Speed Estimation of the Melted Metal into the Electromagnetic Pump Based

on the Differential-Algebraic Method

ADRIAN DANILA, LUMINITA POPA Department Automatics

Transilvania University of Brasov ROMÂNIA

adrian.danila@unitbv.ro, mluminita2001@unitbv.ro

Abstract: - The electromagnetic pump is widely used for melted metals’ conveyance and dosage into the metallurgical processes. In the second chapter of this paper the mathematical model of the single-sided stator electromagnetic pump is presented. Then, in the third chapter, the differential-algebraic approach to estimate the speed of the induced from the stator voltages and currents is described. Based on these theoretical developments, the implementation of the differential-algebraic method to the speed estimation of the melted metal is presented within the fourth chapter of the paper. The effect of the corrupting noise due to real-time measurements onto the speed estimate is then analyzed by statistical means.

Key-Words: - electromagnetic pump, mathematical model, speed estimation

1 Introduction The electromagnetic pump is an induction motor with single-sided or double sided stator; the motor induced is the melted metal. The electromagnetic pumps may be used, [1] to extract the melted metal directly from the basis of the induction furnace with a non mobile structure and to cast and dose the melted metals from the mobile melting pots, Figure 1. The electromagnetic pumps may be easily inserted into complex plants for continuous casting. When used for melted metals conveyance, the electromagnetic pump takes the following advantages in comparison with other conveyance methods: (1) the control by electrical means of the liquid's flow-rate if the pump is supplied from controlled three-phased frequency converters, (2) the costs reduction due to the elimination of purging the channel with tempered metal scraps, (3) the increase of the metal's pureness because the non-metallic elements aren't drawn by the electromagnetic forces, (4) the melting furnace and the melting pot may be at the same level thus their structure is much simpler, and (5) the improvement of the working conditions within the casting departments. The main disadvantage of the electromagnetic pump is the channel maintenance especially to the channel's basis where the material is highly solicited. The control of the conveyance process from the furnace to the melting pot requires the liquid flow-rate to be measured. Due to the measurement conditions – high temperature and the presence of intense electromagnetic fields, the use of the direct flow-rate transducers is not possible. Therefore dedicated procedures are implemented in this case such as the metal’s height

control within the channel based on level sensors placed into the liquid metal. Another way to control the liquid metal flow-rate is to measure the weight-amount of the melting pot while casting the liquid metal. However, the sensors-based methods are of utmost concern in industrial applications and represent a significant cost as well as a reliability issue. Another approach is to determine the induced speed without speed or weight transducers based on stator currents, and voltages sensors and a field-oriented algorithm. This approach is also referred to as the sensorless control of the induction machine. Currents and voltage transducers are far less vulnerable than other sensors and the algorithm may be designed such as it can tolerate the failure of the sensor. Different techniques to estimate the speed of the induced without a speed transducer are generally available in the literature. In this paper, the authors used the ideas presented in [2] to estimate the speed of the melted

Fig.1. An electromagnetic pump for melted metal conveyance.

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metal. In this approach, the mathematical model of the induction motor is used to obtain an algebraic expression for the rotor speed in terms of system’s inputs, outputs and their derivatives.

2 The Mathematical Model of the Single-

Sided Stator Electromagnetic Pump The expressions of the physical variables of the progressive field and the electromagnetic force respectively may be deduced through the Maxwell's equations into the space above the inductor i.e. the air gap of the electromagnetic pump such as for the classical electric machines, [3]. In addition, for the induction electro-magnetic pumps the following phenomena have to be taken into account: the transversal edge effect, and the longitudinal edge effect (static and dynamic). The transversal edge effect consists in the modification of currents distribution through the transversal cross-section of the massive induced because the currents paths freely encircle the magnetic field. The static longitudinal edge effect is due to the stator’s unbalanced windings. The phenomenon consists in a non-symmetrical currents sequence through the stator's windings. If the magnitudes of the phase voltages are known and the electric and magnetic parameters of the pump are estimated through computations, than the direct- and inverse phase sequence of the stator currents may be determined with the following set of equations: ( )( )

BCiidd

CAABii

dd

Uj

IZIZ

UUIXjZ

IXjZ

⋅=⋅−⋅

−=⋅⋅⋅+⋅+

+⋅⋅⋅+⋅

3

43

43

0

0

(1)

the significance of the terms is: 1id Z,Z,Z - the stator direct- and inverse-phase sequence impedances respectively, id I,I - the symmetrical components of

the stator phase currents, 0X - the unbalanced total

reactance and CABCAB U,U,U - the stator phase

voltages. The extended mathematical of the electromagnetic pump is presented in [1]. For the on-line computations i.e. adaptive control and physical variables estimations, the simplified electrical equivalent diagram, Figure 2, is to be used; the significance of the components is as follows:

1Z σ - the leakage impedance with respect to the induced,

mX - the linkage inductance, '

aFe R,R - the equivalent

resistances corresponding to the cast-metal and channel

losses respectively '

2R - the equivalent resistance of the

induced and s - the slip. The simplified diagram is only valid for larger polar step inductors at low frequencies, [1].

3 The Differential-Algebraic Method to

Speed Estimation The expression of the algebraic speed estimator is obtained by differentiating the general space vector model of the induction motor as follows [2].

( )

( )

J

Mi

LJ

Lp

dt

d

iT

L

TpjTdt

d

uL

i

TpjTdt

id

Wh

R

h

R

R

R

R

−Ψ⋅⋅⋅

⋅=

Ω

⋅+

+Ψ⋅⋅Ω⋅⋅−⋅−=Ψ

⋅⋅

+⋅−

−Ψ⋅⋅Ω⋅⋅−⋅=

*21'

2

1

'2

'2

11

1

'2

1

Im

11

1

1

σγ

β

. (2)

Where: '211 ,, Ψiu are the space vectors of the stator

voltages, currents and the rotor flux linkage respectively,

p is the number of pair poles of the pump, '21 , RR are

the inductor and induced resistances, '21 ,, LLL h are the

inductor inductance, the mutual inductance and the

induced inductance respectively, '2

'2 RLTR = is the

induced time constant, ( )'21

2 LLLh ⋅−=σ is the total

leakage factor, Ω is the induced angular speed, WM is

the load torque. The following notations have been introduced for

simplification purposes: ( )'21 LLLh ⋅⋅= σβ and

( ) ( )[ ] [ ] [ ]'2

2111 11 LLTLLR hR ⋅⋅⋅+⋅= σσγ

Fig.2 The simplified equivalent electrical diagram of the electromagnetic pump.

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Fig.4 . The measurements of the melted metal speed observer. Real-time measurements altered by 0.1% corrupting noise – green graph and their mean value.

Differentiating (2) gives, [2]:

( )

dt

d

L

ui

dt

id

Tpj

Tpj

dt

ud

L

dt

id

dt

dTpj

dt

dTpj

Tdt

id

R

R

R

R

R

Ω⋅

⋅−⋅+⋅

⋅Ω⋅⋅−⋅⋅

−

−⋅⋅

+

+⋅−Ω

⋅Ψ⋅⋅Ω⋅⋅−

−Ψ

⋅⋅Ω⋅⋅−⋅=

1

11

1

1

1

1'2

'2

2

12

1

1

1

σγ

σ

γ

β

(3)

Solving the equation above for dtdΩ leads to a

complex function depending on Ω , the machine parameters, stator voltages, stator currents and their derivatives. It is proven in [2] that, in steady-state operation, the real part of the complex function is never stable and cannot be used as a speed observer. However, the imaginary part which is a second degree in Ω is stable and one of the two zeros is equal to the induced speed. In steady-state operation, the first derivative of the speed is zero and the last term in notation (3) becomes zero. Follows,

−

−⋅

+⋅=Ω 1

112

12

Cdt

id

BAD (4)

Where:

⋅−⋅⋅+⋅−=

11

1 11

Li

dt

id

TA

R σγ ,

12i

T

LB

R

h ⋅⋅

=β

,

dt

ud

Ldt

idC 1

1

1 1⋅

⋅+⋅−=σ

γ ,

RTpjD ⋅⋅−= . The expression (4) is a first order estimator of the speed. The coefficients of this expression contain the second and the first order derivatives of the stator currents and the first order derivative of the stator voltage so there is a noise concern. In the followings the influence of the measurement noise over the speed estimate is analyzed.

4 The Implementation of the Algebraic

Speed Observer and Conclusions The basic features of the electromagnetic pump taken into account are given in Table1.

Table 1: The Technical Features of the Experimental Induction Electromagnetic Pump.

No. Denomination Symbole Units Value 1 The rated

supply voltage

N1U V 220

2 The rated frequency

N1f Hz 50

3 The rated flow rate

mNQ skg 22

4 No. of units - - 2 5 Total power

of an unit tP W 9500

6 The induced - - cast-iron 7 The pump’s

slope s 0 10

Fig.3 The supply phase voltage - (1) and the measured phase voltage corrupted by random noise - (2).

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Fig.5 . The measurements errors and their average value for a 0.01% corrupting noise. To emphasize the behavior i.e. the stability of the speed observer and the influence of the corrupting noise over the estimate, a program in the MatLab environment has been implemented. In the first place, the general expresion of the speed observer given in notation (4) has been adapted for the model of the linear induction motors. Afterwards, a normal distributed noise with zero mean and the variance 1=λ has been added to the measured phase voltages and phase currents. The magnitude of the corrupting noise was referred as a percentage of the variabile’s magnitude, Figure 3. In Figure 4 the computation results referred to the speed estimations under 0.01% noise conditions are presented. The errors and the mean value of the measurements are also presented in Figure 5. The plots in Figure 6 referrs to the speed estimations under 0.05% noise conditions. The results show that the measurements are largely

affected by the corrupting noise. The mean values of the measurements and their standard deviation with respect to the noise measurements conditions are presented in Table 2.

Table 2: The Mean Values and Standard Deviation of the Speed Observer with Respect to the Noise

Conditions. No. Noise magnitude Mean Value Standard

Deviation - [%] [m/s] [m/s]2 2 0.10 0.0231 0.0258 3 0.05 0.0223 0.0127 4 0.04 0.0229 0.0098 5 0.03 0.0229 0.0074

As seen from Table 2 the statistical mean value of the speed observer is convergent to the theoretical value of the speed. These results prove that if the statistical characteristics of the corrupting noise are known then the sensorless algebraic speed observer is consistent in the statistical sense. In Figure 7 the probability distribution of the speed observer is depicted.

References:

[1] Peşteanu, O., Jgheab electromagnetic pentru

transportul fontei topite. Universitatea Transilvania din Braşov. Contract de cercetare nr. 126/1984

[2] Li M., Chiasson J., Bodson M., and Tolbert L.M., A Differential-Algebraic Approach to Speed Estimation

in an Induction Motor, In: IEEE Transactions on Automatic Control, Vol 51, No 7, pag 1172-1177.

[3] Chiasson J., Modeling and High-Performance

Control of Electric Machines, New York: Wiley, 2005

[4] Henneberger K., Electrical Machines II, Dynamic Behavior, Converter Supply, Aachen University, 2002

Fig.6 . The measurements errors and their average value for a 0.05% corrupting noise.

Fig.7 . The probability distribution function of the speed observer under 0.01% noise.

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ISSN: 1792-4294 238 ISBN: 978-960-474-203-5