An Electric Arc Furnace Model for Flicker Estimation
JOÃO PAULO M. SOUSA, ZÉLIA M. A. PEIXOTO, MARIO F. ALVES, CELSO P. GARCIA.
Graduate Program in Electrical Engineering
Pontifical Catholic University of Minas Gerais
Av. Dom Jose Gaspar, 500 – Coração Eucarístico
30535-610 - Belo Horizonte - MG
BRAZIL
Abstract  This paper presents an arc furnace model using the chaos theory and differential equations
describing the arc furnace behavior, the flicker effect particularly. The arc voltage is simulated by solving a
differential equation yielding the non-linear dynamic and multivalued v-i characteristics of the arc. In order to
simulate the flicker a low-frequency chaotic signal is generated by Chua’s chaotic circuit to modulate this nonlinear system. A flickermeter was developed to calculate the Pst. A methodology is developed that allows for the
adjustment of the parameters of Chua`s circuit, according to each particular furnace and feeding power system.
Key-words: - Modelling, Arc Furnace, Chua’s Oscilator, Flicker, Flickermeter
1 Introduction
The number of electric arc furnaces installations in
the metallurgy industry has greatly increased in the
last decades. Some reasons for this are the abundance
of scrap iron, the necessity of recycling and the
possibility of producing metallic leagues with high
level of quality at relatively low cost when compared
to other forms of energy [1][2][3]. An electric arc
furnace transforms the electrical energy into thermal
energy in the form of an electric arc to melt the raw
materials held by the furnace. The used heat is
transferred to the load by radiation and convection.
The implantation of an arc furnace generates jobs
and attracts new industrial installations, promoting
the economic development mainly in the regions of
lesser degree of industrial development. However,
these furnaces have a great impact on power quality.
The voltage fluctuation they impose on the supply
system result in the well known flicker phenomena. It
is reported that a low level frequency modulation of
the supply voltage of less than 0.5% can cause
annoying flicker in incandescent lamps when the
frequency lies in the range of 6-10 Hz [1][4][5].
The development of an electric arc furnace model
constitutes a great challenge. Besides the non
linearity and random behavior presented by the arc, it
is necessary that the model presents some means for
setting its parameters according to each particular
supply system feeding the furnace.
This work presents the application of the chaos’
theory, more specifically the Chua’s oscillator, to
modeling the electric arc furnace in time domain. The
voltage signal calculated from the dynamic
differential equations of the furnace [6][7][8] is
modulated by the low frequency chaotic signal of the
Chua’s oscillator output, yielding the random and
nonlinear characteristics presented by the furnace`s
voltage fluctuation, which is responsible for the
resulting flicker.
In order to estimate the flicker produced by the
furnace in a given power system, a flickermeter was
developed using the IEC/UIE Standard [9][10]. An
approximated mathematical expression is used to
predict the flicker level [11][12] and the resulting
flicker value is compared to the value obtained from
the flickermeter. The result of this comparison is
there used to adjust the parameters of the Chua`s
circuit.
The arc furnace model and the flickermeter were
implemented in the EMTP environment, and an
application example used to demonstrate the
proposed methodology.
2 The electric arc
The arc is an electric discharge, where the load
carriers are mainly electrons. After the furnace is
charged with scrap, operation begins by lowering the
electrodes to strike electric arcs between the
electrodes and the scrap. Ignition begins when
instantaneous voltage grows beyond a critical value.
The arc is extinguished at current zero and it is reignited again in the following semi-cycle. The arc
operation is characterized by a low voltage and a high
current.
The arc presents distinct phases during a typical
operation cycle of the furnace. However only the
melting and refining phases are normally considered
[2][4][5].
In the melting phase, after loading the scrap iron,
the three electrodes are lowered, trying to get a stable
arc. When fusing begins the scrap iron presents a
very irregular surface, causing great current
fluctuations since the electrodes regulators are not
fast enough to respond to the current variations. In
this phase, the arc furnace demands very high active
power from the supply system.
In the refining phase a longer and more constant
arc is adopted. This arc, however, depends on the
furnace’s power factor. With a high power factor,
above 0.9, the voltage in the instant of the
extinguishing of the arc is enough to immediate reignition, resulting in a practically steady arc. On the
other hand, for power factors of 0.7 to 0.8, the
voltage during the arc extinguishing are not enough
for its immediate re-ignition, leading to an unstable
arc [4].
3 The electric arc model
The arc furnace model will be developed in three
parts. First, the differential equation that represents
the dynamic and multivalue v-i electric arc
characteristic, is considered. Second, a low frequency
chaotic signal is used to represent the voltage
fluctuations imposed by the arc. Finally, the chaotic
signal is used to modulate the response given by the
differential equations.
3.1 The arc furnace v-i characteristic
The arc furnace v-i characteristic is obtained from the
differential equation that represents the arc dynamics,
based on the energy conservation law [13]. The
electric arc power balance equation is,
p1  p 2  p 3
(1)
Where,
to simplify the model. Thus the arc radius r appears
only as a state variable.
p1  k1 r n
(2)
k1 is an arbitrary constant and n represents the arc
cooling constant. If the environment around the arc is
hot the arc cooling may not depend on its radius at
all, so in this case n=0. If this is not the case and the
arc is long, then the cooling area is mainly its lateral
surface, so n=1. If the arc is short, then the cooling
area is proportional to its cross-section at the
electrodes, so n=2.
The term p2 is proportional to the derivative energy
inside the arc, which is proportional to r 2 ,
p2  k 2 r
dr
dt
(3)
Finally,
p 3  vi 
k3 / r m
r2
i2
(4)
k2 and k3 are arbitrary constants.
In expression (4), the arc column resistance is
assumed inversely proportional to r m , where the
constant m varies from 0…2. This variance reflects
the fact that the arc may be hotter in the interior if it
has a larger radius.
Substitution of equations (2), (3) and (4) into (1)
gives the differential equation of the arc:
k1 r n  k 2 r
k
dr
 m3 2 i 2
dt r
(5)
The arc voltage is given by,
v
i
g
(6)
p1 represents the transmission of energy in form
g is defined as the arc conductance, and given by the
following equation:
of heat to the external environment;
p 2 represents the internal energy of the arc that
directly affects the radius of the arc;
p3 represents the total energy developed in the
arc and converted into heat.
r m 2
g
k3
The cooling effect is function of temperature, but
this dependency is considered not significant in order
(7)
It is possible to represent the different phases of
the arc modifying the parameters m and n. The
complete combination of these parameters for the
different arc phases can be found in [13]. In this
work, this parameters are chosen as m=0 and n=2 to
represent the refining stage and as m=1 and n=2 to
represent the melting stage.
Figure 1 shows the v-i arc characteristic resulting
by solving equations (5) and (6) in time domain, and
figure 2 shows the arc voltage waveform simulation
to refining stage.
3.2 Chaotic Responses in Arc Furnaces
As demonstrated in [8], the electric fluctuation in the
arc furnace’ s voltage is of a chaotic nature, and
chaos theory [2] may be used to model it. The Chua’s
oscillator circuit is used to establish the random and
nonlinear characteristics presented by the furnace’s
voltage fluctuation.
Fig. 3. Chua’s Circuit.
The nonlinear element is the Chua's diode (NR).
This element has a piecewise linear characteristic and
supplies energy to the system guaranteeing the
independent oscillation of the circuit.
Varying the chosen parameters (L, R, C1 and C2),
the circuit can present different dynamics, such as
steady fixed points, periodic oscillations and chaotic
dynamics [6][7].
3.3 The Arc Furnace Model
The current absorbed from the power system bus is
injected as the input model. The electric arc voltage is
obtained from the simultaneous solution of equations
(5) and (6), and the low-frequency chaotic signal that
is generated by the simulation of Chua’s chaotic
circuit. The solution of these equations is modulated
by the chaotic signal producing the final arc furnace
voltage output. The model behaves as a controlled
source, with the system current as an input and the
terminal voltage value resulting for each time step.
Fig. 1. Dynamic v-i characteristics of electric arc.
4 The Flickermeter Model
1000
[V]
750
The IEC flickermeter [9][10][11][14] is used. It has
five basics blocs as show figure 4. They are described
below.
500
250
0
-250
-500
-750
-1000
0.50
Fig. 4. The IEC flickermeter block diagram.
0.54
0.58
0.62
0.66
[s] 0.70
(f ile MFEAP_PADRAO_ajusteFO.pl4; x-v ar t) v :AFORA
Fig. 2. Voltage wave form of electric arc.
It has been shown in [6] that for an autonomous
circuit consisting of resistors, capacitors, and
inductors to exhibit chaos; it has to contain the
following components:
1) at least one locally active resistor;
2) at least one nonlinear element;
3) at least three energy-storage elements.
The Chua’s circuit shown in figure 3 satisfies the
above listed conditions. More details can be found in
[6][7].
4.1 Block 1 - Input voltage adaptor
A first step in the meter is to adapt the measured
voltage to the internal reference level of the
flickermeter. This is done with a transformer
connected between the bus voltage and the first block
of the meter.
An amplifier with adjustable gain (10% to 90%)
and a response time of 1 minute to a step of the RMS
imput voltage value is used to normalize the voltage
signal. The output of this block is a normalized
signal. The time constant is sufficient long to make
the output signal insensitive to fast changing
modulating signal, and short enough to follow slow
voltage changes.
non-linear visual perception eye-to-brain. The filter
(so called sling mean filter) is used to simulate the
brain memory. The output of this block represents the
bulb-eye-brain sensibility i.e. instantaneous flicker
level.
4.2 Block 2 - Squaring operator
The output of block 1 feeds block 2. The purpose is
to square the input signal. As the incandescent light
bulb can be considered as an ohmic load, the output
of block two corresponds to the electrical power
absorbed by the lamp, which is representative of the
flicker.
The block 3 has three filters. Two of these constitute
a band-pass filter, in order to eliminate the DC
components and those components with double
fundamental frequency. The first filter is a RC highpass first order with a cut-off frequency of 0.05Hz.
The second is a 6th order Butterworth low-pass with
a cut-off frequency of 35Hz to a network frequency
50Hz, and 42Hz to network frequency 60Hz. The
third filter is a band-pass filter that emulate the
transfer function of the bulb-to-eye, and has a cut-off
frequency of 8.8Hz that is the max perception bulbto-eye. The transfer function is,

K 1 s
G ( s )   2
2
 s  2s   1



s 


1 


2 





 (1  s )(1  s ) 

3
4 


(8)
with the filter constants as show Table 1.
k

w1
w2
w3
w4
This block perform a statistical evaluation of the
instantaneous flicker perception, represented by the
short term perception of the human eye - Pst. The Pst
is calculated according to the following relation:
Pst   0,1 0,1   1 1   3 3   10 10   50 50
4.3 Block 3 - Filters
Constant
4.5 Block 5 – Statistical analysis
Incandescent lamp
230V/60W
120v/60W
1,74802
25,50853854
57,5221845
14,32434303
7,699101116
137,6017582
1,6357
26,18438937
57,03353489
18,47194905
8,761700849
108,7941076
Table 1. The filter constants.
4.4 Block 4 - Squaring multiplier and sliding
mean filter
This block contains a squaring multiplier and a first
order low-pass filter with 300ms of cut-off frequency.
The purpose of squaring the signal is to simulate a
(9)
where Pi is the flicker level exceded during a
particular percent of the observation time and ki is the
weighting coefficient. These Pi values are taken from
the comulative distribution curve in 0.1, 1, 3, 10 and
50 % of the simulated time.
The flickermeter was calibrated according to
standard [9].
5 ATPDraw implementation
The furnace arc model and the flickermeter were
implemented in the EMTP environment.
The furnace voltage output is the result of the
solution of the differential equation (5) combined
with a chaotic modulation introduced by voltage Vc1
across capacitor c1 in Chua’s circuit [2].
The amplitude of the system’s voltage fluctuation,
and thus, the flicker level, can be adjusted to a higher
or lower value, by modifying the value of a gain
which is the Chua’s circuit output. Adjusting the
values of the elements L and C in Chua’s circuit
controls the flicker dominant frequency.
It is possible to get an estimation of the flicker
level caused by a new installation from statistical
analysis of the flicker caused by a large number of
arc furnace already in operation. The data needed for
this estimation are the parameters of the impedances
in the grid [11][12]. The estimation is given by:
Pst (99%)  K st
Pst (95%) 
X k , grid
X k , grid  X k , furnace
1
 Pst (99%)
1,25
(8)
(9)
The corresponding data for the studied case
(figure 5) are Xk,grid =0,0555 pu and Xk,furnace=0,4154
pu. The furnace factor , which is different for each
arc furnace installation, has a typical value of 75.
Using these data, equation (8) gives a flicker level
estimation given by Pst(95%) =7,072.
The voltage waveforms, at the primary of the
138 kV transformer, can be seen in figure 8.
200
[kV]
150
100
6 Simulation Results
Figure 5 shows the arc furnace transformer connected
to an equivalent power system supply. System data
for this configuration can be found in the Appendix.
The point of common coupling (PCC) corresponds to
the primary of the 138kV transformer.
50
0
-50
-100
-150
-200
2.0
2.2
2.4
2.6
2.8
[s]
3.0
(f ile MFEAP_PADRAO_ajusteFO.pl4; x-v ar t) v :PCCA
Fig. 8. Voltage waveform at the primary.
Fig. 5. Arc Furnace connection to the power system
supply.
Figure 6 shows the furnace current waveform in
the melting phase. The voltage waveforms, at the
secondary of the arc furnace transformer, can be seen
in figure 7.
250.0
[kA]
187.5
A simulation for a period of ten minutes (as per
IEC 61000-4-15) was carried out, and the
flickermeter results on the 138 kV side are indicated
by a Pst equal to 6,9798.
The calculation uses an iterative process. A first
iteration is calculated with the parameters of Chua´s
circuit set to a certain value. If the simulated Pst is
not the same as that estimated by equation (8), the
parameters of the model are adjusted until the two
values converge, as shown the diagram in figure 9.
125.0
62.5
0.0
-62.5
-125.0
-187.5
-250.0
3.2
3.3
3.4
3.5
3.6
3.7
[s]
3.8
(f ile MFEAP_PADRAO_ajusteFO.pl4; x-v ar t) c:XX0025-AFORA
Fig. 6. Arc Current obtained by simulation.
1500
[V]
1000
500
Fig. 9. Adjusting diagram model.
0
-500
7 Conclusion
-1000
-1500
1.5
1.6
1.7
1.8
1.9
[s]
2.0
(f ile MFEAP_PADRAO_ajusteFO.pl4; x-v ar t) v :AFORA
Fig. 7. Voltage waveform in the secondary of the
furnace transformer.
The electric arc furnace model using a combination
of the dynamic/multivalue v-i arc characteristics,
described by a differential equation, combined with a
modulation of chaotic nature, modeled by Chua’s
circuit, was implemented. A methodology to adjust
the arc-power system model to expected flicker levels
was demonstrated, and the results indicated the
validity of the model.
The model was constructed in time domain and
can be easily connected to the electrical net as part of
an integrated power system, allowing for broader
system’s studies.
The authors are working in the implementation of
the model in the EMTP computational environment,
where it will be integrated to a static flicker
compensation system, resulting in a computational
program for the analysis of the impact of the
connection of an arc furnace to the utility network.
Appendix
Parameters of the studied system shown in Fig. 7 are
shown here:
The base pu values were referenced to 100MVA.
Source: Ideal sinusoidal ac voltage source with 138
kV and zero phase shift.
Zthevenin: resistance R=0.3691  and inductance
L=28.0356 mH.
Transformer: Two windings linear transformer.
Nominal power: Pn=40 MVA (three phase).
Winding 1 parameters: V1(Vrms)=138 kV,
R1(pu)=0.0096, L1(pu)=0.275
Winding 2 parameters: V2(Vrms)=33 kV,
R2(pu)=0.0096, L2(pu)=0.275
Magnetization resistance and reactance: Rm(pu)=50,
Lm(pu)=50.
Arc Furnace Transformer: Two windings linear
transformer.
Nominal power: Pn=30 MVA (three phase).
Winding 1 parameters: V1(Vrms)=33 kV,
R1(pu)=0.0049, L1(pu)=0.1404
Winding 2 parameters: V2(Vrms)=700 V,
R2(pu)=0.0049, L2(pu)=0.1404
Magnetization resistance and reactance: Rm(pu)=50,
Lm(pu)=50.
Arc Furnace: (Parameters for corresponding
differential equation) k1=-3000.0, k2=12.5, k3=12.5.
(Chua’s circuit) C1= 0.575 nF, C2=150 F, L=2 H
with a series resistor, R0= 1634.95 , G=6.1164E-4
mho.
Acknowledgements
The authors take the opportunity to thank the PPGEE
and financial aid by CAPES and CEMIG.
References:
[1] R. Bellido, T. Gómes, Identification and
Modelling of a Three Phase Arc Furnace for
Voltage
Disturbances
Simulations.
IEEE
Transactions on Power Delivery, Vol. 12, No. 4,
1997.
[2] O. Ozgun, A. Abur, Flicker Study Using a Novel
Arc Furnace Model. IEEE Transactions on Power
Delivery, Vol. 17, No. 4, 2002.
[3] C. A. S. Leandro, Fabricação de Aços em Fornos
Elétricos. Programa de Educação Continuada,
Siderurgia para não Siderurgistas, Capítulo 7, pp.
123-133. Associação Brasileira de Metalurgia e
Materiais 1999.
[4] T, Zheng, E. B. Makram, An Adaptive Arc
Furnace Model, IEEE Transactions on Power
Delivery, Vol. 15, No. 3, 2000.
[5] G. C. Montanari, M. Loggini, A. Cavallini, ArcFurnace Model for the study of Flicker
Compensation in Electrical Networks, IEEE
Transactions on Power Delivery, Vol. 9, No. 4,
October 1994.
[6] M. P. Kennedy, Three Steps to Chaos – Part I:
Evolution. IEEE Transactions on Circuits and
Systems, Fundamental Theory and Applications,
Vol. 40, No. 10, 1993.
[7] M. P. Kennedy, Three Steps to Chaos – Part II: A
Chua’s Circuit Primer. IEEE Transactions on
Circuits and Systems. Fundamental Theory and
Applications, Vol. 40, No. 10, 1993.
[8] E. Acha, A. Semlyen, N. Rajakovié, A Harmonic
Domain Computtional Package for Nonlinear
Problems and It`s Aplication to Electric Arcs,
IEEE Transactions on Power Delivery, Vol. 5,
No. 3, 1990.
[9] IEC 61000-4-15. Flickermeter – Functional and
Design
Specifications,
IEC
61000-4-15
International
Standard,
Electromagnetic
Compatibility (EMC) – Part 4: Testing and
Measurement Techniques – Section 15, 1st Ed,
1997.
[10] UIE Part 5 Flicker and Voltage Flutuation.
Prepared by de travail GT “Qualité de
l’alimentation” “Power Quality” Working Group
WG 2, 1999.
[11] T. Larsson, PhD. Theses, Voltage Source
Converters for Mitigation of Flicker Caused by
Arc Furnaces, Kungl Teknisha Högskolan, 2003.
[12] A. Robert, M. Couvreur, Recent Experience of
Connection of Big Arc Furnaces with Reference
to Flicker Level. CIGRE 1994, Paper 36-305.
[13] P. E. King, T. L. Ochs, A. D. Hartman, Chaotic
Responses in Electric Arc Furnaces. Journal
Applied Physics, vol. 76, no. 4, pp. 2059-2065,
1994.
[14] M. Rogós, The IEC Flickermeter Model, AGH
Univerity of Science and Tecnology AGH – UST.
March 2003.