Modeling and Control for Tubular Linear Permanent Magnet Synchronous Machines with Gas Springs in

Drilling Applications Shujun Zhang, Lars Einar Norum

Department of Electrical Power Engineering Norwegian University of Science and Technology, Trondheim, 7491, Norway

Abstract- A novel idea employing Tubular Linear Permanent

Magnet Synchronous Motors (TLPMSM) with two gas springs is presented in this paper. The mathematical model of TLPMSM is developed for the aim of controlling the stroke length of TLPMSM for drilling application. The control strategy based on transfer function model is designed. The simulation result of the control system shows the validity of design of controller. The good dynamic and static performance predicts the intended drive performance.

I. INTRODUCTION

In recent years, linear electric machines have been more extensively employed in industrial applications, such as industrial robots, deep-mining, elevator doors, compressors and artificial hearts. These machines have significant advantages in terms of good dynamic characteristics, efficiency, force control, position accuracy, and reliability [1-5]. In this paper, the oscillatory motion of linear electric machines is proposed. It can be effectively utilized in demanding offshore oil industry drilling application. The proposed tubular linear permanent magnet synchronous machine (TLPMSM) with two gas springs is suited as linear hammer, especially for hard rock drilling with mud as drilling fluid. This machine can directly transmit power from electric source to drill bit without any mechanical equipments like gears, bearings, driving shaft, and so on.

In order to facilitate the design optimization and accurate dynamic modeling of linear permanent magnet synchronous machine, a variety of techniques have been developed to realize optimal control performances. To control the position, speed, acceleration and force of linear permanent magnet synchronous machine, modeling, dynamic analysis, and parameter estimation have been done in [6-9]. Prescribed closed-loop speed control method offered an accurate realization in [10, 11]. Sensorless control method for a miniature application has been validated in [12]. An advanced scheme, based on Neural Network, has been proposed in [13] to compensate for sudden variations of the load.

The TLPMSM used in this work has been introduced in [5, 14]. Analysis of the force performance of this machine is carried out in [5]. Therefore, this paper deals with TLPMSM modeling and controller design and predicts the performance for the drilling application.

II. MODELING AND DYNAMIC ANALYSIS

Fig. 1 shows the TLPMSM prototype built for drilling application in offshore oil industry. The machine is connected to the shaft by an external spring. The machine consists of casing with stator winding, piston with permanent magnets, gas spring in each end of the casing. The very high pressure at the ends of the piston results in a high oscillation frequency of the system. The piston moves back and forth at the forced resonant mode by the electromagnetic force. The casing will move at the mode of forced vibration under forces by the gas springs when the piston is moving up and down.

The electromagnetic force is produce by the interaction between the magnetic fields in the airgap from the permanent magnet and current in the stator winding.

In the modeling, the following assumptions have also been made for the purpose of simplification:

1) No magnetic saturation due to large airgap. 2) No fringing of the magnetic circuit. 3) Eddy currents and hysteresis effects are neglected. 4) Air gap field is axial and independent of piston position. 5) No leakage between two gas chambers. 6) Temperature effects are negligible.

A. Machine design description The TLPMSM is axi-symmetric and the outer casing seals

the whole unit. The winding of the coils is made around a plastic casing used as a low friction bearing for the moving

Fig. 1. Tubular linear permanent magnet synchronous machine

piston. The stator winding lies in the middle of the casing. The piston is an assembly of laminated iron disks and permanent magnet disks. Each disk has a hole in the center so that these disks can be held together by a rod of stainless steel. At each end of the piston there is a disk of stainless steel which creates a sealing against the plastic casing.

Two small size springs are connected on each end of the piston in order to keep the piston in the middle of the TLPMSM when TLPMSM is at rest.

B. Gas springs Employing gas springs makes a relatively heavy piston

oscillate at high frequency. The stiffness coefficient of gas spring can be changed by changing the initial pressure of the gas in the chambers, by this the natural frequency of the system can be changed. Therefore, TLPMSM with gas springs can output a higher power than other designs.

The function of force and displacement is defined by:

0g g

gasg g

y yf A p

y y y y

= +

(1)

where gy maximum distance the piston can move y displacement of the piston from the zero position A cross-sectional area of the piston 0p initial pressure for the gas springs adiabatic constant of gas

Fig. 2 gives a characteristic curve of gas springs according to (1). It shows that the forces increase with the displacement of the piston and the characteristic curve of gas springs can be approximately taken as a linear curve when the displacement is very small or gy y . So, the force function can be approximated by:

gas gasf K y (2) where

gasK stiffness coefficient of two gas springs We have found the value of gasK from Fig.2.

C. Force analysis Fig.3 shows the forces on the piston. The forces acting on

the piston include electromagnetic force, force of friction, gravity of piston and forces from gas springs. The force

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02-400

-300

-200

-100

0

100

200

300

400

X: 0.01Y: 42.22

displacement of piston(m)

forc

e(N)

Fig.2. Curve shape of force and displacement of piston

balance can be expressed as follows: ( )22p e fp gas g

d y tm f f f f

dt = (3)

where pm mass of the piston ef electromagnetic force fpf force of friction between piston and casing gasf force from gas springs gf gravity of piston

Same as the piston, we can get the forces of the casing as follows:

( )22

TT gas fT T Tg

d y tm f f f f

dt = (4)

where ( )Ty t displacement of casing from the zero position

Tm mass of TLPMSM fTf friction force between casing and surroundings Tf force from external spring Tgf gravity of TLPMSM

Here, gf will be neglected because another coil will be designed to offset gravity of piston, and then these two springs will hold piston in the middle of casing when piston is at standstill. A strong force will impose on the shaft when the machine works. Therefore, the gravity of TLPMSM Tgf will be neglected when analyzing the forces. The friction force will be proportional to the velocity for simplicity [15].

The electromagnetic force is calculated using the well-known interaction formula between a magnetic field and a moving point charge, in this case embedded in a current-carrying conductor. This relationship is described by the Lorentz's force equation by (5).

( ) ( ) ( )e t i t= gF l B (5) where ( )e tF electromagnetic force vector l vector of length in the direction of current i gB flux density vector of the air-gap

fe

f g

x

yz

0

f gas

y

ffp

Direction of moving

Piston

Fig. 3. Force analysis for the piston (Here, o-x axis designates the origin. The

forces are distributed on o-x axial to make them understood easily.)

Maximum force is generated in this application when l and gB are orthogonal (= 2 ). It is a common practice to ensure this relationship in designs. Therefore the magnitude of electromagnetic force in scalar is given by (6) in TLPMSM.

( ) ( )e gf t N B D i t= (6) where N number of turns of the stator winding gB flux density of permanent magnet D diameter of the coil

D. Forced vibration The system behaves as two spring-mass-damper systems by

analyzing (3) and (4) in this case: one is gas springs-piston-damper; another is Spring-TLPMSM-damper. Motion function of the piston in mathematics is

( ) ( ) ( ) ( )2

2p p gas e

d y t dy tm b K y t f t

dtdt + + = . (7)

where pb friction coefficient between piston and casing When the sinusoidal current ( )mi sin t is supplied to the

coil the electromagnetic force produced by the coil will force the piston move at the mode of forced damped vibration. The frequency of the input current should be exactly same as the natural frequency of spring-piston-damper system to achieve the maximum energy transform.

The TLPMSM moves up and down at the mode of forced damped vibration and the force comes from gas springs. And motion function of TLPMSM in mathematics is

( ) ( ) ( ) ( )2

2T T

T T s T gas

d y t dy tm b K y t K y t

dtdt + + = . (8)

where Tb friction coefficient between casing and surroundings sK stiffness coefficient of external spring

E. Output power to the drill bit Assuming no restrictions on the movement i.e. no hammer

effect, the outputting mechanical power will be only friction. It can be expressed [1] as

( )212mec T Tm

P b y= . (9)

where frequency of resonance motion Tmy maximum value of displacement of TLPMSM When the casing hits the drill bit, the friction coefficient

between casing and surroundings will increase to a very large value rapidly.

III. CONTROL MODEL AND CONTROL ANALYSIS

According to the Kirchhoff's voltage law, the voltage equation in the coil circuit is as follows:

( ) ( ) ( ) ( )emfdi tu t R i t L u tdt= + + (10) Similarly, based on Faraday's law of electromagnetic

induction, which implies that a back electromagnetic force, emfu is induced in a conductor moving at a velocity in a

medium of magnetic flux density gB , and the magnitude of the induced voltage of the conductor is given by

( ) ( ) ( )( )emf g p e d y tu t N B D v t K dt= = . (11) where e gK =N B D ( )pv t velocity of piston And then, the mathematics model of TLPMSM can be

developed from (6-8) and (10-11). Application of the Laplace transformation with zero initial condition to the differential equations gives the transfer function of TLPMSM as

( ) ( )( )

( ) ( ) ( )( )2 2 2

TT

e gas

T T s p p gas e

Y sG s

U sK K

m s b s K Ls R m s b s K K s

= =

+ + + + + +

. (12)

Substituting the values of the various parameters from table 1 into (12), the model of TLPMSM is obtained:

( ) ( )( ) ( )( )

2

2 2

10.606 0.01 54180.1875

30.6418 57220.0004 4.74 0.064 0.001 5722 30.6418

TG s s s

s s s s

=

+ +

+ + + +

(13)

The system is stable because all the roots of the transfer function denominator polynomial have negative real parts. To ensure a robust control a controller is needed here. A PID controller with a control equation is given by (14)

( ) 1( ) ( ) ( )p di

de tu t K e t T e t dtdt T

= + + . (14)

where pK proportional gain dT derivative time iT integral time The transfer function of PID is

( ) 1( ) 1( )c p d i

U sG s K T sE s T s

= = + +

. (15)

TABLE I PARAMETERS OF TLPMSM

Resistance of coil ( R ) 4.74 Inductance of coil ( L ) 0.0004H Mass of TLPMSM ( Tm ) 0.606kg

Friction coefficient of TLPMSM ( Tb ) 0.01Ns m

Stiffness coefficient of spring ( sK ) 54180.1875 N m

Mass of piston ( pm ) 0.064kg

Friction coefficient of piston ( pb ) 0.001Ns m

Stiffness coefficient of gas springs(including two mechanical springs) ( gasK )

5722 N m

Flux density of permanent magnet ( gB ) 21.2 Wb m

Diameter of piston ( D ) 0.016m Number of turns in coil ( N ) 508

With the transfer function of the system and the controller derived, the parameters of the controller have to be determined so that the desired performance of the system would be achieved. Fig. 4 shows schematic block of the system with a compensator.

IV. IMPLEMENTATION OF TLPMSM CONTROL SYSTEM

The control objective is to keep the stroke length constant. A PI controller can be used, but the step response of this system with PI controller exhibits the hidden oscillation according to simulations. Therefore, a compensator shown in equation (16) is designed to control TLPMSM.

( ) 1 147.4048 10.003486 0.25 1c

G ss s

= +

+ (16) The developed system with accepted parameters is

simulated with MATLAB and the result shown in Fig.5 is satisfied.

V. CONCLUSION

The development of a TLPMSM control system with two gas springs has been presented. Compared with the other machine for drilling application, the use of TLPMSM as described, has the advantages of high efficiency, strong force and high frequency. The mathematic model of TLPMSM is established and a control scheme is developed based on this model. The simulation result shows that the system achieves good dynamic and static performance and the control strategy is feasible for light load. This system will be applied to drilling application in the offshore in future.

Fig. 4. Schematic model for control

Fig. 5. Step response of the closed-loop system

ACKNOWLEDGMENT

This work was supported by Research Council of Norway (NFR) and Resonator AS.

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