], self-resistant control [85], and so on. Zhang et al. [86] used improved active disturbance rejection control (ADRC) based on a generalized differentiator (GD) to realize the control of the hydro-generator speed control system. This algorithm achieved better results in both tracking and disturbance response, as shown in Figure 9.

When the rotational speed rises to a certain value, the excitation controller is put into operation. Under this condition, the excitation control and speed control work together on the unit. The excitation control and speed control are designed as a coupled system, and, in the early 1970s, Yu [87] and others first used linear optimal control theory to design the integrated generator controller, which further improves the stability of the system and the anti-disturbance ability. Subsequently, some nonlinear control theories were applied, such as differential geometry method [88], self-resistant method [89], inverse system method [89], and robust control [90], etc., in the integrated control of a hydraulic turbine, which achieved better control effect than linearized methods. In recent years, with the development of advanced algorithms, excitation control algorithms such as sliding mode control [84], PID control based on particle swarm algorithm [91], neural network adaptive control [92], and adaptive grayscale predictive control [93] have appeared one after another.

5.2. The Influence of Control Signal on Physical Fields

The control methods directly affect the motor speed and excitation current, which in turn leads to changes in the electromagnetic field distribution. Further, the motor structure field and temperature field will change due to the change of electromagnetic field, and the motor will experience different degrees of vibration and temperature rise. From the point of view of motor operation efficiency and service life, it is necessary to find a suitable control method to make the vibration and temperature rise of the motor as small as possible. Therefore, it is necessary to analyze the action mechanism of the control method affecting the physical field of the motor.

Many scholars have studied this issue, mostly focusing on small motors in automobiles. First, the control strategy affects the vibration aspect. Yu et al. [94] explored the impact of motor control strategy on electromagnetic vibration of an electric vehicle powertrain. They built a motor control model and a mechanical structure model to analyze the impact of two commonly used motor control strategies on the vibration performance of electric vehicles and provide solution ideas for electric vehicle vibration from the perspective of optimal control, as shown in Figure 10. Shi et al. [95] established a complete analytical model of the control strategy to the motor vibration noise response, and compared and analyzed the impact of harmonics affected by the control strategy on the vibration noise of permanent magnet synchronous motors, and the computational results were experimentally verified. Bahri et al. [96] proposed the switched reluctance machine (SRM) electromagnetic mechanics model based on the Shi et al. model to analyze the effect of two current control strategies on the stator structure vibration. The results show that the adaptive current control with fixed switching frequency has a lower vibration level.

Second, the control strategy affects the temperature rise aspect. Zhang et al. [97], in the study of thermal dynamics of automobile motors combined with the model predictive control strategy, proposed a predictive energy management system (EMS) system considering electromagnetic thermal control to avoid overheating of the motor while ensuring fuel economy. Wu et al. [98] built a field-circuit coupling simulation model of an embedded permanent magnet synchronous motor and carried out the field-circuit coupling calculations under current and weak magnetic speed control. They analyzed the temperature field distribution of the motor under different working conditions. Liang et al. [99] investigated the effects of the pulse-width modulation (PWM) control strategy on the quality of the motor current, the reduction of the motor losses, the motor temperature rise, and the suppression of the operation noise from the perspectives of experiments and simulations, and compared them with the traditional strategy, which confirms the effectiveness of the PWM control strategy.

Currently, most of the studies on the effect of control on the physical field of motors are focused on small motors, and studies on large hydroelectric generators are lacking. Large hydroelectric generator physical fields are mostly considered to be related to structural characteristics. However, from the study of small motors, it can be found that the effect of control strategy on the physical field is real and cannot be ignored. If vibration and temperature rise can be reduced by control methods, it will bring great convenience and save a large amount of cost.

6. Discussion

This review delves deeply into the intricate multi-physics phenomenon exhibited by hydroelectric generators, an area of study that is both fascinating and complex. It sheds light on several critical aspects of generator performance, each influenced by the interplay of multiple physical fields. At the heart of this exploration lies the electromagnetic vibration characteristics of the generator. This is not a standalone phenomenon but rather one that emerges from the dynamic interaction between the electromagnetic field and the structural field. Understanding these vibrations is crucial, as they can significantly impact the operational stability and lifespan of the generator. The review meticulously examines how variations in the electromagnetic field can induce vibrations in the generator’s structure, and vice versa, highlighting the need for a comprehensive analysis that encompasses both fields.

Moving beyond electromagnetic vibrations, the review also focuses on the ventilation and heat dissipation performance of the generator. This aspect is particularly important given the high temperatures generated during operation. The coupling effect of the flow field and the temperature field plays a pivotal role here. The flow field, driven by the cooling systems within the generator, interacts with the temperature field to determine the efficiency of heat dissipation. A thorough understanding of this coupling effect is essential for optimizing the generator’s cooling system, ensuring it operates within safe temperature limits, and preventing potential thermal failures.

Furthermore, the review delves into the thermal stress distribution within the generator. Thermal stress arises due to the uneven distribution of temperature within the generator’s components, leading to differential thermal expansion and contraction. This phenomenon is further complicated by the interaction of the electromagnetic field, temperature field, and structural field. This review emphasizes the importance of considering all these fields simultaneously when analyzing thermal stress and highlights that a holistic approach is necessary to accurately predict and mitigate thermal stress-related issues in hydroelectric generators.

Another critical area explored in the review is the influence of electromechanical control signals on the multiple physical fields of the generator. Electromechanical control systems are integral to the operation of hydroelectric generators, regulating various parameters such as speed, power output, and voltage. However, these control signals can also have a profound impact on the physical fields within the generator. Changes in control signals can alter the electromagnetic field, flow field, and temperature field, leading to changes in generator performance and potentially inducing unwanted vibrations or thermal stresses.

7. Conclusions and Outlook

In this review, the complex multi-physical phenomena of hydro-generators are discussed in depth. At the same time, the key influence of multi-physical field interaction on generator performance is revealed. Despite the comprehensive analysis presented in the review, much remains to be explored in this field. Collaborative efforts are urgently needed to advance our understanding of the multi-physics phenomena in hydroelectric generators. This collaboration should encompass experiments, numerical simulations, and model development.

• The magnetic-structure coupling analysis of the electromagnetic vibration of hydroelectric generators is still limited to the two-dimensional electromagnetic field. While this simplification has allowed for some initial insights, it falls short of capturing the full complexity of the three-dimensional electromagnetic field. The influence mechanism of the three-dimensional field on electromagnetic vibrations deserves further study. Researchers should focus on developing more sophisticated models and simulation tools that can accurately represent the three-dimensional nature of the electromagnetic field and its interaction with the generator’s structure.
• The method of numerical calculation of the thermal stress of hydroelectric generators is still not perfect. Current methods often overlook the influence of the wind field, which plays a crucial role in heat dissipation and therefore affects thermal stress distribution. To address this gap, researchers need to include the wind field in their thermal stress analyses. This will require the development of new numerical methods that can comprehensively consider the influence of magnetic-thermal-fluid-structure coupling. These methods should be able to accurately predict thermal stress distributions under various operating conditions, providing valuable insights for generator design and optimization.
• The influence mechanism of the hydroelectric generator control strategy in the multi-physics field remains unclear. Control strategies have a profound impact on the physical fields within the generator, but our understanding of this impact is still limited. Researchers need to investigate how different control strategies affect the electromagnetic field, flow field, and temperature field. This will require a combination of experimental studies and numerical simulations to elucidate the complex interactions between control signals and the physical fields.

In conclusion, the multi-physics phenomenon in hydroelectric generators is a rich and complex area of study that offers numerous opportunities for research and innovation. Future research should focus on three-dimensional electromagnetic vibration analysis, thermal stress analysis under the influence of magneto-thermal fluid-structure coupling, and the influence of electromechanical control signals on physical field analysis. By collaborating across disciplines and using experiments, numerical simulations, and model development, researchers can deepen our understanding of these phenomena and pave the way for the design and optimization of more efficient, reliable, and sustainable hydroelectric generators.