Nonlinear Modeling and Control of Active Magnetic Bearings for A Flywheel Energy Storage System

Chih-Keng Chen Department of Mechanical and Automation

Dayeh University Changhua, Taiwan (R.O.C) ckchen@mail.dyu.edu.tw

Trung-Dung Chu Department of Mechanical and Automation

Dayeh University Changhua, Taiwan (R.O.C)

R0111042@mail.dyu.edu.tw

Abstract—In this study, a structure of a five degree of freedom flywheel energy storage system (FESS) is introduced. A nonlinear model of active magnetic bearing (AMB) system in the FESS is obtained by Lagrange’s equation. In this model, the current in each coil is treated as a state variable and the control input is the voltage applied to each coil, this approach offers more advantages than current control input approach. PID controllers with decentralized structure are proposed to control the nonlinear multiple-input multiple-output (MIMO) system. Dynamic behavior of the flywheel in magnetic bearings and performance of the controller is discussed in simulation results.

Keywords – Modeling; control; active magnetic bearings; flywheel energy storage system;

I. INTRODUCTION In recent years, when the need for cost effective, long life

cycle, high efficient and environment friendly energy storage systems become more important, flywheel energy storage system (FESS) has emerged as a viable solution for this problem [1].

FESS stores energy mechanically by rotating a rotor at high speed. If the rotor speed is increased, this system can store more energy. To reduce the losses from friction, the rotor is suspended in a vacuum chamber by a set of magnetic bearings.

Magnetic bearing is an important part in FESS, because it enables noncontact operation and can guarantee performance of the system at high speed without lubrications. Magnetic bearings can be classified into passive magnetic bearings (PMBs), which use permanent magnets and active magnetic bearings (AMBs), which use electromagnets. AMBs offer more flexibility and advantages than PMBs, i.e., the system can work in a wide range of speed or load changing. However, modeling and control of AMBs are challenges because AMBs have unstable behavior and are nonlinear mechatronic systems.

Most of control design approaches for AMBs are based on the linearized model about a nominal operating point [2][3]. The behavior of linear model is acceptable when the operating point is close enough to the linearized point. In order to ensure the system’s performance in a wide range of working conditions, a nonlinear model should be consider in controller design [4].

In this study, a five mechanical degree of freedom (DOF) FESS is introduced. A nonlinear electromechanical model of this system was derived from Lagrange’s equation by using symbolic computation package Maple®. A decentralized PID control strategy is applied to control the nonlinear multi input-multi output system. Finally, numerical simulation results are presented to demonstrate dynamic behaviour of the system and the controllers’ performance.

II. DESIGN OF FLYWHEEL SYSTEM

ab

zΩ

1x

2x

2y

1y

Gxxφ

yφ

Gy

Figure 1. Flywheel energy storage system

The rotor of FESS is suspended by an axial bearing and two radial bearings. Structure of flywheel energy storage system is showed on Figure 1.

The axial bearing consists of two electromagnets and a permanent magnet. The permanent magnet is used to compensate the gravity force active on the rotor. The force vector of electromagnet acts through the center of gravity (COG) and has no influence on the radial motion therefore the axial axis can be controlled by an individual controller.

Each radial bearing consists of four electromagnets (of the same parameters) controlled separately. Radial position sensors are integrated into the bearings (collocated).

The physical parameters of this FESS are showed on tables below.

2014 Sixth International Conference on Intelligent Human-Machine Systems and Cybernetics

978-1-4799-4955-7/14 $31.00 © 2014 IEEE

DOI 10.1109/IHMSC.2014.76

284

TABLE I. ROTOR PARAMETERS

Parameters Symbol Value Unit Mass m 98 [ ]Kg

Polar moment of inertial pJ 1.25 2Kgm� �� �

transversal moment of inertial tJ 1.38 2Kgm� �� �

Distance from COG to radial bearing 1

a 0.138 [ ]m

Distance from COG to radial bearing 2 b 0.138 [ ]m

TABLE II. AXIAL AND RADIAL MAGNETIC BEARING PARAMETERS

Parameters Symbol Value

Unit Axial Radial Axial Radial

Air permeability 0μ 0μ 4πe-3 4πe-3 [ ]/H m

Nominal air gap AT RT 0.6e-3 0.5e-3 [ ]m Cross section

area AA RA 2.1e-3 0.6e-3 2m� �� �

Number of coil AN RN 150 208 -

Coil resistance AR RR 0.2 0.515 [ ]Ω

Sensor gain AVK RVK 1000 1000 [ ]/V m

III. MODELING OF ROTOR DYNAMICS IN AMB SYSTEM

A. Elctromechanical model In this section, a model of AMB with single mechanical

degree of freedom (Figure 2) is introduced to illustrate the Lagrange’s equation approach for an electromechanical system.

xu +xu −

xi +xi −

RRRN RR

z,z

T x+ T x−x

RN

Figure 2. Single DOF AMB model

2

2 2

1 ,21 1 ,2 2

0,0.

M

E x x x x

M

E

Ke mx

Ke L q L q

VV

+ + − −

=

= +

==

�

� � (1)

Energy contributions of this system are showed in Equation (1), where MKe , MV are the kinetic and potential energy of mechanical part, EKe and EV are the kinetic and potential energy of electrical part, the electrical charge in each coil, ,x xq q+ − is generalized coordinates of electrical

part, x is the displacement of the rotor, and ,x xL L+ − are coil inductances. The relation of coil inductance with air gap RT and the coil characterizing parameters is described in Equation (2).

( )

( )

2

0

2

0

,2

.2

R Rx

R

R Rx

R

N ALT x

N ALT x

μ

μ

+

−

=−

=+

(2)

The dissipation of copper losses in the coils is

2 21 12 2R x R xP R q R q+ −= +� � . (3)

The dynamic equation of single DOF AMB model can be derived from Lagrange’s equation

d Pdt

∂ ∂ ∂� � − + =� ∂ ∂ ∂ �Q

s s s� �L L , (4)

where s is the generalized coordinate vector [ ], , ,T

x xq q x+ −=s (5) Q is a vector of generalized external forces (control input voltage and mechanical force)

[ ], , ,Tx x xu u F+ −=Q (6)

and L is the Lagrangian function M E M EKe Ke V V= + − −L . (7)

B. Rotor dynamics Consider a rigid rotor depicted in Figure 1 with

assumptions, there is no axial flexural and rotation angle should be small (except for spinning rotation Ω ).

The displacements at radial AMB1 and AMB2 are described in the mechanical generalized coordinates MAs

[ ]1 2 1 2, , , , .TMA x x y y z=s (8)

However, for more convenient when investigate in the influence of gyroscopic effect and unbalance effect on the rotating rotor, another generalized coordinates ( MGs ) fix on the rotor’s center of gravity is used. The relationship between

MAs and MGs can be written as

1 0 0 01 0 0 00 0 1 00 0 1 00 0 0 0 1

G

y

MA MG G

x

G

xab

yab

z

φ

φ

� �−� �� � � � � � = =� � − � � � � � � � �

s Cs . (9)

The equations of motion are obtained by the introduced approach. The generalized coordinates for electromechanical model are

E

MA

� �= �

� �

ss

s , (10)

where Es is the vector of electrical charges in the coils, for this model we have ten coils (eight coils of radial AMBs and two coils of axial AMB). Es can be written as

285

1 1 2 2 1 1 2 2

, , , , , , , , ,T

E x x x x y y y y z zq q q q q q q q q q+ − + − + − + − + −� �= � �s . (11) Assume that the rotor is rotating at an initial angular

velocity Ω . When the influence of gyroscopic effect and unbalance mass effect are taken into account, the translational and rotational kinetic energy of the rotor in MGs are

( )( ) ( )

( ) ( )

2 2 2 2 2_

2 2 2_

_ _

12

sin cos ,

1 1 2 ,2 2

,

M Trans G G G

G G

M Rot t x y p x y

M M Trans M Rot

Ke m x y z

m x t y t

Ke J J

Ke Ke Ke

ε

ε

φ φ φ φ

= + + + Ω

+ Ω − Ω + Ω� �� �

= + + Ω + Ω

= +

� � �

� �

� � � (12)

where mε is the eccentric mass. The kinetic energy of mechanical part MKe can be transform to MAs by using Equation (9).

The other terms of energy contribution in the system can be derived by the approach proposed in single DOF model and can be described as

102

1

102

1

1 ,2,

0,1 ,2

E i ii

M

E

i ii

Ke L q

V mgzV

P R q

=

=

=

==

=

�

�

�

�

(13)

where i is the thi row of vector .Es The generalized nonconservative forces are

1 1 2 2 1 1 2 2

1 2 1 2

[ , , , , , , , , , ..

.. , , , , ] .x x x x y y y y z z

Tx x y y z

u u u u u u u u u u

F F F F F+ − + − + − + − + −−=Q

(14)

Applied the Equation (4) and Equation (7) , the equations of motion of the system can be derived in a standard form of differential equation

,=Js U� (15) where 20 20×∈J � is the inertial matrix and 20 1×∈U � is the vector of nonlinear function. These equations are solved in MATLAB/Simulink by C MEX S-function block.

IV. CONTROL DESIGN

xu −

xu +

xi +

xi −

x

1 1, ,x xi i x

+ −

0i

cu

cu

Figure 3. Control diagram of the single DOF AMB

The obtained model treated the current in each coil as a state variable. It is more reality and also allows us to synthesis the controllers with the control input is voltage. [5]

has shown that voltage input control have more benefit then classical current input control, e.g., improved the robustness, high efficient and low cost (pulse width modulation implement, sensorless control). However, the plant model and controller is more complicated, and current control loops is required.

Control diagram of a single DOF magnetic bearing is showed on Figure 3. Because two electromagnets is symmetric, the control signal is supposes to a bias current 0i in both coils of positive and negative force direction respectively.

To control MIMO model of FESS, PID controllers with decentralized structure are proposed. Decentralized control strategy is independent control for each axis; this structure is common practice in industrial AMB system [7].

V. SIMULATION RESULTS In this section, dynamic behaviors of the system and

control performance are discussed in simulation results. The simulation’s parameters are showed on TABLE I. and TABLE II.

0 0.5 1 1.5 2

-2

0

2

4

6

8

10x 10

-6

Time(s)

Dis

plac

emen

t (m

)

x1

x2

y1

y2

Figure 4. Displacements of the rotor

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

Time(s)

Cur

rent

(A

)

ix1+

ix1-

ix2+

ix2-

iy1+

iy1-

iy2+

iy1-

Figure 5. Current inside each coil

To demonstrate the rotor dynamics clearly, a low bias current 0 0.1i = A is used. The rotor is rotating at an initial speed 15000Ω = rpm, the controller’s parameters,

38, 8, 3i dKp K K= = = and the initial displacement

286

( )1 0 0.01x = mm. Under the influence of gyroscopic effect, the control force in 1x direction will effect to the others. After about 1 s, the rotor’s displacements will be kept at the nominal value (Figure 4).

Figure 5 shows the responses of current loop control. After the transient time of the displacements, the current inside each coil is kept at the bias current 0.i

The rotor’s behavior in axial direction is showed on Figure 6 with 61, 42, 3i dKp K K= = = and ( )0 0.4z = −

mm.

0 0.5 1 1.5 2-5

-4

-3

-2

-1

0

1

x 10-4

Time (s)

Dis

plac

emen

t (m

)

Figure 6. Response of the rotor in axial direction

-1-0.5

0

0.51

x 10-6

-1

-0.5

0

0.5

1

x 10-6

0

500

1000

1500

2000

xG

(m)yG (m)

Ω (

rpm

)

Figure 7. Orbital tube of the rotor's displacement under unbalance effect

Effect of unbalance mass highly depends on the speed of rotor. Figure 7 shows the 1 2,x x displacement when the rotor’s speed is increased gradually. The resonance of unbalance effect and the natural frequency of the system occurred when the rotor speed at the first critical speed (about 1000 rpm).

To make the system become more robust when the unbalance effect is taken into account, the control parameter

55, 38, 1i dKp K K= = = and bias current 0 5.5i = A are proposed. When the speed of rotor reached 15000 rpm, the

current disturbances is fed into the coil 1 1,x y+ + are 5.5 A and 3 A. The orbital of rotor in plane ( )1 1,x y is showed on Figure 8. An unreal circle is used in plotting to prove that even under the influence of unbalance effect, gyroscopic effect and large disturbances inside the current loops, the rotor is still suspended with noncontact with bearings (displacements of rotor is smaller than 0.01 mm).

-1 -0.5 0 0.5 1

x 10-5

-1

-0.5

0

0.5

1x 10

-5

x1 (m)

y 1 (m

)

Figure 8. Orbital of rotor in the plane of radial magnetic bearing 1

VI. CONCLUSION In this study, a nonlinear electromechanical model of a

five degree of freedom FESS is obtained. The PID controller with decentralized structure is applied to control the rotor’s displacements. The simulation results showed that the controller can guarantee the rotor stays close at the desired displacement even when disturbance and dynamic effect of rotating are taken into consider.

ACKNOWLEDGMENT The work was supported by the Bureau of Energy,

Ministry of Economic in Taiwan, Republic of China, under the project number 102-D0624.

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