Research on Control of Levitation Force and Torque of a Maglev Device for Water-Turbine Generator Set

Abstract

Hydropower generation is clean, pollution-free, and renewable, and has good social and economic benefits, so it is given priority for development throughout the world. The capacity of hydropower stations is increasing to 1000 MW from 700 MW. As the p value on the bearing reaches a new height, coupled with the original risk of easy damage, the thrust bearing faces new technical challenges. Maglev technology is studied and applied to a large vertical-shaft hydro-generator set to solve the bearing problem. The maglev device is designed, and the working principle is expounded, using active-control repulsive-suspension technology. The levitation-force addition and the torque cancellation are realized by controlling the frequency of the excitation power supply. The dynamic mathematical models of levitation force and torque are derived. Combined with the design and theoretical analysis, the vector-control strategy is developed and the simulation analysis is completed. According to the results, the controller is improved to enhance the response performance. Finally, a control experiment is carried out on the prototype, and the results verify the effectiveness of the design and control strategy.

1. Introduction

According to China’s National Energy Administration, the country generated 2485.3 billion KWH (kilowatt hours) of renewable energy in 2021, of which 1340.1 billion KWH was generated by water conservancy. It can be seen that hydropower accounts for a considerable proportion of about 54% in the field of renewable-energy generation [1]. At present, hydropower generation is given priority for development throughout the world. However, in hydropower stations that have been successfully put into use, the capacity of each unit is no more than 700 MW. In fact, the Baihetan and Wudongde hydropower stations are under construction, with a 1000 MW single capacity in China. One of the problems to study is thrust bearing, as its damage will affect the safety and stability of the unit [2,3]. The thrust bearing of the 1000 MW unit has a higher p value than the 700 MW unit, so greater load-capacity thrust bearings are required [4]. We can use new materials, technology, and structures to improve the thrust bearing. On the other hand, we can use maglev technology to reduce the load of thrust-bearing, to decrease the p value on the bearing. Meanwhile, it can improve the service life of thrust bearings even if it is applied to small-capacity units such as those that are 700 MW or less.

2. Structure and Principle of the Maglev Device

A disc electromagnetic levitation-reduction device is proposed for the thrust bearing of the large vertical-axis hydropower unit, as shown in Figure 1. The maglev load-reduction device is mainly composed of a primary stator and a secondary rotor plate. The stator consists of iron cores, excitation coils, and a back iron, and the rotor is a whole-disc conductor plate. The stator is fixed to the external support structure, and the rotor is fixed on the main shaft of the hydropower unit and rotates synchronously. When the primary coils carry an alternating current, a rotating magnetic field will be generated in the air gap. Eddy currents will be induced in the conductor plate [5]. The eddy field interacts with the primary magnetic field, and a space electromagnetic force is generated. The normal force is a repulsion force, which is used as the levitation force to reduce the load for the thrust bearing. The force density is limited by the saturation characteristics of the core with the conventional coils [6]. To improve the force density, the primary stator can be designed as the coreless structure of the superconducting coils. When lift force is generated, there is also torque, and the lift force and torque should be concertedly controlled [7], to ensure the stability of the thrust-bearing load and avoid electromagnetic or mechanical coupling to the power-generation system.

3. Torque Offsetting Design

For the maglev device, the levitation force is useful but the tangential force is useless, which causes an electromagnetic and mechanical-coupling effect between the maglev load-reduction device and the original hydrogenerator system. Through design and control, zero torque and a not-reduced levitation force can be achieved. Referencing the mechanical properties of an induction motor, the torque is written by

$$T=C_T{\varphi }_m{I}_2{}^{\prime }\cos {\theta }_2$$

(1)

where $C_T$ represents a coefficient composed of motor structure parameters, ${\varphi }_m$ represents the main flux of the air gap, and $I_2{}^{\prime }\cos {\theta }_2$ represents the active component of the secondary rotor current. If we make the power-supply voltage constant, ${\varphi }_m$ will be constant too, and the torque is proportional to $I_2{}^{\prime }\cos {\theta }_2$, with the direction depending on the relative moving direction of the exciting field and the secondary rotor.

Based on the above theory, a disc structure with positive and negative halves is proposed with two equal and opposite electromagnetic torques offsetting each other, and the same normal lift forces added are shown in Figure 2, which is the overhead view of Figure 1. The two halves of the maglev device are designed with the same structural parameters, and the excitations can be controlled to obtain the above effect. On the other hand, the hydrogenerator has a certain velocity by itself determined by the waterhead and water flow. Considering the original rotating speed, we control the moving of positive exciting magnetic field ahead of the rotor and the negative field lagging behind the rotor to obtain opposite torques and consistently control the relative velocities of the two halves, respectively, via the rotor plate.

Keeping the other structural and electrical parameters exactly the same, the frequencies of the positive ${\omega }_1$ and negative exciting magnetic field ${\omega }_2$ should meet

$${\omega }_1-{\omega }_2=\frac{2\pi n_{hg}p}{30}$$

(2)

where $n_{hg}$ represents the rotating speed of the hydrogenerator, p represents the pole-pair number of the disc device.

4. Calculation and Relationship of the Levitation Force and Torque

4.1. Analytic Calculation of the Levitation Force and Torque

According to Maxwell’s tensor theorem, the electromagnetic repulsion of the primary and the secondary is written as [8]

$$F_l=-\frac{{\mu }_0}{4}\left(\frac{{\left|B_1\right|}^2}{{\mu }_0{}^2}-{\left|H_1\right|}^2+\frac{{\left|B_2\right|}^2}{{\mu }_0{}^2}-{\left|H_2\right|}^2\right)$$

(3)

$$F_t=\frac{1}{2}\mathrm{Re}(B_2{H}_2{}^{*}-B_1{H}_1)$$

(4)

where $B_1$ and $H_1$ and $B_2$ and $H_2$ represent the normal magnetic flux density and the tangential magnetic field intensity of the primary upper surface and the lower surface, respectively.

Based on the multilayer electromagnetic physical model often used for a linear-induction motor, the normal levitation force and torque can be calculated by the surface impedance [9]

$$F_l=\frac{{\mu }_0}{4}{J}_1{}^2\left(1-\frac{k^2}{{\mu }_0{}^2{\omega }^2}{\left|z_2\right|}^2\right)$$

(5)

$$F_t=0.5J_1{}^2\mathrm{Re}\left(z_2\right)/v_s$$

(6)

where $J_1$ represents the equivalent surface linear-current amplitude, $\omega$ represents the slip frequency, $z_2$ represents the surface impedance of the air-gap layer, and $v_s$ represents the slip velocity of the primary and the secondary.

We built a prototype sample FEA (finite-element analysis) model to verify the accuracy of the above analytical formula. The results from the FEA and the analytical method are compared in Figure 3. With the primary and secondary parameter changing, such as the exciting current amplitude, slip frequency, the thickness of the conduction plate, as well as the length of the air gap, we can see that the normal lift-force curves from the two ways are very close to each other, and the difference is about 10% or less. The analytical formulas are verified to be valid to estimate the normal force at design time.

4.2. Relationship of the Levitation Force and Torque

For the device, the lift force and the torque are related to the primary exciting current, slip frequency, air gap, and the material and thickness of the secondary conduction plate. By studying the ratio of the lift force to the torque, the relationship of the lift force and the torque is uncovered, and a foundation for the forces controlling is laid. To avoid solving the complex magnetic field and highlight the key parameters, we consider a limiting case, that the exciting traveling-wave magnetic field rotates at an infinitely high speed. The skin effect of the eddy current will be obvious in the conductor plate, and the normal component of the magnetic field is almost zero on the surface of the conduction plate. We can assume that there is only a tangential component for the field. After a Fourier transform, the tangential field can be expressed as

$$H\left(x\right)={\mathrm{R}}_{\mathrm{e}}\left[H_a\left(k\right)\exp (ikx-\omega t)\right]$$

(7)

where $H(x)$ represents the tangential component of the magnetic field intensity, $k$ represents the $2\pi /\tau$, $\tau$ represents the pole pitch, $\omega$ represents the slip frequency, and $\omega =kv$, $v$ represents the slip line velocity.

According to Maxwell’s tensor theorem, the normal magnetic pressure per unit area in the air gap is written as

$$F_l=H^2/8\pi ={\left|H_a\left(k\right)\right|}^2/16\pi$$

(8)

where H represents the valid value, and $H_a(x)$ represents the amplitude of the magnetic field intensity. In addition, the tangential force per unit area can be expressed as

$$F_t=P_{\sigma }/v=k{\delta }_d{\left|H_a\left(k\right)\right|}^2/16\pi$$

(9)

where $P_{\sigma }$ is the energy loss per unit area of the secondary conduction plate, and ${\delta }_d$ represents the depth of propagation of the electromagnetic waves in the secondary conductor []. So $F_l/T$ approximates to [10]

$$F_l/T=\frac{\tau }{\pi R}{\left(\mu \sigma \omega \right)}^{1/2}$$

(10)

In addition, the structure parameters also affect the vertical and horizontal components of the magnetic field and the electromagnetic forces, such as the tooth-to-slot ratio, the shape and width-to-depth ratio of the slot, the thickness of the secondary plate, and the length of the air gap. All the above analysis is verified by FEA, and the results are consistent with the analytic calculation [6].

For further analysis and verification, we manufactured a prototype shown in Figure 4, and the test results are obtained, which are compared with those from FEA, as shown in Figure 5. We use the ratio of the normal force to the tangential force $F_l/F_t$ instead of $F_l/T$ to get the same order of magnitude. From the curves, we can see that the $F_l/F_t$ is proportional to ${\omega }^{1/2}$ under the high frequency, and it is greater than ${\omega }^{1/2}$ under the low frequency, which verifies the above theoretical analysis. The cut-off point is about at 30 Hz, so (10) can be rewritten as

$$F_l/T=\lambda \frac{\tau }{\pi R}{\left(\mu \sigma \omega \right)}^{1/2}$$

(11)

where λ is used as a coefficient to correct the ratio for the low frequency. Under the frequencies higher than 30 Hz, λ is about 1.

In addition, the prototype is designed and manufactured based on the experience of the traditional linear motor, as shown in Figure 4. The FEA model is further optimized through the structure parameters and the magnetic circuit to get a bigger normal force and a smaller torque. For example, we can use deeper slots and increase the effective area of the magnet to get the bigger normal component of the air-gap field and the ratio $F_l/T$, with the cooling condition permitting. From Figure 5, we can see that $F_l/T$ is improved by about 20% from the prototype to the optimized FEA model, with the curves in solid lines.

5. Control of the Normal Force and the Torque

5.1. Dynamic Mathematical Model of the Torque

Based on the torque-offsetting-design idea, we need to implement coordinated control of the normal force and the torque. According to the theory of coordinate transformation, we can obtain, respectively, the voltage equations of the stator side and the rotor side under a rotating-coordinate system by

$$\{\begin{array}{l}{u}_{sd}=R_s{i}_{sd}+p{\psi }_{sd}-{\omega }_{dqs}{\psi }_{sq}\\ u_{sq}=R_s{i}_{sq}+p{\psi }_{sq}+{\omega }_{dqs}{\psi }_{sd}\\ u_{rd}=R_r{i}_{rd}+p{\psi }_{rd}-{\omega }_{dqr}{\psi }_{rq}\\ u_{rq}=R_r{i}_{rq}+p{\psi }_{rq}+{\omega }_{dqr}{\psi }_{rd}\end{array}$$

(12)

and the flux equations are

$$\{\begin{array}{l}{\psi }_{sd}=L_s{i}_{sd}+L_m{i}_{rd}\\ {\psi }_{sq}=L_s{i}_{sq}+L_m{i}_{rq}\\ {\psi }_{rd}=L_m{i}_{sd}+L_r{i}_{rd}\\ {\psi }_{rq}=L_m{i}_{sq}+L_r{i}_{rq}\end{array}$$

(13)

where the subscripts s and r represent the stator and the rotor, respectively, d and q represent the d axis and q axis, respectively, and $L_s$, $L_r$ and $L_m$ represent the equivalent self and mutual induction as well as the referred-to calculation, respectively.

We use the rotor-flux orientation, and there are

$${\psi }_{rd}={\psi }_r,{\psi }_{rq}=0$$

(14)

Through the voltage equations and flux equations, we can get the dynamic mathematical model of the torque as

$$T_e=n_p\frac{L_m}{L_r}{i}_{sq}{\psi }_r$$

(15)

and there are

$$\{\begin{array}{l} {\psi }_r=\frac{L_m}{1+T_{r}p}{i}_{sd}\\ i_{sq}=\frac{T_r{\psi }_r}{L_m}{\omega }_s\end{array}$$

(16)

where $T_r$ represents the time constant, p represents the differential symbol, and ${\omega }_s$ represents the slip frequency. So, $T_e$ can be controlled by adjusting $i_{sq}$ and ${\psi }_r$, and ${\psi }_r$ can be controlled by $i_{sd}$. When ${\psi }_r$ is made constant, $T_e$ is determined by $i_{sq}$ depending on ${\omega }_s$.

5.2. Dynamic Mathematical Model of the Normal Lift Force

Based on the principle of electromagnetic induction, the normal force is produced by the interaction between the exciting current and the induced current. The normal force between a single exciting coil and a secondary coil can be represented by the secondary induced eddy current and structure parameters, as [11]

$$F_l=\frac{{\mu }_0{I}_r{}^{2}l}{2\pi \delta }$$

(17)

where $I_r$ represents the induced current on the secondary, $l$ represents the length of a closed coil, and $\delta$ represents the distance of the both coils. As Reiz concludes, when the primary coil carried the current, which will induce almost the same shaped eddy current as in the secondary conductor plate [12]. Assuming all the primary coils are located on the primary surface and all the induced eddy current is located on the surface of the plate, we can use a model of the electromagnetic force generated by the interaction of two identical current-carrying coils to simplify the analytical solution. For the proposed induction device, the normal force can be expressed as

$$F_l=C_e\frac{{\mu }_{0}l{W}_1{}^2{I}_r{}^2}{4\pi \delta p}$$

(18)

where $C_e$ represents the coefficient on the structure (and can be tested), $l$ represents the average length of the primary or the secondary closed coil, and $W_1$ represents the number of turns in series per phase.

Since

$$i_r{}^2=i_{rd}{}^2+i_{rq}{}^2$$

(19)

the dynamic mathematical model of the lift force can be obtained by

$$F_l=C_e\frac{{\mu }_{0}l{W}_1{}^2}{4\pi \delta p}{\left(\frac{L_m}{L_r}\right)}^2\left[{\left(\frac{{\psi }_r}{L_m}-i_{sd}\right)}^2+i_{sq}{}^2\right]$$

(20)

From the above model under the dq-coordinate system, the lift force depends on $i_{sd}$, $i_{sq}$ and the flux linkage of the rotor is ${\psi }_r$, which is up to and lagging behind $i_{sd}$, so $i_{sq}$ is related to ${\omega }_s$ and ${\psi }_r$. Thus, we can control the lift force by controlling $i_{sd}$ and $i_{sq}$.

5.3. Control Strategy of the Normal Force and the Torque

From Equation (16), ${\psi }_r$ is up to $i_{sd}$, and ${\psi }_r$ lags behind $i_{sd}$ under transient changing. ${\psi }_r$ is steady when $i_{sd}$ is constant, considering Equation (20), the normal force only depends on $i_{sq}$ and can be expressed as

$$F_l=C_e\frac{{\mu }_{0}l{W}_1{}^2}{4\pi \delta p}{\left(\frac{L_m}{L_r}\right)}^2{i}_{sq}{}^2$$

(21)

So, we can adjust $i_{sd}$ to control ${\psi }_r$ and adjust $i_{sq}$ to control $F_l$ by keeping ${\psi }_r$ constant. At the same time, the positive torque and the negative torque shown in Figure 2 are equal and opposite, because there are two opposite slip angular frequencies, with one leading and one lagging like ${\omega }_{s1}=-{\omega }_{s2}$, and ${\omega }_{s1}$ is the slip frequency from the positive half coils, and ${\omega }_{s2}$ is from the negative half coils. For the two halves, ${\psi }_r$ is the same, and the resultant torque is zero, thus, there is no need to control the torques.

A control strategy for the positive control shown in Figure 6 (a negative-control block diagram likehis, and the difference is ${\omega }_{s1}$ or ${\omega }_{s2}$) is developed to realize the decoupling control of the lift force and the flux linkage for the positive half [13,14]. The control of the primary $i_{sd}$ and $i_{sq}$ are designed as the inner control loops, while the lift force and the flux linkage are the same as the outer control loops.

The axial load of the bottom thrust bearing is used to control the lift force, and the axial load force is measured by a tension–pressure transducer. The given value of the lift force as the input bias comes from the given value $F_s{}^{*}$ and the tested value $F_s$ of the axial force, so $F_s$ is the positive feedback. The input bias is the control signal to control the primary q-axis current $i_{sq}$. For the flux-linkage loop, the input bias from the given value and the observation value of ${\psi }_r$ are used to control $i_{sd}$. The given value of ${\psi }_r$ can be estimated and obtained by Equations (11) and (15). Meanwhile, the real-time slip frequencies ${\omega }_{s1}$ and ${\omega }_{s2}$ are acquired from the primary currents and the flux linkage, and the synchronous angular frequencies, ${\omega }_1$ and ${\omega }_2$, of the positive and negative parts are obtained too. When the speed of the rotor varies, ${\omega }_{s1}$ and ${\omega }_{s2}$ have not changed yet, so the synchronous angular frequency ${\omega }_1$ and ${\omega }_2$ follow the changing of the rotor to control the lift force and keep the torques balanced.

6. Simulation and Experimental Verification of Control Strategy

Simulation models of the proposed device and the whole control system are built to verify the control strategy. The system-simulation model mainly includes the module of the load-reduction device, power inverter, controller, pulse generator, flux observer, observation modules, and so on [15,16,17].

6.1. Simulation Verification of Control Strategy

Assuming that the axial load of the thrust bearing is 780 kN, the actual bearing load will be 180 kN after load reduction, and the required lift force should be 600 kN. First, to verify the role of the flux linkage ${\psi }_r$, which started at 0 time and changed from 3.2 Wb to 2.7 Wb at 0.5 s, and the simulation results of the lift force, the forward force and the resultant torque, because of the change of the flux linkage, were obtained, as shown in Figure 7.

From the results, we can see that the lift force can be set up in a very short time, and the proposed device has a short startup time and small overshoot. The building of the flux linkage needs a period of time because of lagging behind the current. The flux linkage affects the torque but not the lift force. It will cause the half torque to drop and the resultant torque to fluctuate. Finally, after a brief adjustment, the resultant torque became zero. So, the results verified the theoretical analysis and control strategy.

During the operation of the hydropower-generation system, there must be a fluctuation in rotor speed because of the water turbine. To verify the feature against the rotor-speed fluctuation, assuming the rotor speed changed from 107 rpm to 125 rpm at 0.5 s, the simulation waves of the lift force and torque are obtained, as shown in Figure 8.

Since the current is controlled directly, by the above waves, we can see the slip frequency is invariable during the speed fluctuation, and the frequencies of the positive and negative coils can follow the speed fluctuation. So, the lift force and the torque remain invariable, and it states that the device system has good adaptability to rotor-speed change. The two half torques almost cancel out and produce no additional mechanical disturbance to the generator.

Since the axial-pressure sensor feeds back the lift-force signal indirectly, there must be a disturbance phenomenon during the transmission. To obtain the response to the disturbance, a pulse wave force of a 0.01-s width is applied in the axial direction at 0.4 s, and the simulation results are shown in Figure 9. When the given value of the pressure keeps constant, the axial-load disturbance will cause the mutation of the input bias, and $i_{sq}$ follows the mutation. The lift force follows the mutation of $i_{sq}$, because there is no delay inertia between $i_{sq}$ and the lift force. So, any electrical or mechanical disturbance will affect the lift force, and the controllers need to be optimized to improve the disturbance-rejection performance.

To reduce the fluctuation of the torque because of the flux linkage and improve the disturbance-rejection performance of the lift force, cascade control is used in the lift-force loop and the flux-linkage loop. The cascade control includes a main controller and an assistant controller. The axial-force mutation is implemented at 0.4 s, the flux linkage is adjusted at 0.6 s, and the simulation results of the lift force and the resultant torque are described in Figure 10.

From the results, we can see that the overshoot of the flux linkage is eliminated, and the fluctuation of the resultant torque is greatly reduced, under the cascade control. At 0.4 s, the influence of the axial force mutation is also weakened by the lift force. So, the added cascade control improved the control performance of the system more than the single PI controller.

6.2. Experimental Verification of Control Strategy

A prototype is manufactured for the experimental measurement in the laboratory, as shown in Figure 4, and the control module is built, as shown in Figure 11. To test the lift force, torque, and the performance of the control system with the rotor rotating, a permanent magnet servo motor is used to drive the rotor.

The positive and negative windings are excited, respectively, and the tested lift force is shown in Figure 12. We can see that the lift force increased, and the rotor speed rose, when the positive coils were applied the exciting current, because there is the torque from the positive coils, so the rotor is driven further. When we continue to add the exciting currents to the negative coils, the lift force continues to increase but the resultant torque is almost zero. To easily measure the torques on both sides, the test is carried out with the rotor blocked, and the results are shown in Figure 13. The torques of the two halves are equal and opposite, and the joint torque is almost zero. All of the results verify the effectiveness of the design and the control strategy.

It should be noted that due to the existence of mechanical stress and the limitation of machining accuracy, the actual test data of the suspension force and the torque are not accurate. At the same time, due to the limitation of the laboratory conditions, the experimental test has only completed the preliminary test, and more adequate experimental tests need to be carried out next.

7. Conclusions

Maglev technology is studied to apply to a large vertical-shaft hydro-generator set (1000 MW) to solve the bearing problem. A controlled maglev-reduction device is designed for the thrust bearing, and normal force is used as the lift force, so torque is useless. In this paper, through the theoretical study on electromagnetic force, a control strategy combined with a structure design is proposed. The dynamic mathematical models of the lift force and the torque are established, and the simulation is implemented. From the simulation, it states the torque and the lift force can realize decoupling control, and the control system has good following and disturbance resistance. By applying cascade controllers, the overshoot characteristics have also been significantly improved. In the lab, the preliminary tests were carried out, and the results confirm the validity of the design and control strategy.