Modeling and Analysis of a Deﬂection Type Permanent Magnet Synchronous Wind Generator System

: A novel type of multi-degree-of-freedom (multi-DOF) deﬂecting-type permanent-magnet synchronous wind generator (PMSWG) is constructed to improve the reliability and utilization of wind energy. The basic working principle of the multi-DOF deﬂecting-type permanent-magnet synchronous generator (PMSG) is introduced, and its structural size is experimentally and theoretically determined. Subsequently, the multi-DOF deﬂecting-type PMSG was used to operate a complete wind turbine. A prototype and three-dimensional (3D) model of the wind turbine is simulated, allowing one to analyze the aerodynamics of the turbine and power generation performance. The electromagnetic ﬁeld analysis is performed via analytical methods, followed by a 3D ﬁnite element and torque analyses. Furthermore, the wind turbine power generation characteristics curves are obtained through simulation software. Finally, transient analysis of post deﬂection is demonstrated. The before and after deﬂection values of the generator voltage, current, ﬂux linkage, and induced voltage are compared and analyzed, relying on simulations and experiments. Additionally, the wind tunnel experiment is used to compare voltage variation with wind direction. The comparison reveals that the wind generator phase voltage remains maximized with wind direction variation. The results conﬁrm that the proposed PMSWG has excellent performance and future research potential.


Introduction
Wind energy is deemed a prospective energy source, owing to its clean and economic characteristics. The current target of the World Wind Energy Association is to generate 30% of worldwide electricity using wind power by 2025. The wind generator has thus become a common research subject [1][2][3]. Currently, most common wind power generators are squirrel cage asynchronous generators, permanent-magnet synchronous generators (PMSG), and doubly-fed asynchronous generators. The squirrel cage asynchronous generator wind turbine has low efficiency, as it cannot adjust the speed to obtain maximum wind energy. The gearbox therein increases the weight of the generator and reduces its efficiency. The electrical energy produced by a doubly-fed asynchronous generator is directly connected to a power grid through a boost transformer. Power electronic devices are used in the speed control system of generators to generate power harmonics. Moreover, the lagging reactive power from a grid must be absorbed, as it increases the power factor and burdens the grid. Therefore, reactive power compensation must be performed to improve the power factor. A PMSG In Section 2, basic structural parameters and the mechanism of the generator are addressed. Section 3 presents electromagnetic field modeling via both the analytical method and three-dimensional (3D) finite element analysis. Simultaneously, both rotation and deflection torques are analyzed. Section 4 illustrates the operation of the generator as a wind turbine, and an investigation of its aerodynamics and power generation performances. Section 5 presents simulations and experimental evaluations. Finally, the discussion and concluding remarks are presented in Sections 6 and 7, respectively.

Structural Parameters and Mechanism of Multi-DOF Deflecting-Type PMSWG
A wind generator is one of the core components of a wind turbine. Traditional wind generators focus on a single wind direction, and wind energy efficiency substantially reduces when the wind direction changes. A deflecting-type PMSWG performs wind wheel deflection through a control system, which can align the wheel according to the wind direction and increase the rotation speed of the wheel. This control strategy can increase the wind energy use rate of the wind turbine, by performing rotation with multiple degrees of freedom; thus, the turbine not only rotates, but also deflects to a certain angle. Moreover, the wind turbine is directly connected to the generator, saving materials, and simplifying the structure of the turbine. In addition to the advantages of a simple structure, high power generation efficiency, and small size, the proposed novel wind generator exhibits large output torque, simple control structure, and stable power generation.

Generator Structure
The deflecting-type PMSG comprises a stator, a rotor, and windings. The inner core of the rotor contains a magnetic conductive material and a magnetic separator, comprising a non-magnetic conductive material, having the separator placed outside the inner core. The permanent magnets are arranged along the output radial direction between two adjacent magnetic separators that form staggered distributions of the North (N) and South (S) poles. The permanent magnets are sintered from NdFeB material with radial magnetization and a spherical shape, whereas the stator is cylindrical with two types of windings. The stator is made of M19-24G_2DSF0.950 material, and the manufacturing technique is almost the same as traditional PMSG. The distributed winding can operate the generator in a generating state, which is embedded in the pear-shaped groove, creating convenient embedding lines for heat dissipation and a high utility rate, with no insulation layer between the windings. The other type, concentrated winding, is evenly and symmetrically distributed on both sides of the generator middle plane, enabling the generator to operate in an electric state. The stator magnetic yoke composed of magnetic material comprises evenly divided stator pole boots with stator slots between them. The concentrated windings are placed on the stator pole boots. When the generator is used in the electric state, the concentrated windings conduct external power supply and deflect the rotor. Figure 1 shows the basic structural diagram of the generator.

Determination of Main Parameters
The designed wind generator is suitable for small power applications, owing to its simple structure and installation. The proposed generator can also perform grid-connected operations. The generator uses a battery to store electricity; hence, it requires several batteries to obtain a high rated output voltage. The output voltage selected herein is not significantly high to reduce the cost of the complete machine. Specifically, the designed generator has low voltage and high current, with the rated power and rated voltage set to 10 kW and 120 V, respectively.
According to wind speed variability, the generator is required to work in a wide range. Moreover, as no gearbox mechanism is used, the shaft is directly connected to the wind wheel coupling. Thus, the speed of the generator directly depends on the wind speed, and the generator must have a low rated speed. The starting wind speed of the generator is 2.5-3 m/s, while the operating wind speed range is 2.5-18 m/s, and the rated speed of the generator is set at 600 r/min. Compared with other permanent magnet materials, the NdFeB has a strong antidemagnetization ability and excellent magnetic properties, which is more suitable for use in small and medium-sized permanent-magnet wind turbines. Considering both experimental requirements as well as an economical and easy-to-assemble principle, we proposed a four-pole generator model rather than a multi-pole model (more than four poles). Table 1 lists the parameters of the generator.

Determination of Main Parameters
The designed wind generator is suitable for small power applications, owing to its simple structure and installation. The proposed generator can also perform grid-connected operations. The generator uses a battery to store electricity; hence, it requires several batteries to obtain a high rated output voltage. The output voltage selected herein is not significantly high to reduce the cost of the complete machine. Specifically, the designed generator has low voltage and high current, with the rated power and rated voltage set to 10 kW and 120 V, respectively.
According to wind speed variability, the generator is required to work in a wide range. Moreover, as no gearbox mechanism is used, the shaft is directly connected to the wind wheel coupling. Thus, the speed of the generator directly depends on the wind speed, and the generator must have a low rated speed. The starting wind speed of the generator is 2.5-3 m/s, while the operating wind speed range is 2.5-18 m/s, and the rated speed of the generator is set at 600 r/min. Compared with other permanent magnet materials, the NdFeB has a strong anti-demagnetization ability and excellent magnetic properties, which is more suitable for use in small and medium-sized permanent-magnet wind turbines. Considering both experimental requirements as well as an economical and easy-to-assemble principle, we proposed a four-pole generator model rather than a multi-pole model (more than four poles). Table 1 lists the parameters of the generator.

Electromagnetic Field Modeling and Torque Analysis
The magnetic field characteristics of a generator are crucial in generator analysis. The proposed generator's magnetic field model is established through the analytical method and finite element analysis. Moreover, the accuracy of the results of the two modeling methods is compared [21][22][23].

Analytical Method Modeling
According to the boundary conditions of magnetic media in different regions of the rotor space, the rotor can be divided into three regions, as shown in Figure 2. This is in the region between the rotor permanent magnet and stator winding, termed as Region 1, where both interact with each other to form an air gap magnetic field. The rotor permanent magnet forms Region 2, whereas the inside of the rotor permanent magnet is termed Region 3.

Electromagnetic Field Modeling and Torque Analysis
The magnetic field characteristics of a generator are crucial in generator analysis. The proposed generator's magnetic field model is established through the analytical method and finite element analysis. Moreover, the accuracy of the results of the two modeling methods is compared [21][22][23].

Analytical Method Modeling
According to the boundary conditions of magnetic media in different regions of the rotor space, the rotor can be divided into three regions, as shown in Figure 2. This is in the region between the rotor permanent magnet and stator winding, termed as Region 1, where both interact with each other to form an air gap magnetic field. The rotor permanent magnet forms Region 2, whereas the inside of the rotor permanent magnet is termed Region 3. The magnetization characteristic in the three regions is given as follows: where B 1 , B 2 , and B 3 are the magnetic induction intensity in region 1, 2, and 3, respectively, H 1 , H 2 and H 3 are the magnetic field intensities in region 1, 2, and 3, respectively, and μ 0 and μ r are the permeability of vacuum and relative permeability of the permanent magnet, respectively, and M 0 is the residual magnetization.
No current flows through any of the three spatial regions of the rotor; hence, the magnetic fields in all three regions satisfy the following equations: The relationship between the magnetic field density and scalar magnetic potential is H k = −Φ k ; in spherical coordinates, this relation can be expressed as: The magnetization characteristic in the three regions is given as follows: where B 1 , B 2 , and B 3 are the magnetic induction intensity in region 1, 2, and 3, respectively, H 1 , H 2 and H 3 are the magnetic field intensities in region 1, 2, and 3, respectively, and µ 0 and µ r are the permeability of vacuum and relative permeability of the permanent magnet, respectively, and M 0 is the residual magnetization.
No current flows through any of the three spatial regions of the rotor; hence, the magnetic fields in all three regions satisfy the following equations: Energies 2020, 13, 1524 6 of 22 The relationship between the magnetic field density and scalar magnetic potential is H k = −Φ k ; in spherical coordinates, this relation can be expressed as: The Laplace equation for a scalar magnetic potential in spherical coordinates is given as follows: Therefore, the general solution is: where Y m l (θ, ϕ) denotes the spherical harmonic function, P m l (cos θ) is the associated Legendre function, and A m l and B m l are unknown coefficients. According to the existing condition of the three regions, the boundary conditions of the motor are set as follows.
At infinity outside the rotor (r = ∞), the magnetic induction intensity is 0. That is: At the boundary between the air-gap region and the permanent magnet radius (r = R r ), as well as at the boundary between the rotor core and the permanent magnet body radius (r = R b ), the tangential components of the magnetic field and magnetic induction intensities are continuous. That is: B 3r|r=R r = B 2r|r=R r , H 3θ|r=R r = H 2θ|r=R r , H 3ϕ|r=R r = H 2ϕ|r=R r .
At the rotor core (r = 0), the magnetic induction intensity is finite: Thus, the expression of the air-gap flux density of a permanent magnet can be expressed as follows: As the generator is radially magnetizing, we exclusively focus on the analysis of B r . Moreover, if we exclusively consider the fundamental wave, the air-gap component B r changes in the direction of ϕ with the sinusoidal wave. Figure 3 shows the B r plot, wherein two positive and negative peaks are observed across the equator, which agrees with the structure of the four-rotor magnetic poles. When θ = 90 • , the air-gap magnetic density is the largest; thus, the closer the position is to the equator of the rotor magnetic pole, the greater the magnetic density. The amplitude of B r is approximately 0.5 T.

Finite Element Analysis Modeling
The analytical method can only calculate fundamental wave components of the magnetic field under ideal conditions, ignoring many unimportant conditions. Therefore, Finite Element Analysis (FEA) software was used to model and analyze the magnetic field more accurately.
The finite element model of the generator was established, and the material properties, boundary conditions, and excitation sources of the model were set. The specific simulation parameters are shown in Table 2. Figure 4 presents the finite element solution model of the generator.  The model parameters, which include the number of iteration steps calculated, were set. Additionally, the convergence error value was solved, and the percentage of units was updated at each iteration. Finally, a static simulation of the generator was performed. Based on an adaptive solution, the system error was small, and the target requirement could be achieved. Figures 5 and 6 show the magnetic density curve of the generator and the magnetic dense cloud plot of the rotor.
According to Figure 7, the air-gap component B r changes in the direction of φ with the sinusoidal wave; Figure 7 is nearly identical to Figure 3, obtained through the analytical method. Figure 8 compares the air-gap magnetic flux obtained using the finite element and analytical methods when θ = 90°. Based on Figure 8, the variation trend of air-gap magnetic flux is nearly identical for both methods, and the amplitudes are similar. The curve appears smooth because the analytical method neglects some factors, having less impact on the model. The finite element model simulates the change in the air-gap magnetic field under regular motor operation; thus, several harmonics exist in time and space. Consequently, the curve of the finite element method appears to be more irregular.

Finite Element Analysis Modeling
The analytical method can only calculate fundamental wave components of the magnetic field under ideal conditions, ignoring many unimportant conditions. Therefore, Finite Element Analysis (FEA) software was used to model and analyze the magnetic field more accurately.
The finite element model of the generator was established, and the material properties, boundary conditions, and excitation sources of the model were set. The specific simulation parameters are shown in Table 2. Figure 4 presents the finite element solution model of the generator. Step Size Step

Finite Element Analysis Modeling
The analytical method can only calculate fundamental wave components of the magnetic field under ideal conditions, ignoring many unimportant conditions. Therefore, Finite Element Analysis (FEA) software was used to model and analyze the magnetic field more accurately.
The finite element model of the generator was established, and the material properties, boundary conditions, and excitation sources of the model were set. The specific simulation parameters are shown in Table 2. Figure 4 presents the finite element solution model of the generator.  The model parameters, which include the number of iteration steps calculated, were set. Additionally, the convergence error value was solved, and the percentage of units was updated at each iteration. Finally, a static simulation of the generator was performed. Based on an adaptive solution, the system error was small, and the target requirement could be achieved. Figures 5 and 6 show the magnetic density curve of the generator and the magnetic dense cloud plot of the rotor.
According to Figure 7, the air-gap component B r changes in the direction of φ with the sinusoidal wave; Figure 7 is nearly identical to Figure 3, obtained through the analytical method. Figure 8 compares the air-gap magnetic flux obtained using the finite element and analytical methods when θ = 90°. Based on Figure 8, the variation trend of air-gap magnetic flux is nearly identical for both methods, and the amplitudes are similar. The curve appears smooth because the analytical method neglects some factors, having less impact on the model. The finite element model simulates the change in the air-gap magnetic field under regular motor operation; thus, several harmonics exist in time and space. Consequently, the curve of the finite element method appears to be more irregular. The model parameters, which include the number of iteration steps calculated, were set. Additionally, the convergence error value was solved, and the percentage of units was updated at each iteration. Finally, a static simulation of the generator was performed. Based on an adaptive solution, the system error was small, and the target requirement could be achieved. Figures 5 and 6 show the magnetic density curve of the generator and the magnetic dense cloud plot of the rotor.

Rotation Torque
The torque calculation considers the flux linkage before and after virtual displacement as constant, while the system is considered free of loss. Thus, the formula for estimating the total energy stored in the solution domain is given as: According to Figure 7, the air-gap component B r changes in the direction of ϕ with the sinusoidal wave; Figure 7 is nearly identical to Figure 3, obtained through the analytical method.

Rotation Torque
The torque calculation considers the flux linkage before and after virtual displacement as constant, while the system is considered free of loss. Thus, the formula for estimating the total energy stored in the solution domain is given as:

Rotation Torque
The torque calculation considers the flux linkage before and after virtual displacement as constant, while the system is considered free of loss. Thus, the formula for estimating the total energy stored in the solution domain is given as:

Rotation Torque
The torque calculation considers the flux linkage before and after virtual displacement as constant, while the system is considered free of loss. Thus, the formula for estimating the total energy stored in the solution domain is given as: Energies 2020, 13, 1524 where V denotes the volume of the field, and H represents the magnetic field intensity. According to the virtual displacement principle, the rotor is assumed to have a small angular displacement ϕ along the equator. Thus, the torque in the direction receiving ϕ by the rotor is: where the electromagnetic torque T ϕ is the rotation torque along the equator. Figure 9 shows the rotation torque of the generator, which is approximately symmetric and changes periodically.
Energies 2020, 13, x FOR PEER REVIEW 9 of 22 where V denotes the volume of the field, and H represents the magnetic field intensity. According to the virtual displacement principle, the rotor is assumed to have a small angular displacement φ ′ along the equator. Thus, the torque in the direction receiving φ ′ by the rotor is: where the electromagnetic torque T φ is the rotation torque along the equator. Figure 9 shows the rotation torque of the generator, which is approximately symmetric and changes periodically.

Deflection Torque
The controller can control the generator to change the operation state if deflection is required. Through a brief adjustment, the generator operates in the electric state. At this time, the energy storage power supply is used to drive the generator to deflect, the wind wheel is aligned with the wind direction, and the controller ensures that the generator operates in the generating state. Further, the working principle is the same when the generator operates in the electric state. That is, it follows a similar principle as that of the permanent-magnet deflection-type motor, in which both working modes rely on the interaction between the magnetic fields generated by the current coil and the permanent magnet. A current is fed to several coils to generate N-pole magnetic fields to enable deflection. Conversely, an opposite current is applied to several other coils, which consequently generate S-pole magnetic fields. Following the principle of like poles repel and unlike poles attract, the generated N-and S-poles are repulsed and attracted. Tables 3 lists the values of deflection torques at different angles. Based on these tables, when the deflection angle increases, the distance between the rotor and the stator increases, which causes a decrease in the deflection torque.

Deflection Torque
The controller can control the generator to change the operation state if deflection is required. Through a brief adjustment, the generator operates in the electric state. At this time, the energy storage power supply is used to drive the generator to deflect, the wind wheel is aligned with the wind direction, and the controller ensures that the generator operates in the generating state. Further, the working principle is the same when the generator operates in the electric state. That is, it follows a similar principle as that of the permanent-magnet deflection-type motor, in which both working modes rely on the interaction between the magnetic fields generated by the current coil and the permanent magnet. A current is fed to several coils to generate N-pole magnetic fields to enable deflection. Conversely, an opposite current is applied to several other coils, which consequently generate S-pole magnetic fields. Following the principle of like poles repel and unlike poles attract, the generated Nand S-poles are repulsed and attracted. Table 3 lists the values of deflection torques at different angles. Based on these tables, when the deflection angle increases, the distance between the rotor and the stator increases, which causes a decrease in the deflection torque.  Figure 10 shows the 3D model of the multi-DOF deflecting-type permanent-magnet synchronous wind turbine. The turbine usually comprises of a wind wheel, vertical shaft, bearing, bearing seat, tower, footing, coupling, multi-DOF deflecting-type PMSG, controller, and an inverter [24][25][26]. According to Figure 10, the wind turbine has a double-layer wheel structure, and each layer has three shafts. Figure 11a shows a two-dimensional structural model of the double-layer wind wheel, whereas Figure 11b shows a 3D structural model of the one-layer wind wheel. Table 4 lists the specific parameters of the wind turbine obtained from the measured data of the wind turbine prototype from the wind power laboratory at Hebei University of Science and Technology.

30°
0.74123 0.71937 Figure 10 shows the 3D model of the multi-DOF deflecting-type permanent-magnet synchronous wind turbine. The turbine usually comprises of a wind wheel, vertical shaft, bearing, bearing seat, tower, footing, coupling, multi-DOF deflecting-type PMSG, controller, and an inverter [24][25][26]. According to Figure 10, the wind turbine has a double-layer wheel structure, and each layer has three shafts. Figure 11a shows a two-dimensional structural model of the double-layer wind wheel, whereas Figure 11b shows a 3D structural model of the one-layer wind wheel. Table 4 lists the specific parameters of the wind turbine obtained from the measured data of the wind turbine prototype from the wind power laboratory at Hebei University of Science and Technology.     Figure 10 shows the 3D model of the multi-DOF deflecting-type permanent-magnet synchronous wind turbine. The turbine usually comprises of a wind wheel, vertical shaft, bearing, bearing seat, tower, footing, coupling, multi-DOF deflecting-type PMSG, controller, and an inverter [24][25][26]. According to Figure 10, the wind turbine has a double-layer wheel structure, and each layer has three shafts. Figure 11a shows a two-dimensional structural model of the double-layer wind wheel, whereas Figure 11b shows a 3D structural model of the one-layer wind wheel. Table 4 lists the specific parameters of the wind turbine obtained from the measured data of the wind turbine prototype from the wind power laboratory at Hebei University of Science and Technology.

Performance Analysis
A flow field analysis was performed using the CFX module of the ANSYS Workbench, by simulating the operation of the wind turbine in practical operations. The boundary condition setting is shown in Figure 12. The wind enters through the inlet, with speeds ranging from 2.5 m/s to 13 m/s. The pressure outlet has a static pressure of 0 Pa. The left and rights sides are set as a symmetrical structure, and the top and bottom are set to the wall. After the boundary conditions are established, the meshing is performed. Figure 13 is a schematic diagram of meshing. During the division process, the selected mesh type, grid layout, and algorithm will directly affect the accuracy of the numerical simulation. This study adopts the automatic mesh generation method, which is a relatively classic mesh division method in software. The specific solver parameters are presented in Table 5.
simulation. This study adopts the automatic mesh generation method, which is a relatively classic mesh division method in software. The specific solver parameters are presented in Table 5.
In the calculation process, the wind speed of the inflow airflow was maintained constant at 5 m/s, and the tip speed ratio changed between 0-1 with a variation in the corresponding parameters. Figure 14 shows the variation curve of the wind energy utilization coefficient with respect to the tip speed ratio, which ranges between 0.1 and 0.9; the wind energy utilization rate first increases and subsequently decreases with increasing tip speed ratio. The wind energy utilization coefficient becomes maximum at a tip speed ratio of 0.5. The wind turbine can maintain the maximum utilization efficiency of the wind energy if the rotation speed is stabilized at this value.    simulation. This study adopts the automatic mesh generation method, which is a relatively classic mesh division method in software. The specific solver parameters are presented in Table 5.
In the calculation process, the wind speed of the inflow airflow was maintained constant at 5 m/s, and the tip speed ratio changed between 0-1 with a variation in the corresponding parameters. Figure 14 shows the variation curve of the wind energy utilization coefficient with respect to the tip speed ratio, which ranges between 0.1 and 0.9; the wind energy utilization rate first increases and subsequently decreases with increasing tip speed ratio. The wind energy utilization coefficient becomes maximum at a tip speed ratio of 0.5. The wind turbine can maintain the maximum utilization efficiency of the wind energy if the rotation speed is stabilized at this value.     In the calculation process, the wind speed of the inflow airflow was maintained constant at 5 m/s, and the tip speed ratio changed between 0-1 with a variation in the corresponding parameters. Figure 14 shows the variation curve of the wind energy utilization coefficient with respect to the tip speed ratio, which ranges between 0.1 and 0.9; the wind energy utilization rate first increases and subsequently decreases with increasing tip speed ratio. The wind energy utilization coefficient becomes maximum at a tip speed ratio of 0.5. The wind turbine can maintain the maximum utilization efficiency of the wind energy if the rotation speed is stabilized at this value. Figure 15 shows the torque coefficient comparison curves of the three blades in one rotation period where the torque coefficient curves are roughly sinusoidal, and the sum of the torques of the three blades is considerably more stable than the individual blade torques. Figure 16 shows the torque coefficient comparison curves of the single-and double-layer wind wheels. The torque coefficient of the double-layer wind wheel is stronger and more stable than that of the single-layer wind wheel.
Energies 2020, 13, x FOR PEER REVIEW 12 of 22 Figure 14. Tip speed ratio-wind energy utilization coefficient curve. Figure 15 shows the torque coefficient comparison curves of the three blades in one rotation period where the torque coefficient curves are roughly sinusoidal, and the sum of the torques of the three blades is considerably more stable than the individual blade torques. Figure 16 shows the torque coefficient comparison curves of the single-and double-layer wind wheels. The torque coefficient of the double-layer wind wheel is stronger and more stable than that of the single-layer wind wheel.   Figures 17a and b show the top view and 3D model for different stagger angles of the two wind wheel layers, respectively. The rotation space is symmetric; hence, cutting the rotation period by half will allow a 180° rotation. The wind wheel has three blades; thus, the stagger angle of each blade is 60°. We considered 0°, 20°, 40°, and 60°stagger angles to analyze the torque coefficients; the corresponding results are depicted in Figure 18, which shows that the torque coefficient stabilizes with increasing stagger angle, and the torque coefficient is best when the stagger angle reaches 60°.    Figure 15 shows the torque coefficient comparison curves of the three blades in one rotation period where the torque coefficient curves are roughly sinusoidal, and the sum of the torques of the three blades is considerably more stable than the individual blade torques. Figure 16 shows the torque coefficient comparison curves of the single-and double-layer wind wheels. The torque coefficient of the double-layer wind wheel is stronger and more stable than that of the single-layer wind wheel.   Figures 17a and b show the top view and 3D model for different stagger angles of the two wind wheel layers, respectively. The rotation space is symmetric; hence, cutting the rotation period by half will allow a 180° rotation. The wind wheel has three blades; thus, the stagger angle of each blade is 60°. We considered 0°, 20°, 40°, and 60°stagger angles to analyze the torque coefficients; the corresponding results are depicted in Figure 18, which shows that the torque coefficient stabilizes with increasing stagger angle, and the torque coefficient is best when the stagger angle reaches 60°.    Figure 15 shows the torque coefficient comparison curves of the three blades in one rotation period where the torque coefficient curves are roughly sinusoidal, and the sum of the torques of the three blades is considerably more stable than the individual blade torques. Figure 16 shows the torque coefficient comparison curves of the single-and double-layer wind wheels. The torque coefficient of the double-layer wind wheel is stronger and more stable than that of the single-layer wind wheel.   Figures 17a and b show the top view and 3D model for different stagger angles of the two wind wheel layers, respectively. The rotation space is symmetric; hence, cutting the rotation period by half will allow a 180° rotation. The wind wheel has three blades; thus, the stagger angle of each blade is 60°. We considered 0°, 20°, 40°, and 60°stagger angles to analyze the torque coefficients; the corresponding results are depicted in Figure 18, which shows that the torque coefficient stabilizes with increasing stagger angle, and the torque coefficient is best when the stagger angle reaches 60°.   Figure 17a,b show the top view and 3D model for different stagger angles of the two wind wheel layers, respectively. The rotation space is symmetric; hence, cutting the rotation period by half will allow a 180 • rotation. The wind wheel has three blades; thus, the stagger angle of each blade is 60 • .
We considered 0 • , 20 • , 40 • , and 60 • stagger angles to analyze the torque coefficients; the corresponding results are depicted in Figure 18, which shows that the torque coefficient stabilizes with increasing stagger angle, and the torque coefficient is best when the stagger angle reaches 60 • .
The wind turbine deflection mode was analyzed in Figure 19, to verify the performance reliability of the multi-DOF deflecting-type PMSWG. The comparison results of the wind energy utilization at different deflecting angles are shown in Figure 20. The wind energy utility decreases as the deflection angle increases; however, the PMSG can still operate in a rational range.  The wind turbine deflection mode was analyzed in Figure 19, to verify the performance reliability of the multi-DOF deflecting-type PMSWG. The comparison results of the wind energy utilization at different deflecting angles are shown in Figure 20. The wind energy utility decreases as the deflection angle increases; however, the PMSG can still operate in a rational range.  The wind turbine deflection mode was analyzed in Figure 19, to verify the performance reliability of the multi-DOF deflecting-type PMSWG. The comparison results of the wind energy utilization at different deflecting angles are shown in Figure 20. The wind energy utility decreases as the deflection angle increases; however, the PMSG can still operate in a rational range.   The wind turbine deflection mode was analyzed in Figure 19, to verify the performance reliability of the multi-DOF deflecting-type PMSWG. The comparison results of the wind energy utilization at different deflecting angles are shown in Figure 20. The wind energy utility decreases as the deflection angle increases; however, the PMSG can still operate in a rational range.

Rotor Deflection Motion
The proposed generator performs deflection in multiple DOF, so that the rotor can achieve eccentric motion. The air-gap distribution between the stator and the rotor becomes unequal and changes as the rotor deviates from its position when eccentric motion begins. Figure 21 depicts the eccentric motion of the rotor. Here, we assume that the spherical coordinates of the rotor are (α, β, γ) before rotor deflection. Following rotor deflection, the deflection displacement is significantly smaller compared with the rotor radius displacement. At this point, the expression of the air-gap length in any plane (such as the OCZ plane) in the original coordinate system is given as: where θ is the angle at which the rotor center is deflected relative to the z-axis. Based on Figure 22, as the OC axis is obtained by rotating the XOY plane around the z-axis at angle φ, the length of OC can be expressed as follows: |OC| = ∆x sin(φ) + ∆y cos(φ).

Rotor Deflection Motion
The proposed generator performs deflection in multiple DOF, so that the rotor can achieve eccentric motion. The air-gap distribution between the stator and the rotor becomes unequal and changes as the rotor deviates from its position when eccentric motion begins. Figure 21 depicts the eccentric motion of the rotor.

Rotor Deflection Motion
The proposed generator performs deflection in multiple DOF, so that the rotor can achieve eccentric motion. The air-gap distribution between the stator and the rotor becomes unequal and changes as the rotor deviates from its position when eccentric motion begins. Figure 21 depicts the eccentric motion of the rotor. Here, we assume that the spherical coordinates of the rotor are (α, β, γ) before rotor deflection. Following rotor deflection, the deflection displacement is significantly smaller compared with the rotor radius displacement. At this point, the expression of the air-gap length in any plane (such as the OCZ plane) in the original coordinate system is given as: where θ is the angle at which the rotor center is deflected relative to the z-axis. Based on Figure 22, as the OC axis is obtained by rotating the XOY plane around the z-axis at angle φ, the length of OC can be expressed as follows: |OC| = ∆x sin(φ) + ∆y cos(φ). Here, we assume that the spherical coordinates of the rotor are (α, β, γ) before rotor deflection. Following rotor deflection, the deflection displacement is significantly smaller compared with the rotor radius displacement. At this point, the expression of the air-gap length in any plane (such as the OCZ plane) in the original coordinate system is given as: where θ is the angle at which the rotor center is deflected relative to the z-axis. Based on Figure 22, as the OC axis is obtained by rotating the XOY plane around the z-axis at angle ϕ, the length of OC can be expressed as follows: Energies 2020, 13, 1524

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Substituting Equation (1) in Equation (2), the length of the air-gap at any position of the rotor core represented in spherical coordinates can be expressed as follows: g 01 (θ, ϕ)= (∆x sin(ϕ) + ∆y cos(ϕ))· sin(θ) + ∆z cos(θ)]. (18) Hence, the air-gap post rotor deflection expression is: where g 0 is the equally distributed air-gap length between the stator and the rotor before rotor deflection.

Transient Analysis Post Deflection via Simulation
The operation of the generator automatically adjusts when the wind direction changes; hence, the generator has to operate in the electric state for a short duration. The working conditions at rotor deflections of 0°, 15°, and 30° were compared. During the simulation, the unstable state at the startup was ignored. There are three phase windings on stator, which are phase A, phase B and phase C. Considering phase A as an example, the comparison plots of the phase voltage, phase current, flux linkage, and induced voltage are shown in Figures 23, 24, 25, and 26, respectively.

Transient Analysis Post Deflection via Simulation
The operation of the generator automatically adjusts when the wind direction changes; hence, the generator has to operate in the electric state for a short duration. The working conditions at rotor deflections of 0 • , 15 • , and 30 • were compared. During the simulation, the unstable state at the startup was ignored. There are three phase windings on stator, which are phase A, phase B and phase C. Considering phase A as an example, the comparison plots of the phase voltage, phase current, flux linkage, and induced voltage are shown in Figure 23, Figure 24, Figure 25, and Figure 26, respectively.
In Figures 23 and 25, the amplitude and phase of the voltage and flux linkage remain nearly equal before and after a deflection. Therefore, the influence of harmonics on voltage and flux linkage is minimal. According to Figures 24 and 26, the amplitudes of current and induced voltage decrease, however, the induced voltage curve is relatively smoother than that of the current; thus, harmonics have a smaller effect on the induced voltage than the current. As can be observed here, harmonics can not only affect generator characteristics, but also reduce service life. Therefore, controlling harmonics is essential.
Hence, the air-gap post rotor deflection expression is: where g 0 is the equally distributed air-gap length between the stator and the rotor before rotor deflection.

Transient Analysis Post Deflection via Simulation
The operation of the generator automatically adjusts when the wind direction changes; hence, the generator has to operate in the electric state for a short duration. The working conditions at rotor deflections of 0°, 15°, and 30° were compared. During the simulation, the unstable state at the startup was ignored. There are three phase windings on stator, which are phase A, phase B and phase C. Considering phase A as an example, the comparison plots of the phase voltage, phase current, flux linkage, and induced voltage are shown in Figures 23, 24, 25, and 26, respectively.   In Figures 23 and 25, the amplitude and phase of the voltage and flux linkage remain nearly equal before and after a deflection. Therefore, the influence of harmonics on voltage and flux linkage is minimal. According to Figures 24 and 26, the amplitudes of current and induced voltage decrease, however, the induced voltage curve is relatively smoother than that of the current; thus, harmonics have a smaller effect on the induced voltage than the current. As can be observed here, harmonics can not only affect generator characteristics, but also reduce service life. Therefore, controlling harmonics is essential.

Experimental Transient Analysis Post Deflection
We used an experimental platform to analyze the working conditions at rotor deflections of 0°, 15°, and 30°. Figure 27 shows the corresponding experimental setup, and 2D models of the stator winding scheme, 3D model of the rotor, and 3D model of the stator are shown in Figures 28, 29, and 30, respectively.  In Figures 23 and 25, the amplitude and phase of the voltage and flux linkage remain nearly equal before and after a deflection. Therefore, the influence of harmonics on voltage and flux linkage is minimal. According to Figures 24 and 26, the amplitudes of current and induced voltage decrease, however, the induced voltage curve is relatively smoother than that of the current; thus, harmonics have a smaller effect on the induced voltage than the current. As can be observed here, harmonics can not only affect generator characteristics, but also reduce service life. Therefore, controlling harmonics is essential.

Experimental Transient Analysis Post Deflection
We used an experimental platform to analyze the working conditions at rotor deflections of 0°, 15°, and 30°. Figure 27 shows the corresponding experimental setup, and 2D models of the stator winding scheme, 3D model of the rotor, and 3D model of the stator are shown in Figures 28, 29, and 30, respectively.

Experimental Transient Analysis Post Deflection
We used an experimental platform to analyze the working conditions at rotor deflections of 0 • , 15 • , and 30 • . Figure 27 shows the corresponding experimental setup, and 2D models of the stator winding scheme, 3D model of the rotor, and 3D model of the stator are shown in Figure 28  In Figures 23 and 25, the amplitude and phase of the voltage and flux linkage remain nearly equal before and after a deflection. Therefore, the influence of harmonics on voltage and flux linkage is minimal. According to Figures 24 and 26, the amplitudes of current and induced voltage decrease, however, the induced voltage curve is relatively smoother than that of the current; thus, harmonics have a smaller effect on the induced voltage than the current. As can be observed here, harmonics can not only affect generator characteristics, but also reduce service life. Therefore, controlling harmonics is essential.

Experimental Transient Analysis Post Deflection
We used an experimental platform to analyze the working conditions at rotor deflections of 0°, 15°, and 30°. Figure 27 shows the corresponding experimental setup, and 2D models of the stator winding scheme, 3D model of the rotor, and 3D model of the stator are shown in Figures 28, 29, and 30, respectively.     Only the phase voltage and induced voltage were compared, because the phase current can be calculated from the phase voltage, and the flux linkage cannot be directly measured using an oscilloscope. The oscilloscope results showed excessive harmonics; hence, we eliminated these harmonics using a low-pass filter in Matlab. Phase A is considered as an example here, and the impact of rotor deflection on phase and induced voltages are shown in Figures 31 and 32, respectively.   Only the phase voltage and induced voltage were compared, because the phase current can be calculated from the phase voltage, and the flux linkage cannot be directly measured using an oscilloscope. The oscilloscope results showed excessive harmonics; hence, we eliminated these harmonics using a low-pass filter in Matlab. Phase A is considered as an example here, and the impact of rotor deflection on phase and induced voltages are shown in Figures 31 and 32, respectively.   Only the phase voltage and induced voltage were compared, because the phase current can be calculated from the phase voltage, and the flux linkage cannot be directly measured using an oscilloscope. The oscilloscope results showed excessive harmonics; hence, we eliminated these harmonics using a low-pass filter in Matlab. Phase A is considered as an example here, and the impact of rotor deflection on phase and induced voltages are shown in Figures 31 and 32 Only the phase voltage and induced voltage were compared, because the phase current can be calculated from the phase voltage, and the flux linkage cannot be directly measured using an oscilloscope. The oscilloscope results showed excessive harmonics; hence, we eliminated these harmonics using a low-pass filter in Matlab. Phase A is considered as an example here, and the impact of rotor deflection on phase and induced voltages are shown in Figures 31 and 32, respectively. Figures 31 and 32 show that the variation trends of phase and induced voltage remain almost identical before and after deflection. Moreover, both parameters exhibit a sinusoidal change over time. The voltage and back electro-motive force are slightly distorted post deflection, owing to harmonics; hence, the proposed generator can operate normally within the maximum deflection range.
Only the phase voltage and induced voltage were compared, because the phase current can be calculated from the phase voltage, and the flux linkage cannot be directly measured using an oscilloscope. The oscilloscope results showed excessive harmonics; hence, we eliminated these harmonics using a low-pass filter in Matlab. Phase A is considered as an example here, and the impact of rotor deflection on phase and induced voltages are shown in Figures 31 and 32, respectively.  Figures 31 and 32 show that the variation trends of phase and induced voltage remain almost identical before and after deflection. Moreover, both parameters exhibit a sinusoidal change over time. The voltage and back electro-motive force are slightly distorted post deflection, owing to harmonics; hence, the proposed generator can operate normally within the maximum deflection range.

Wind Tunnel Experiment
The wind tunnel experiment was used to study the relation between voltage variation and wind direction. The direction of the wind in the tunnel changed from −30° to 30°. The generator used is shown in Figure 27 to simulate a wind turbine (with bearing, bearing seat, tower, footing, coupling, controller, and inverter included). Figure 33 shows the wind turbine prototype, and the basic parameter of the wind tunnel model is shown in Table 6.
The generator power generation characteristic curves are obtained in Figure 34.

Wind Tunnel Experiment
The wind tunnel experiment was used to study the relation between voltage variation and wind direction. The direction of the wind in the tunnel changed from −30 • to 30 • . The generator used is shown in Figure 27 to simulate a wind turbine (with bearing, bearing seat, tower, footing, coupling, controller, and inverter included). Figure 33 shows the wind turbine prototype, and the basic parameter of the wind tunnel model is shown in Table 6.  Figures 31 and 32 show that the variation trends of phase and induced voltage remain almost identical before and after deflection. Moreover, both parameters exhibit a sinusoidal change over time. The voltage and back electro-motive force are slightly distorted post deflection, owing to harmonics; hence, the proposed generator can operate normally within the maximum deflection range.

Wind Tunnel Experiment
The wind tunnel experiment was used to study the relation between voltage variation and wind direction. The direction of the wind in the tunnel changed from −30° to 30°. The generator used is shown in Figure 27 to simulate a wind turbine (with bearing, bearing seat, tower, footing, coupling, controller, and inverter included). Figure 33 shows the wind turbine prototype, and the basic parameter of the wind tunnel model is shown in Table 6.
The generator power generation characteristic curves are obtained in Figure 34. Figures 34a-c show the speed, current and torque curves of the generator, respectively.   The generator power generation characteristic curves are obtained in Figure 34. Figure 34a-c show the speed, current and torque curves of the generator, respectively.   Figure 35 shows the voltage comparison curves post deflection. The voltage nearly maintained a peak value when the rotor deflection angle changed with the change of wind direction. Moreover, the voltage variation was substantial when no deflection occurred in the rotor. Thus, the wind tunnel experiment confirmed that the multi-DOF deflection-type PMSWG can significantly improve wind utility. The power and efficiency comparison curves post-deflection are presented in Figures 36 and  37, respectively. Both power and efficiency can improve by approximately 8 % to 10 % when compared with no deflection in the rotor, making it more attractive than other wind generators.  Figure 35 shows the voltage comparison curves post deflection. The voltage nearly maintained a peak value when the rotor deflection angle changed with the change of wind direction. Moreover, the voltage variation was substantial when no deflection occurred in the rotor. Thus, the wind tunnel Energies 2020, 13, 1524 20 of 22 experiment confirmed that the multi-DOF deflection-type PMSWG can significantly improve wind utility. The power and efficiency comparison curves post-deflection are presented in Figures 36 and 37, respectively. Both power and efficiency can improve by approximately 8% to 10% when compared with no deflection in the rotor, making it more attractive than other wind generators.

Discussion
To improve the utilization rate of wind energy, some researchers have proposed new wind turbine models, while others have analyzed different control and frequency regulation strategies [7][8][9][10][11][12]. In addition, several studies have focused on multi-DOF generators in a different field, such as the magnetic field, and a modeling analysis, rotor dynamics characteristics, and multiphysics coupling analysis. This article draws on the essence of the above two research areas. We use the characteristics of multi-DOF theory into wind turbine analysis to propose a high quality novel wind turbine model. The results from transient analysis with deflection relying on simulations and experiments and wind tunnel experiments have shown that multi-DOF PMSWG can effectively

Discussion
To improve the utilization rate of wind energy, some researchers have proposed new wind turbine models, while others have analyzed different control and frequency regulation strategies [7][8][9][10][11][12]. In addition, several studies have focused on multi-DOF generators in a different field, such as the magnetic field, and a modeling analysis, rotor dynamics characteristics, and multiphysics coupling analysis. This article draws on the essence of the above two research areas. We use the characteristics of multi-DOF theory into wind turbine analysis to propose a high quality novel wind turbine model. The results from transient analysis with deflection relying on simulations and experiments and wind tunnel experiments have shown that multi-DOF PMSWG can effectively

Discussion
To improve the utilization rate of wind energy, some researchers have proposed new wind turbine models, while others have analyzed different control and frequency regulation strategies [7][8][9][10][11][12]. In addition, several studies have focused on multi-DOF generators in a different field, such as the magnetic field, and a modeling analysis, rotor dynamics characteristics, and multiphysics coupling analysis. This article draws on the essence of the above two research areas. We use the characteristics of multi-DOF theory into wind turbine analysis to propose a high quality novel wind turbine model. The results from transient analysis with deflection relying on simulations and experiments and wind tunnel experiments have shown that multi-DOF PMSWG can effectively

Discussion
To improve the utilization rate of wind energy, some researchers have proposed new wind turbine models, while others have analyzed different control and frequency regulation strategies [7][8][9][10][11][12]. In addition, several studies have focused on multi-DOF generators in a different field, such as the Energies 2020, 13, 1524 21 of 22 magnetic field, and a modeling analysis, rotor dynamics characteristics, and multiphysics coupling analysis. This article draws on the essence of the above two research areas. We use the characteristics of multi-DOF theory into wind turbine analysis to propose a high quality novel wind turbine model. The results from transient analysis with deflection relying on simulations and experiments and wind tunnel experiments have shown that multi-DOF PMSWG can effectively improve the utilization rate of wind energy. This result provides a reference for optimizing the structure of wind generators and improving their operational stability.
Due to the fact that this wind turbine represents a novel type of structure, two improvements can be performed in the future. First, the structure needs to be improved to realize multi-DOF movement, stability, and collaboration of every component in the structure. Second, the wind turbine can only work in off-grid conditions. The grid-connection process should be considered, to use the wind turbine generation system connected to a grid.

Conclusions
This study focused on a deflecting-type PMSG used as a wind turbine. The model parameters and the mechanism of the generator were determined through simulation and experimental analyses. The magnetic field model and torque were simulated, and its air-gap magnetic field characteristics were calculated. Both the aerodynamics and power generation performance of the complete wind turbine were considered. Transient analysis post deflection was performed relying on the simulation and experimental results; further, the wind tunnel experiment was used to compare voltage variation according to the wind direction. The feasibility of the generator design was verified. In addition to the calculated data, the simulation and experimental results provide a theoretical basis and supporting data for the optimization of wind power generation systems.