ers « back to results for ""
Below is a cache of http://www.cim.mcgill.ca/~ialab/ev/TM18-4_as_published.pdf. It's a snapshot of the page taken as our search engine crawled the Web.
The web site itself may have changed. You can check the current page or check for previous versions at the Internet Archive. Yahoo! is not affiliated with the authors of this page or responsible for its content.
Modeling and control of an electric arc furnace Modeling and Control of an Electric Arc Furnace
Benoit Boulet, Gino Lalli and Mark Ajersch
Centre for Intelligent Machines
McGill University
3480 University Street, Montréal, Québec, Canada H3A 2A7

Abstract
Electric arc furnaces (EAFs) are widely used in steelmaking and in
smelting of nonferrous metals. The EAF is the central process of
the so-called mini-mills, which produce steel mainly from scrap.
Typical EAFs operate at power levels from 10MW to 100MW.
The power level is directly related to production throughput, so it
is important to control the EAF at the highest possible average
power with a low variance to avoid breaker trips under current
surge conditions. For efficient power control, good dynamic
models of EAFs are required. This paper solves the electrical
circuit of an EAF with a floating neutral, proposes a dynamic
model for the EAF, and investigates simple proportional electrode
current and power control.
1 Introduction
Electric arc furnaces (EAFs) are widely used in steelmaking and in
smelting of nonferrous metals. The EAF is the central process of
the so-called mini-mills, which produce steel mainly from scrap.
Typical EAFs operate at power levels from 10MW to 100MW.
The power level is directly related to production throughput, so it
is important to control the EAF at the highest possible average
power with a low variance to avoid breaker trips under current
surge conditions. For efficient power control, good dynamic
models of EAFs are required [1].











2
Physical Model of Arc Furnace
Figure 1 shows the physical model of the electric arc furnace. In
this particular EAF model, there are three electrodes that are
moved vertically up and down with hydraulic actuators. Each of
these electrodes has a diameter of roughly 1.5m, weighs
approximately 40 tons and is 1 to 2 stories tall. In theory, the ore
is melted with a huge power surge from the electrodes. The actual
product is denser than the scrap and thus falls to the bottom of the
furnace creating the matte. Above the matte lies the slag where the
electrode tips are dipped. The tremendous heat created by these
electrodes causes the ore to liquefy and separate. Thereupon more
raw materials are placed in the furnace and the process repeats
itself.
2.1 Arcing
Arcing is a phenomenon that occurs when the electrodes are
moved above the slag. As the electrode approaches the slag,
current begins to jump from the electrode to the slag, creating
electric arcs. Depending on the magnitude of the input voltages of
the electrodes, the arcing distance can vary. Usually, arcing occurs
in a region within centimeters of the slag (approximately 10-
15cm). Therefore, the EAF model must take into account the
instances when x
1
, x
2
, x
3
are negative (i.e. the electrodes are
suspended above the slag). Figure 1 above shows the sign
convention used in the project.
3
Solving the Electrical Circuit of the EAF
To solve any electrical model, assumptions are made to facilitate
the derivation. Similarly, the EAF electrical circuit requires
several assumptions before reaching the final equations. The first
step in the analysis of the electrical circuit is to use Kirchoffs
Current Law (KCL) to equate currents and voltages. Figure 2
shows 4 nodes, one for each of the electrodes and the fourth
representing the virtual ground at the matte (V
m
). Using these
nodes, it is possible to determine the current in each electrode with
respect to each voltage and the conductance coefficients, using its
position as the input. Proper assumptions can facilitate derivations
thus calculating the following equation involving matrices:

[ ] [
][ ] [ ]
i
i
j
i
i
I
G
x
B
=
+

Equation 1: Current Matrix Model
Here, I
i
is a 3x1 matrix with electrode currents, G
ij
is a 3x3
conductance matrix and B
i
is a 3x1 constant matrix.
3.1
Assumptions
For the EAF circuit, several assumptions were made. It must be
noted that this is a three-phase circuit with a double configuration.
The outer resistances (inter-electrode resistances) form a delta-
circuit with the three nodes. The inner resistances (slag-to-matte
resistances) form a wye-connection with V
m
as a virtual ground
(floating neutral). Figure 2 shows the electrical model for the EAF
with the chosen direction of currents.

To simplify calculations, the inter-electrode resistances are
equivalent and represented by R. As for the slag-to-matte
resistances, tests showed that these resistances displayed inverse
linear relations with respect to their position. Consequently, by
taking the slag-to-matte conductances, the inverse function
becomes a linear relationship, which makes for simpler
calculations. The slag-to-matte conductances G
i
, where i
represents the electrode, can be written as:

i
i i
s
G
c x G
=
+

Equation 2: Slag-to-Matte Conductance
Figure 1:
Physical Model of EAF

MATTE
SLAG
1
m


3

2
AR
C

1
+ve X
-ve X
0-7803-7896-2/03/$17.00 ©2003 IEEE
3060
Proceedings of the American Control Conference
Denver, Colorado June 4-6, 2003 where c is the conductance coefficient (in Siemens/m), x
i
is the
immersion depth of the electrode in the slag (in m) and G
s
is the
total conductance of the slag (in S). In other words, G
s
is the
conductance of the slag when the electrodes are positioned at the
surface of the slag. Using these assumptions and KCL, it is now
possible to solve the EAF electrical circuit.
3.2 Nodal
Equations
The following sets of equations can be obtained by
applying KCL to each of the four nodes displayed in Figure 2.

1
12
1
2
1
1
13
1
3
(
)
(
)
(
)
m
I
G V
V
G V
V
G V
V
= + +
2
12
1
2
2
2
23
2
3
(
)
(
)
(
)
m
I
G V
V
G V
V
G V
V
= + +
3
3
3
13
1
3
23
2
3
(
)
(
)
(
)
m
I
G V
V
G V
V
G V
V
= +
1
1
2
2
3
3
(
)
(
)
(
) 0
m
m
m
G V
V
G V
V
G V
V + + =

Equation 3: KCL using Conductances

Equation 4 represents the expression for V
m
, which will be used to
replace V
m
in the KCL equations above.

1 1
2 2
3 3
1
2
3
m
G V
G V
G V
V
G
G
G
+
+
=
+
+

Equation 4: Expression for V
m
3.3 Current
(I
I
) Calculations
Considering that the currents of the three electrodes will behave in
a similar manner, it is not necessary to display in full detail the
complete derivation for all three currents. The derivation of I
2
and
I
3
therefore follows from I
1
. The final expression for the total
current I
1
flowing through electrode 1 is shown in Eqn 5, with the
position inputs properly factored. I
1
is equal
to:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
1 1
2 1
3 1
1 2 2
1
2
1
1 3 3
1
3
2
1 2
2 2
3 2
1
3
1 3
2 3
3 3
2
1
2
3
2
1
2
2
2
3
s
s
s
s
s
TOT
s
s
s
s
V c G
G
V c G
G
V c G
G
c c x V V
x
c c x V V
x V c G
G
V c G
G
V c G
I
G
x V c G
G
V c G V c G
G
V V
V
G
GG
+ + +
+ + +
+ + =
+
+
+
+

+

Equation 5: Current in Electrode 1 (I
1
)
Similarly, Equations 6 and 7 represent the total currents
I
2
and
I
3
respectively.
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
1
1 1
2 1
3 1
1 2
2 2
3 3
2 3 3
2
3
2
1 2 1
2
1
2
3
1 3
2 3
3 3
2
1
2
3
2
2
1
2
2
3
s
s
s
s
s
TOT
s
s
s
s
x
V c G
G
V c G
G
V c G
V c G
G
V c G
G
V c G
G
c c x V
V
x
c c x V
V
I
G
x
V c G V c G
G
V c G
G
V
V
V
G
GG +
+
+
+
+
+ +
+ +
+ =
+
+ +
+ + +

Equation 6: Current in Electrode 2 (I
2
)


(

)

(

(

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(
)
















+

+









+

+

+

+

+

+

+

+

+

+

+

+2
G
))

+

=

s
s

s

s

s

s
s

s

s

TOT

GG
G
V
V
V
V
V
x
c
c
V
V
x
c
c
G
G
c
V
G
G
c
V
G
G
c
V
x
G
G
c
V
G
G
c
V
G
c
V
x
G
c
V
G
c
V
G
G
c
V
x
G
I
3
2
2
2
1
2

3

2
1
2

3
2

3
2

1

3
1

3
1
3

3

3
2

3
1

3
2
3

2

2
2

1

2

1

3

1
2
1
1
1

3


Equation 7: Current in Electrode 3 (I
3
)

where

1
1
2
2
3
3
3
TOT
s
G
c x
c x
c x
G
=
+
+
+

Equation 8 represents the Current Matrix Model shown by Eqn 1

(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
1 1
2 1
1 3
2 3
1 2
2 2
3 1
1 2 2
1
2
3 2
3 3
1 3 3
1
3
1 2
2 2
1
1 1
2 1
2
3 3
3 1
3
2
2
2
2
2
1
s
s
s
s
s
s
s
s
s
s
s
TOT
V c G
G
V c G
G
V c G
G
V c G
V c G
G
V c G
G
V c G
G
c c x V
V
V c G
V c G
G
c c x V
V
V c G
G
V c G
G
I
V c G
G
V c G
G
I
V c
G
V c G
I
+ +
+
+ + +
+
+
+
+
+
+
+
+
+

=

(
)