wETEWREYHRDHSFTUJDFGJTYULKFDGC,HMGC

1. Automatica, Vol.15,pp. 137 148
PergamonPressLtd. 1979.Printedin Great Britain
InternationalFederationof AutomaticControl
Electric Arc Furnace Modelling
and Control*
S. A. BILLINGS, F. M. BOLAND and H. NICHOLSON+
Electric arc.f'urmwe control may be improved considerably using a dmtl impedance/
current control strategy, a derivative regulator, a temperature weighting adaptive
controller and estimates of the molten steel temperature and carbon content provided
by an extended Kalman filter.
Key Word Index Adaptive control; identification; industrial control; Kalman filters; modelling; state
estimation.
Abstract Problems associated with the production of
special steels during the melting and refining stages in the
electric arc furnace are discussed. Three-phase models of an arc
impedance and arc current controlled furnace are devel-
oped by combining the results of an identification study with
analytically derived models. Current and impedance con-
trol strategies are compared and a dual impedance/current
controlled regulator is proposed. Adaptive control of the elec-
trode regulator is discussed and a temperature weighting adap-
tive controller is designed to compensate for changes in the
arc characteristics over the period of a melt. The problems
presented to the development of a mathematical model of the
refining process, by the complex metallurgical nature of the
process, are considered and the need for a compromise
between implied accuracy and complexity of the model is
discussed. A model of the process containing both determinis-
tic and stochastic components is presented and techniques for
evaluating the statistics of both the process and observation
noise vectors are considered. The implementation of the
extended Kalman filter for state estimation is considered and
a technique for controlling filter divergence is presented. The
results of simulation studies indicate that estimates of the
states can be obtained to the accuracy required for control of
the industrial process.
1. INTRODUCTION
SINCE its invention by Heroult in 1905 the
electric arc furnace has become an increasingly
important steelmaking process. The availability
of cheap scrap steel and the increased demand
for high quality steel products have contributed
to the rapid expansion of electric steelmaking
throughout the world. The fundamental problem
in the electric arc furnace is the production of a
specified steel at the lowest possible cost. Many
factors contribute to this cost and the operation
of the furnace can best be analysed by segment-
ing the production cycle into three stages consist-
*Received March 27 1978; revised August 28 1978; revised
October 15 1978. The original version of this paper was
presented at the 5th IFAC/IFIP Conference on Digital
Computer Applications to Process Control which was held in
The Hague, Netherlands during June 1977. The published
Proceedings of this IFAC Meeting may be ordered from: The
North Holland Publishing Co., P.O.B. 103, Amsterdam
West, Netherlands. This paper was recommended for publi-
cation in revised form by associate editor A. Longmuir.
tThe authors are with the Department of Control
Engineering, University of Sheffield, Mappin Street, Sheffield
SI 3JD.
ing of (i) scrap mix selection and load scheduling,
(ii) control during the melting cycle, and (iii)
refining control. A schematic diagram of the
production cycle is illustrated in Fig. 1.
Next
order
Leos~r
costmix~ _
strategy -
Order
specifications
$crop
Electric
power
ELectricpower
oxygen
furnaceadditions
FIG. 1. The composite steelmaking process in the electric arc
furnace.
137
Control of the scrap charge to achieve a least-
cost-mix has been studied extensively within the
industry and solutions are available which make
use of linear programming techniques. These are,
however, limited in application by the absence of
suitable process models relating the yield at melt-
out to the customer specification. Since both the
problems of scrap mix selection and load
scheduling for maximum demand control have
been the subject of many previous studies they
will not be discussed here. The present research
has been directed towards increasing the el'-

2. 13S S.A. BII.I~IN(is, f:. M. I~(.~I~AN[)and H. NI(II()I.SON
ficiency of steel production in the electric arc
furnace by designing improved regulators for
power input control during the melting and re-
fining stages and developing a state-estimator
suitable for use in a computer-aided refining
control system.
Although the basic concepts of arc furnace
design have remained virtually unchanged for
almost half a century, a considerable amount of
research has been directed towards improving the
design of electrode regulators (Driller, 1954;
Kolkwiewiez, 1967; Roebuck, 1969; Morris and
Sterling, 1975) Electrode and refractory consum-
ption have been studied (McGee and
Ravenscroft, 1959) and refractory index control
has been investigated (Schwabe, 1962~. The prob-
lems of power system loading and voltage flick-
er have also been studied extensively ISchwabe.
1958, Payne, 1959; Robinson, 19521.
The available information on high power, high
current electric arc discharges is limited
(Bowman, Jordan and Fitzgerald. 1969; Bowman
and Fitzgerald, 1973), and few researchers have
investigated the properties of arcs in a pro-
duction furnace.
Direct digital control of arc furnace operation
has to date been limited to supervisory duties
(Fielder, Tippet and Whitwell, 1965}, although
the feasibility of automatic power input control,
(McGee and Ravenscroft, 1959) and on-line op-
timal control (Nicholson and Roebuck, 1970,
1971) has been demonstrated.
Various authors (Morris and Sterling, 1975;
Nicholson and Roebuck, 1972) have developed
single phase models of the electrode control
system but very few researchers have studied the
influence of the arc characteristics upon the con-
troller response, or the selection of a controlled
variable.
Computer control of the production of steel in
the electric arc furnace has, to date, been largely
concerned with decisions for optimum plant and
scrap useage with a few applications to the
control of batch production in individual fur-
naces. Gosiewski and Wierzbicki (1970) have
shown that optimal control of power input dur-
ing the melting stage could give significant re-
ductions in energy consumption. Static optimi-
zation of the stainless-steelmaking process in the
arc furnace has been investigated by Lipszy
(1966), Calanog and Geiger (1967) and optimal
control of steehnaking was investigated by
Woodside and colleagues (1970).
2. CONTROL DURING THE MELTING CYCLE
The present study relates to a three-phase arc
impedance controlled 135 tonne 35MVA pro-
duction electric arc furnace. The furnace consists
of a refractol'y lined shell with serxlcc dool> al~d
a tapping Launder. Three electrodes pass !i~-
rough holes in the roof ,ahich can be s,aung aside
in a horizontal plane to permit scrap charging
from an overhead basket. Electrical powe~ >
supplied to the furnace electrodes through bus-
bars and water-cooled flexible cables from Ih¢
furnace transformer. Heat is transferred to the
scrap steel from electric arcs drawn between the
tips of the electrodes and the metallic charge. A
schematic diagram of an arc impedance con-
trolled furnace is illustrated in Fig. 2.
The wide range of control requirements l)r
electric arc furnaces can be summarised as ma-
ximum heat utilization of the electrical energy
input consistent with production constraints. The
constraints arise from the restrictions imposed by
the electrical supply authorities and the large
economic incentives to mmimise refractory and
electrode erosion.
Throughout the period of a melt the arc length
varies erratically due to scrap movement within
the furnace and some form of control is required
to maintain the desired power input level. The
existing control philosophy (Nicholson and
Roebuck, 1972) is based upon maintaining a
preset arc impedance throughout the melting
cycle and long-term control of power input is
achieved by the selection of suitable voltage taps
on the furnace transformer secondary winding.
Each electrode is positioned individually by an
electrode position controller which attempts to
maintain a reference arc impedance compatible
with the long-term power input schedule.
A typical melt in the electric arc furnace lasts
for approximately four hours, and consists of
charging the furnace with scrap steel (lst basket l
and applying power until the unoccupied volume
of the furnace can accept more steel scrap.
Further scrap is added (2nd basket) and power is
applied until the furnace is again capable of
accepting a further charge of scrap (3rd basket).
Power is again applied until the steel is molten.
The power input is then reduced and a sample of
the steel is taken for analysis. The production
process then enters the refining cycle, as de-
scribed in Section 3.
2.1 Single-phase modelling
The servomechanism for positioning the elec-
trodes in the furnace under investigation is typi-
cal of many installations and consists of an
amplidyne Ward-Leonard regulator operating on
an arc impedance error signal. The single-phase
electrode position controller has been modelled
using transfer function relationships, (Billings and
Nicholson, 1975) assuming zero interaction be-

3. Electric arc furnace modelling and control 139
i .... National grid
system
II
I Intermedia% I . . . . . . . . . . .
Flexible
' ' I v°l~ *°P " 1 /
[ changer _1 Curren~¢ ~__[
Furnace - ]transformer
F--
transformer J
[ Curre~
Voffage } = signal
signal ,7 ,~
Arci~*~l I . -I I
medsu(Ing ~ eo.~er Prime
1 circuit"
J I_,ampllr'er I [ mover.
,nL _
4- 7
i
Furnac
ba'th
Pneumatic
counter-
balance
FIG. 2. A schematic diagram of an arc impedance controlled
furnace.
~:;:;a??,;d
It ~K21 I '+s'3 I ~ I IJ=K'I I '+s~
I o.ro.r
Stobilising fieldl ' s [
Mast
dynamics
Arc discharge [ -,. [ ( ..^ I
Amptidyne
[•Oifferential
field
FIG. 3. Block diagram of the electrode position controller.
tween the regulators, and can be represented as
shown in Fig. 3.
The single-phase furnace transmission system,
including the high power arc discharge can be
represented as (Billings, 1975)
v,.=Dh{(1 - WRc~)(R°l +Rc,)- WX~}[Z,, I-'
=D'h
(2.1)
FDh
i~ = FRaO _ io = - WDh = -Kh (2.2)
where
[ { 3
F= {-l°) 3 {R,2+R,3) ~. (RtkR,--XkX,)
k,l= l
ket,
M = E2{( - R,3 - R,z/2 - x/3X2/2) 2 + (~3R,e/2
- X2/2 - X3)2},
[Ztt[ = {(R~ t + Ret)2 + X~} ,/2 = {(R,, }2+ X~}'/2

4. 14i) S. /. BII.I_INGS, F. M. BOI/~NI) and tt. NI(H()I.S()I
R,i represent arc resistances. R<i and X i the
system line resistance and reactance, E line vol-
tage. t,,,, i and h the change in measured secon-
dary voltage, arc current (I) and arc length
respectively, D the arc discharge coefficient, D'
the discharge coefficient. K the arc gain and the
superscript 0 indicates nominal rabies.
Combining the models of the electrode con-
troller, arc discharge and transmission system
provides a complete mathematical description of
the furnace control system which can be used to
investigate arc furnace operation and control.
2.2 Identification of the furnace control system
Although the theoretically derived model pro-
ved to be representative of furnace operation it
was considered that a more concise description of
the electrode regulator and further information
about the process could be obtained from an
identification study. The basic aims of the identi-
fication included, identification of properties of
the arc discharge, investigation of the interaction
between the regulators of an arc impedance con-
trolled furnace, and identification of a low-order
representation of the electrode-position controller.
The diversity of the identification requirements
entailed designing several experiments and re-
cording a large amount of data. This was usually
done in an iterative manner so that the initial
experiments added to the knowledge of the pro-
cess and suggested the form of future experiments
(Billings and Nicholson, 1975).
Normal operating data for a typical melt were
recorded and analysed to determine the discharge
coefficient and arc gain. During the first basket,
the discharge coefficient D' was found, typically
to be 3764V/m with an arc gain K of 859kA/m.
The values of D' (1653V/m) and K (367kA/m)
estimated during refining were notably lower
than those experienced during the first basket
melt because of the ionization of the furnace
atmosphere.
Dynamic volt-ampere characteristics of the arc
discharge in a production furnace during refining
and at the beginning of the melt are illustrated in
Fig. 4. The two distinct slopes of the characteris-
tic over each arc cycle correspond to a two-
resistance model of the arc. Inspection of the
dynamic characteristics clearly shows the insta-
bility of the arc at the beginning of the melt and
indicates the dependence of the arc characteristics
on the furnace environment (i.e. temperature).
The injection of a 127 bit 33.3ms PRBS'into
the amplidyne control field of phase-two
electrode-position controller, with the furnace in
normal operation during refining, enabled the
interaction between the regulators to be investi-
gated. An on-line correlator aab /hey, u~cd ~,
identify the impulse responses between lhc ~-
rious inputs and outputs of the three-phase sy>-
tem. Analysis of the estimated cross-correlogratns
illustrated in Fig. 5 clearly indicates that there is
minimal interaction when operating under arc
impedance control, and this was confirmed by
Arc _ 281.9 V
voltage
I I t t Ij
(Refining) "
Arc 323.4V
voltage -
I 1 1
4Jl
1 - - I' i l
41, . . . . .
(Ist Basket)
FIG. 4. Dynamic arc characteristics,
Arc
current
i
i
62 KA
the results of several on-line step disturbance
tests.
Because the electrode regulator is an auto-
nomous closed-loop system it was necessary to
break the feedback loop by turning the power off
and extinguishing the arc to satisfy the identi-
fiability conditions for the estimation of the re-
gulator forward-loop transfer function. Initially,

5. Electric arc furnace
step inputs were applied to the open-loop system
to check the linearity of the system, estimate the
system gain and assess the characteristics of the
noise. PRBS sequences of amplitude 64mA were
then injected into the amplidyne control field and
motor speed and mast movement recorded. All
the signals were recorded on an F.M. tape re-
corder and digitized off-line prior to analysis
using an identification package SPAID (Batey
and colleagues, 1975; Billings, Sterling and Batey,
1977a).
~u.2Cr)
modelling and control 141
The results of the identification are illustrated in
Fig. 6.
Numerous model order tests including the de-
terminant ratio test. F-test, pole-zero cancellation
and tests for independence and operations on the
residuals were applied to check the validity of the
model. Identification using different data se-
quences produced notably consistent estimates
and verified the time invariance of the model.
A comparison of step responses of the analyti-
cal and identified models, illustrated in Fig. 7,
shows the similarity of the response of the two
models derived independently.
;% ',,
I I I T
3.0
sec
CCF of phase 2 coRtro[ field currerlt
end phase 2 most position
~o.,(T)
<
E
E
0
Lo
r--
be
oJ
I I I r
3.0
sec
CCE of phase 2 control field current
and phase I mast position
FIG. 5. Cross correlation comparison.
Generalised least squares parameter estimation
was applied to the modified data sequences for
increasing model orders and time delays in the
range suggested by the deconvoluted system im-
pulse response. The final process model relating
mast position and control field current was found
to be
z "2 r(0.2498z- 1 + 0.3079z- 2 + 0.09533z- 3)10 - 3
Y'--l--3.547Z 1+4.826Z-2_2.9967Z 3+0.7177Z 4-
2.3 Controller design
The identification of the furnace control system
was motivated by the requirement to gain further
knowledge and insight of the process and to
design suitable closed-loop controllers. Before
any controller can be expected to give improved
plant control it must make use of information
which previous designs ignored. This has been
achieved in the case of the arc furnace by using
the results of the identification study to develop a
dual impedance/current control strategy (Billings
and Nicholson, 1977b) and a temperature weight-
ing adaptive controller (Billings and Nicholson,
1977c).
2.3.1 A dual impedance/current control strategy
Although many measurable quantities appear
intuitively attractive as control variables in the
arc furnace, most of these including phase power
and arc voltage have to be excluded because of
practical limitations, and arc impedance or oc-
casionally arc current control are usually implem-
ented. The ability of current control to combine
the corrective action of all the electrode re-
gulators to clear disturbances, may or may not
result in less accumulated power discrepancy
compared with non-interacting impedance con-
trol, and the two strategies were compared to
assess which is the most efficient.
Three-phase models of current and impedance
controlled regulators were derived to facilitate a
comparison between the two control strategies.
The models were based on the identified differ-
ence equation representation of the electrode
position controller and utilised the results of the
identification of the arc discharge and interaction
between the phases to extend the single phase
transmission system model.
The three-phase transmission system models
can be represented by Billings and Nicholson,
(1977bL
Ut= gkk(Z- 1)Ut. (2.3) ::[= G4.D'~+ (isKk]h~ 1, I. 2.3 12.4)

6. 142 S. :. t'}111 IN(iS, F. M. BOLA.NI)and It. N (Ht)I ~,~ ",,
1.076 input
L! UU JULI/JItL/U L/ U! LIUVLILIUL/ I/U L1IIfUI UUI ]I313
14,64
14.64 Predicfed oufpuf
2.533 Residuals
10.59 Deferministic prediction errors
FK;. 6, Identification of the electrode position controller (k = 2, H- 3),
2.887cm
Most
position
z~
- 2
z~ Identified model
Non- linear model
FI(;. 7. A comparison of closed loop step responses.
I
4.0sec
for the arc impedance controlled furnace and
] j j= 1,2,3 (2.5)
j=l~
I0
where
_g 2
Cz . . . . . . . . . . . . . . . . . . .
g',ck= Aki k k = 1,2, 3 (2.6)
1 c
for the current controlled furnace, where % c.k are
error signals proportional to the change in arc
impedance and the change in arc current re-
spectively, Bi, c%, i,j=1,2,3, are constants. Ak is
a scalar which is selected such that the error
current in the amplidyne control field is initially
equal for both current and impedance controlled
regulators operating to correct a defined distur-
bance. Combining equations (2.4), (2.5) and (2.6)
with the identified difference equation model of
the electrode regulator given by (2.3) and using
hk= gkk(Z- 1)% k = 1, 2, 3 gives the three-phase arc
impedance and arc current controlled furnace
models respectively.

7. Electric arc furnace modelling and control 143
Simulation of the three-phase models using a
performance index defined as the integral of the
sum of the three absolute arc power deviations
indicated that independent electrode positioning
results in less accumulated power discrepancy
compared with interacting control. However, in-
teracting current control was found to be be-
neficial under short circuit conditions, when the
current magnitude should be reduced in the
shortest possible time. A controller which com-
bined both these characteristics should therefore
improve melting efficiency and result in a greater
transformer life expectancy particularly when the
short-circuit currents exceed the normal operat-
ing currents.
A dual control strategy is proposed which
consists of operating under the normal imped-
ance control during normal melting conditions
and reverting to current control if any arc
current approaches its short-circuit value.
Because the short-circuit current is usually signi-
ficantly larger than the normal operating current
the dual impedance/current control strategy can
be readily implemented on the furnace and
should result in a minimum accumulated power
discrepancy and rapid correction of short circuits.
If the dual strategy is to be operated to
maximise the heat utilization of the electrical
energy input then the time during which the
power input is not at the predetermined level
must be kept to a minimum. This can best be
achieved by improving the transient behaviout of
the electrode regulator and a proportional-
derivative controller has been designed to reduce
the position rise time and overshoot (Billings,
1975).
2.3.2 A temperature weighting adaptive con-
troller
Conditions are far from static in the electric
arc furnace, and variations in the arc characteris-
tics over the period of a melt affect the overall
loop gain and sensitivity of the electrode position
controller in such a manner that the system
performance varies between the two extremes of
highly overdamped and unstable responses.
Identification of the arc characteristics on a pro-
duction furnace showed that the arc gain and
discharge coefficient can vary by a factor of at
least two over the period of a melt. Adaptive
control to maintain the overall loop gain at some
predetermined value is therefore highly desirable.
Consider the effects of a step disturbance in arc
length over the period of a melt. When the
furnace is cold, at the beginning of the melt, a
step change in the arc length of amplitude h' will
be sufficient to cause a current change i, thus
i = -K~o~ah'.
Towards the end of the melt, when the furnace is
hot and the atmosphere ionized, a step change in
the arc length of amplitude h" will be required to
cause a similar change in the arc current
i = - K both"
where h">h', and hence Kco~d>Khot. Since the
discharge coefficient D' is a function of the arc
gain K, D'=KQ where Q is a constant, the
relationship between the arc impedance error
function and the change in arc length ~;=(G4Q
+Gs)Kh will vary considerably over the period
of a melt.
The arc consists essentially of a gas at high
temperature, known as the arc column or arc
plasma, in which current is carried by electrons
and ions. If the temperature of the arc increases,
the gas becomes more thermally ionized and the
electrical conductivity increases such that the
required current is able to flow with less potential
drop (Maecker, 1964). The arc resistance is there-
fore heavily dependent on the arc temperature,
and Edels (1961) has shown that the electrical
conductivity can be ultimately expressed as a
function of temperature only. Thus when the
furnace is 'cold', at the beginning of a melt, a
defined change in arc length will cause a greater
change in arc impedance than at the end of the
melt when the temperature is high (typically
1600°C) and the atmosphere ionized. The elec-
trode position controller is tuned for one defined
relationship between arc impedance and arc le-
ngth but, because of the temperature dependence
of arc impedance this relationship will only be
valid for one temperature, T* say. However, it
can be shown, (Billings and Nicholson, 1977c)
that the relationship between the arc impedance
error e for a defined change in arc length h at
temperatures T* and T' is of the form
eT* L(T*)
~r, A(T')
- - = QT' V h. (2.7)
Thus if the arc impedance error at temperature
T' is multiplied or weighted by the constant Qr',
the relationship between e and h will remain
constant throughout the melt and independent of
temperature. Similar relationships can be found
for all temperatures leading to a complete weight-
ing function Qr.
The weighting function Qr, which can be ob-
tained by measuring the temperature at the hot
spots and arc length over the period of only one
melt, could be implemented as part of a direct
digital control scheme or by using diode function
generators. In either case the controller is re-
latively uncomplicated and could readily be in-

8. 144 S.A. BII,I IN(IS. F. M. BOt.AND and H. NI~,'ItOI.S()N
stalled as part of the existing arc regulator to
maintain a consistent control action through-out
the melt. A schematic diagram of the proposed
arc regulator and the temperature weighting
adaptive controller is illustrated ill Fig. ~.
-f
Electrode position controller
z-2(Z.Sz-I+3.0z-2+09 5z-3)10-4
I-3.5z-I +4.8z-2_2 99z-3+Q717z-4
Mast
position
I Multiplier ,,
Temperatureweighting
adaptive corrtroller
Arc discharge and ~"
transmission /
system
Arc impedonce F
measuringcircuit
I
Weigh-I"
3 I I
] impedonce
I error
emperoture
tl(i. N. A schematic diagram of lilt: tclllpcraturc wcighlmg
adaptixc controller.
Although the energy rate input schedule is
operated according to a recommended code of
practice, the original proposal (McGee and
Ravenscroft, 1959), which suggested that the
transformer voltage taps should be changed when
the furnace refractories exceeded a specified tem-
perature, has received little attention until re-
cently because of the lack of suitable temperature
measuring devices. However, recent research has
suggested that a temperature controlled energy
rate input schedule could improve considerably
the heat utilization of the electrical energy input.
Direct measurements of the refractory tempera-
ture or an estimate of this temperature obtained
using optimal filtering techniques could therefore
be used for both the temperature weighting adap-
tive controller and an electrothermal control
scheme.
3. STATE-ESTIMATION DURING THE REFINING
CYCLE
The refining process, during the production of
medium and low alloy steels in the electric arc
furnace, starts with liquid metal at a lcnlperattlrc:
of about 1580 C, with a carbon concentration of
about 0.5"i, greater than that required by the
order specification. The process ends when the
carbon content has been reduced to the desired
level and the process temperature has been raised
to a level necessary t~r satisfactory pouring into
ingot moulds. In addition to the carbon and
temperature specifications, the steel i,~ also re-
quired to meet up to eleven end-point chemical
specifications. Concentrations of specific alloying
elements may be changed by making alloy ad-
ditions, and impurities are removed by the in-
jection of gaseous oxygen and the establishment
of a suitable slag. The energy requirement for
steelmaking is supplied almost wholly by electric
power through the electrodes, but the heat libe-
rated by exothermic reactions must also be con-
sidered. Accurate continuous measurement tech-
niques are not axailable l)r most of the process
variables and the high temperature ill excess of
1600:C, and the highly corrosive nature of the
molten steel are but two of the man~ difficulties
encountered when designing furnace instrumen-
tation. It is desirable that all of the target specifi-
cations are met simultaneously so that there is no
oxygen, electricity or lime expended on end-point
corrections. Thus even if continuous measure-
ments were available, some predictive algorithm
would still be necessary so that the process could
be controlled to meet simultaneously the end-
point constraints.
3.1 Model./ormulution
Both analytical and statistical techniques have
been employed to develop models of steelmaking
processes. However, considering the difficulties
encountered with the collection of adequate data
on the wide range of steels produced and the
inherent inflexibility of statistically evolved mo-
dels from the process control viewpoint, the
former approach was adopted in the present
study. It is generally accepted that at steelmaking
temperatures the rates of the major refining re-
actions are determined by mass transport which
can be described mathematically by Fick's laws
(King, 1963). However, a theoretical evaluation
of the diffusion rates requires knowledge of the
areas and thicknesses of phase boundaries and
the diffusivities and chemical activities of the
system species. Investigations of many of these
complex process phenomena have been reported
in the literature and progress has been made in
the understanding of the heat and mass transfer
occurring during steelmaking (Szekely and
Themlis, 1971}. By combining the results of these
investigations with data collected from the pro-
cess a mathematical model was formulated to

9. Electric arc furnace modelling and control 145
describe the dynamics of the concentrations of
the chemical species in the slag and metal phases,
the mass balances for both phases and a thermal
balance for the process.
Casts refined under a variety of operating
modes were simulated using this model and the
studies revealed that from the control viewpoint
the model could not be considered to have a
practical application. In particular the following
limitations were exposed.
(i) Complete process data were required at
melt-out. The data included a full chemical ana-
lysis of the slag which is not available during
normal operation.
(ii) The level of accuracy required of the mo-
del, for use in process control, was only attain-
able for those casts during which none of the
standard interruptions to the process, necessary
to make additions and allow some of the slag to
flow out, were made.
(iii) Even when the conditions in (i) and (ii)
were satisfied unforeseen perturbations on the
refining trajectories were found to result when
solid scrap fell into the bath from the furnace
banks. There was an apparent need for a com-
promise between the implied accuracy and the
complexity of the process model. To this end the
dimension of the state vector was reduced by the
introduction of a set of random variables to
replace the effects of the states associated with
the slag phase and the slag to bath weight ratio.
The resulting model consists of four state equa-
tions describing the dynamics of the process
variables: x l =concentration of carbon, x2
= concentration of manganese, x3 = concentration
of iron oxide and x4 =temperature of the molten
steel. The process is forced by two control inputs,
u~ representing the rate of oxygen iniection and
u2 the electric power input. The state equations
have the form (Boland and Nicholson, 1977)
2i=fi(x,u, fl) i=1 ..... 4 (3.1)
where fl is the vector of model parameters and
contains the random variables replacing unmod-
elled states.
Using the Euler integration formula and as-
suming the model was separable into stochastic
and deterministic parts, the following vector dif-
ference equation description of the process was
obtained
Xk+I = F(Xk, Uk,fl) + GkWk (3.2)
where wk is a white Gaussian noise sequence with
statistics, Wk~N(O, Qk) and ~ is the vector of
expected values of model parameters. The co-
variance matrix Qk was assumed to be diagonal
with elements q~ (i= 1..... 4) which represent a
measure of the uncertainty associated with the
parameter set. Approximate values for these ele-
ments were obtained over the normal operating
range of the process, by analysis of the maximum
probable error for each difference equation as
given by
(Err°ri)2-j=, ~OflQ i=1 ..... 4 (3.3)
Since some of the parameters occur in two or
more of the state equations the noise matrix
Gk was constructed to account for the resulting
correlation of the uncertainties associated with
the components of F. The approach adopted was
to determine (Boland and Nicholson, 1976) the
dominant parameter or relationship common to
two components F~ and F~ and then defining F~
as a function of Fj to give Gij = ?Fj~Fj.
3.2 Measurements
The use of waste gas analysis equipment to
provide an indirect measurement of the carbon
concentration, xx, is an established technique in
the Basic Oxygen sector of the steelmaking in-
dustry (Dennis, John and Porter, 1969). At time,
t, the measurement obtained from the gas ana-
lysis is given by
xl (t) = x 1(0)-~ V(r)dr (3.4)
where V(z) is the decarburization rate estimated
from the measurements of the flow rate and
composition of the waste gases. An analysis
(Boland and Nicholson 1977) was made of the
achievable accuracy using this technique and this
demonstrated that the uncertainty associated
with this mgasu_rement was too large for it to be
used in the control of the arc furnace process. An
expression for determining the variance of this
uncertainty was obtained which permits the car-
bon measurement to be written as
yl (t) = x~(t) + t,l (t) (3.5)
where vl(t) is a zero mean Gaussian process with
variance E(vZ(t))=rll(t), which varies in the ra-
nge 0<rl~ <0.0044.
Because of the high temperature and corrosive
nature of the process, a direct measurement of
the temperature of the molten steel can only be
obtained by use of a disposable thermocouple.
However, as mentioned in Section 2.3.2, measure-
ment of the temperature of the furnace hot spots
does provide an indirect measure of the process
temperature. It was considered reasonable to
assume that a continuous indirect measurement

10. 146 S.A. BIIA,INGS, F. M. BOLAND and H. NICtlOLSON
of the temperature to an accuracy of standard
deviation 10 K is achievable by use, for example.
of lhermocouples embedded in the lining of the
furnace. This indirect measurement of the tem-
perature may be written as
y2(t) = Xa.(t)+ F2(t) 13.6~
where v2(t) is a zero mean Gaussian process with
variance E(v2(t)) = r2z(t)= 100.
erroneous process model and divergence from the
true states would result. A study was made
(Boland and Nicholson, 1976), of a number ~1
simple techniques for control of filter divcrgence.
These techniques employ the well known fact
that, in theory, the innovations process st defined
in (3.7d) is a white, zero mean Gaussian process
with covariance
E(zkz~r)=(HkP~ 1H~+Rkl. (3.8)
3.3. Kalman filtering
The discrete-time form of the extended Kalman
filter, as described by the following equations,
was implemented
XR
k+1 = F(x~, Uk) (3.7a)
p~+ k T T
l=OkPkOk + GkQkGk (3.7b)
_ pk H T
Kk+,-- k+l k+l(Hk+,(Hk~lP~ 1HT,1
+Rt~+ 1) 1
Zk+l =)k+l--Hk+IX~+1
(3.7c1
(3.7d)
xk+l =X~+ @-Kk+lZk+1
k+l l (3.7e)
pk+l =(l_Kk+lHk + )pk+l(l_Kk +IHk+ ~)T
k+l 1
+Kk+IRk+ T
1K~+1 (3.7f)
where @' is the transition matrix associated with
the linearised process equations, and P~, appro-
ximates to the covariance matrix of the un-
certainty on the estimates of x. The notation
(')~+~ denotes the estimate of(.) at time (k+l)
obtained for measurements over the interval
[0, k~]. The matrices H k and Rk are the measure-
ment and measurement noise covariance matrices
respectively, and from (3.6) and (3.7)
0) f; )
1 0 0 and Rk=E(vkv~}= ~ 0
Hk=H= 0 0 0 r22 "
The practical considerations influencing the im-
plementation of Kalman filters in aerospace ap-
plications are well known (Huddle, 1970). Of
particular importance is the problem of filter
divergence which can result from the effects of
errors in the description of the system and the
statistics of the noise processes. In the present
application, the effects of the unforeseen va-
riations in the thermal and chemical behaviour of
the process described in Section 3.1 were con-
sidered to be of major importance. It was ap-
parent that unless some means of accounting for
these changes was included in the estimation
procedure the filter would tend to track the
A procedure (Boland and Nicholson, 1977) was
developed which tests the consistency of the
statistics of the smoothed innovation associated
with the temperature measurement and when
divergence is suspected it effects control by in-
creasing the uncertainty associated with the ther-
mal dynamics. This increase in q44, which is
maintained until the innovations are again con-
sistent with their statistics, has the effect of
shifting the emphasis within the filter from the
model to the measurements.
3.4 Simulation study
The trajectories of the states xl and .v4 tot a
simulated cast are illustrated in Fig. 9. The effects
of variations in furnace behaviour were intro-
duced by assuming that the temperature changed
abruptly by -10 C at t=10min and t=25min
and the rate of heat loss was assumed to double
over the interval 33 rain <t<43min. Control of
divergence was effected by setting q4.~=2.25F4
(Xk,Uk) when the innovations failed the consis-
tency test; normally %,~-=0.0625F~ (:%uD. The
parameter set associated with the filter model
was perturbed from that employed in the process
model by the use of a Gaussian random number
generator. The results illustrated in Fig. 9 were
obtained using a discrete time interval of 10 sec
and an observation interval of 1 min. The appro-
ximate final variances at tk=47min were, with
divergence control
o-~ (i=1 ..... 4)=(0.65× 10 -~,0.2x 10 4, 0.25, 8.4).
Hence, in terms of weight percentages the results
of simulation studies indicate that, e~en in the
presence of the increased uncertainty introduced
by the divergence control procedure, there is a
better than 80'~ii probability that the estimates
satisfy the industrial accuracy requirements of
about ±0.50.; on the chemical states and +5 C
on the temperature state.
4. CONCLUSIONS
Problems associated with the production of
special steels in the electric arc furnace have been
considered. Analysis of the production cycle as a

11. Electric arc furnace modelling and control 147
I0
05
(a)
FJffer model
° °
Observ ~ o
I 1 I i I
I0 20 30 40 50
1.0
tO
,~ 0.5
~ . Divergence controller active
~ ~~'_~x.-,~ ~ / Process
- x--~ Filter ""k~.~"I
~ Filter + div. control ~ ' ~
I I L t 1
Io 20 30 40 50
Time, min
1650
. 1600
F-
1550
(c)
Observations o o o
Process o o A~ o ~ v~ ~ °~
~ o Filter model
o
°° I I. I I
I0 20 50 40
I
50
1650
16(~:
I'-"
1550
(d)
Process
:--x Filter
Filter ÷div. control
I I I I I
10 20 :50 40 50
Time, min
FIG. 9. Simulated performance of the state estimator. O filter
with divergence control; x filter without divergence control.
three-stage process has exposed the difficulties
involved in obtaining an optimal steelmaking
strategy for the electric arc furnace.
Short-term dynamic control of power input to
the steel has been considered and the results of
an identification study to investigate the interac-
tion between the regulators, estimate the propep
ties of the arc discharge and identify a model of
the electrode position controller have been pre-
sented. A dual impedance/current control st-
rategy, a temperature weighting adaptive con-
troller and proportional-derivative regulator have
been designed using the results of the identifi-
cation, and aspects of implementation on a pro-
duction furnace have been discussed briefly.
The development of a mathematical model of
the refining process has been shown to be re-
stricted by the complex metallurgical nature of
the process and on the deficiency of existing
plant instrumentation. The need for a compro-
mise between complexity and implied certainty of
the model has been discussed. The extended
Kalman filter has been presented as an efficient
method of combining the a priori information
about the process in the form of a dynamical
model with the incomplete error-corrupted pro-
cess measurements. Problems of filter divergence
due to modelling errors have been considered
and the results presented indicate that estimates
of the states can be obtained to the accuracy
required for the design of a refining control
strategy.
Acknowledgement--The authors express thanks to the British
Steel Corporation for permission to undertake this investi-
gation and appreciate the interest shown in the work by M.
Foster, J. Gifford and R. Roebuck (BSC).
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