Abstract— Field measurements of voltage and current is the

most effective way for characterizing the electric response of an EAF that describe the nonlinear behavior of AC EAF loads. Sufficient measured information can be adopted to develop an appropriate EAF model. In this paper, two classic methods based on measured data, harmonic current injections and equivalent harmonic voltage sources, for the EAF load modeling are reviewed. For comparison, two advanced methods based on actual recorded data, cubic spline interpolation and radial basis function neural network (RBFNN), are also proposed to model the EAF load. A steel plant power system with EAF loads is used for field measurements and computer simulations. Comparisons between the results of measured data and simulations for the four EAF models are being made according to the voltage/current waveforms and voltage-current characteristics. It is shown that the advanced models yield better performance than classic models of the EAF.

Index Terms—Electric arc furnace, harmonics, cubic spline interpolation, neural network

I. INTRODUCTION

THE electric arc furnace (EAF) employs high temperatures

produced by the low-voltage and high-current electric arc which exists between the electrode and melting material Nowadays, EAFs are designed for very large power input ratings and due to the nature of both the electric arc and the meltdown processes, these devices cause significant electric distortion such as harmonics, interharmonics and flickers in the supply network. Electric utilities and their customers pay much attention to mitigate such power quality (PQ) problems associated with EAF loads. Obtaining time response of the EAF is important for investigating the impact of the nonlinear and time-varying load on power systems. Meanwhile, an accurate EAF model is also necessary for the study of the usefulness of different solutions for PQ problems caused by EAFs.

Traditionally, most research efforts of EAF models focuses on steady-state modeling that used to describe the operation of EAFs during refining stage. Because the arc melting process is a non-stationary stochastic phenomenon, an accurate deterministic model for EAF load usually is very difficult to obtain. The factors that affect the EAF operation usually include the melting or refining materials, the electrode position, the electrode arm control scheme and the supply system voltage and impedance. Modeling the EAF

The authors are with the Department of Electrical Engineering, National

Chung Cheng University, Min-Hsiung, Chia-Yi 621Taiwan, R.O.C. (e-mail: wchang@ee.ccu.edu.tw).

load depends on several parameters (e.g. arc voltage, arc current, and arc length) determined by the position of electrode [1]. Thus it is of importance to know the response of EAFs to develop an appropriate EAF model for PQ study. In reality, some physical responses of EAFs are not easy to derive from the theoretical study and assumptions are often made for simplifying the modeling task.

The most effective and simplest manner for obtaining the responses of EAF loads is the field measurements of EAF voltage and current. Two classic EAF models based on field measured data, harmonic current injection model (HCIM) and harmonic voltage source model (HVSM), are therefore reviewed in this paper. The fast Fourier transform (FFT) is commonly adopted in HCIM and HVSM to determine the harmonic components of EAF voltage and current. According to the harmonic components of EAF voltage or current obtained by FFT, one can use an equivalent harmonic voltage source or harmonic current for describing the behavior of the EAF. However, there is a drawback occurred when applying HCIM or HVSM. When the contents of chosen harmonic components of an equivalent harmonic voltage or current are insufficient, the EAF model is inaccurate. In order to describe the response of the EAF load accurately, developing a more effective model is necessary. In the paper two other advanced methods based on recorded data for EAFs modeling: the cubic spline interpolation and an improved neural network based methods, are also described. Furthermore, an actual steel plant power system for field measurements and computer simulations is illustrated for comparisons of the four EAF models.

II. CLASSIC MODELS FOR ELECTRIC ARC FURNACE

Voltage or current distortions of EAFs are resulted from nonlinear voltage-current (i.e. v-i) characteristics and random behaviors of such loads. Many steady-state EAF models have been developed for PQ studies. The simplest one is to model the EAF by an inductor in series with a resistor [2]. A nonlinear resistance model uses the piecewise linearization method to describe the furnace with the assumed v-i characteristic [3], [4]. In [5] and [6], the EAF is modeled as a time-varying voltage source and this arc voltage is defined as a nonlinear function of the arc length. Classic models based on mathematical equations, Mayr and Cassie, are adopted to model EAF in [7] and [8], where Mayr and Cassie equations are solved to obtain the nonlinear conductance of the EAF. A harmonic-domain solution method of nonlinear differential equation is used in [9], where the EAF model is developed from the energy balance equation and is actually a nonlinear differential equation of the arc radius and the arc current.

Modeling Voltage-Current Characteristics of an Electric Arc Furnace Based on Actual Recorded Data:

A Comparison of Classic and Advanced Models G. W. Chang, Senior Member, IEEE, Y. J. Liu, and C. I Chen, Student Members, IEEE

©2008 IEEE.

However, some physical parameters are not easy to obtain and used for the aforementioned EAF models. The field measurement of EAF voltage and current is most effective way to obtain the necessary parameters for EAF modeling. The following gives two classic methods for modeling nonlinear loads, harmonic current injection and equivalent harmonic voltage source. The two classic methods also adopt the actual measured voltage and current data as the model input.

A. Harmonic Current Injection Model (HCIM)

Harmonic current injection is a commonly used method for modeling nonlinear loads in the harmonic study. In the model, the harmonic components of measured arc current of the EAF load are obtained by performing FFT and then are injected into the power network through the furnace transformer secondary side. The harmonic current source for describing EAF load is represented by its Fourier series as

)(cos)(sin)(1 1

nn n

nnnarc tnbtnati θωθω +∑ ∑++=∞

=

∞

= (1)

where the Fourier coefficients may change randomly during each fundamental cycle and the coefficients are selected as a function of the measured arc current. Figure 1 shows the equivalent circuit of the HCIM used in the study.

1i 2i ni

)(tiarc

Fig. 1. Harmonic current injection model.

B. Harmonic Voltage Source Model (HVSM)

In HVSM, the EAF is modeled as harmonic voltage source behind a series impedance, cableZ , which is composed of the furnace transformer secondary cable to the electrode. The measured arc voltage in the study is taken at the furnace transformer secondary side and the voltage drop across the cable is calculated by the arc current and the impedance of cable. Then, a complete HVSM is derived from the difference of measured arc voltage and voltage drop across the cable. Figure 2 shows the equivalent model of the EAF load based on HVSM and is represented by (2)-(4).

)(tVarc

∑

measuredV

cableV

cableZ

+

−+ −

Fig. 2. Harmonic voltage source model.

∑ +=∞

=1)sin()(

nnnmeasured tnVtV θω (2)

cablearccable ZtItV ×= )()( (3)

)()()( tVtVtV cablemeasuredarc −= (4)

III. ADVANCED MODELS FOR ELECTRIC ARC FURNACE

Literature survey shows that there have been several advanced models developed for the EAF. A model with Chaos theory is described in [10] and shows a good performance for flicker evaluation. An adaptive three-phase EAF model based on a set of linear differential equations is presented in [11], which also models the supply system and the control system. In [12], a three-phase EAF model in the frequency domain for steady-state iterative harmonic analysis (IHA) is introduced. In this paper, two more curve-fitting based techniques, cubic spline interpolation and radial basis function neural network, are described below.

C. Cubic Spline Interpolation Model (CSIM)

CSIM method is based on building up the cubic polynomials of the nonlinear arc conductance for describing the behavior of the EAF. A cubic spline is a spline constructed of piecewise third-order polynomials which pass through a set of data points without knowing slopes. Figure 3 shows one fundamental cycle of measured EAF conductance. Give the measured waveform defined on ],[ ba and a set of

nodes bxxxa n =<<<= L10 , where n is the highest number of sample points. To construct a cubic spline interpolated conductance function, G , the cubic polynomials for each interval of the function is given bellow.

32 )()()()( iiiiiiii xxdxxcxxbaxG −+−+−+= (5)

where 1 , ,2 ,1 ,0 −= ni K .

Fig. 3. Measured conductance of the EAF with its interpolated curve.

A cubic spline interpolated G is a function that must satisfies the following conditions:

(a) )()( iii xfxG = and )()( 11 ++ = iii xfxG ,

for 1 , ,2 ,1 ,0 −= ni K ,

(b) )()( 111 +++ = iiii xGxG , for 2 , ,2 ,1 ,0 −= ni K ,

(c) )()( 1'

11'

+++ = iiii xGxG , for 2 , ,2 ,1 ,0 −= ni K ,

(d) )()( 1"

11"

+++ = iiii xGxG , for 2 , ,2 ,1 ,0 −= ni K .

An additional boundary condition also must be considered in cubic spline interpolation. Depending on different types of boundary conditions imposed at ax =0 and bxn = , two widely used boundary conditions, natural and clamped boundaries, are defined as follows.

(i) 0)()( "0

" == nxGxG (natural)

(ii) )()( and )()( ''0

'0

'nn xfxGxfxG == (clamped)

The natural boundary condition generally gives less accurate results than the clamped condition close to the endpoints of the interval ],[ 0 nxx , unless the function f

nearly satisfies 0)()( "0

" == nxfxf [13]. Although the clamped boundary leads to more accurate approximations; however, for this type of boundary condition to hold, it is necessary to have either the values of the derivative at the endpoints. An alternative to the natural boundary condition that does not require knowledge of the derivative of the function f is the not-a-knot condition which is also

adopted in this paper; such condition requires that )("' xG

be continuous at 1x and at 1−nx . By applying aforementioned conditions defined in (5) and finding the coefficients ia , ib , ic , and id for the cubic polynomials

iG , the approximation process of cubic spline interpolation with not-a-knot conditions for EAF conductance modeling is constructed.

D. Radial Basis Function Model (RBFM)

By means of the ability to solve the problems of function learning, the artificial neural network (ANN) is regarded as a powerful and efficient method for nonlinear device modeling applications. It is believed that many nonlinear and complex problems can be solved with proper adjustments of number of hidden layers, number of neurons in each layer, initial values of weights, choices of updating algorithms, and choices of transfer functions (or activation functions). Recently, the RBFNN has attracted much attention in function learning and modeling because of its efficiency and local adjusting ability of the basis functions [14]. Figure 4 shows the structure of RBFNN, where the input/output relationship is built upon a set of radial basis functions to compute the Euclidean distances between the inputs and the centers of radial basis functions, as given in (6).

Mkcxfxz kk , ,2 ,1 ),()( K=−= (6)

In (6), x represents the input vector from 1x to Nx in

Fig. 4, kz is the output of each neuron in the hidden layer,

and kc is the center of kth neuron. Thus the overall output of the network is

Fig. 4. Structure of radial basis function neural network.

∑ ⋅==

M

kkk xzwy

1)( (7)

where kw is the weight between kth neuron and output.

Since the difference e(p) between the desired )( pd and

estimated )( py signal at pth iteration is given by

)()()( pypdpe −= , (8)

the objective function is defined as

22 ))()(()()( pypdpepE −== (9)

The parameters of the network can be estimated by using the gradient descent method which satisfies

0)(

=kdw

pdE and 0

)(=

kdc

pdE. (10)

In general, the activation function f in (6) is selected to

be Gaussian function as defined in (11), where kσ is the standard deviation representing the variety of input data relative to the center kc .

)exp()(2

2

k

kk

cxcxf

σ

−−=− (11)

By applying the measured current and voltage of the EAF into the network as input and estimated vectors, and finding the optimal parameters iteratively, the approximation of transfer function denoting the EAF v-i relationship is obtained, and vice versa.

IV. RESULTS

To compare the performance of the described four methods for the EAF modeling, Fig. 5 shows a one-line diagram of the steel plant power system used for illustration. In the system, a 50-ton three-phase AC arc furnace connected to an 11.4 kV/460 V EAF transformer rated at 33 MVA is investigated. To observe the mitigation of electric distortions caused by the EAF, devices such static var compensator (SVC) and passive filter banks are also modeled.

Fig. 5. Single-line diagram of the studied steel plant power system.

(a) (b)

(c) (d)

(e) (f)

Fig. 6. Modeling results with the classic methods: (a) voltage waveform with HCIM; (b) voltage waveform with HVSM; (c) current waveform with HCIM; (d) current waveform with HVSM; (e) v-i characteristic with HCIM; (f) v-i characteristic with HVSM.

(a) (b)

(c) (d)

(e) (f)

Fig. 7. Modeling results with the advanced methods: (a) voltage waveform with CSM; (b) voltage waveform with RBFM; (c) current waveform with CSM; (d) current waveform with RBFM; (e) v-i characteristic with CSM; (f) v-i characteristic with RBFM.

In the study, the EAF v-i characteristic is tested to verify the described models by sampling the EAF voltage and current data at phase A with a fixed sampling rate of 3840 Hz for one fundamental cycle during the refining stage. Table I gives the contents of harmonic voltage and current used for EAF model. Figures 6 and 7 display the modeling results of voltage, current waveforms, and v-i characteristics based on the classic and advanced methods.

By observing Fig. 6, it is seen that the classic methods cannot accurately match actual measured EAF current and

voltage. This is because that the studies of the EAF system only consider harmonics up to 9th and some of the other harmonic components are ignored while modeling EAF loads [15]. Therefore, either HCIM or HVSM does not yield a good function approximation for the v-i characteristic and leads to more modeling errors.

On the other hand, the current waveforms are completely matched by adopting the advanced methods due to the direct usage of currents to be the inputs of the models. Hence, the accuracy of the estimated voltage signals is the only thing of

concern. By observing Figs. 7(a) and 7(b), it is found that the curve fitting of voltage waveforms is much accurate than that of the classic ones. However, due to the fitted parameters used in the CSIM are slightly inadequate (only 64 sampling points are used to describe one fundamental cycle of the measured waveform in this study), the estimation error of v-i characteristic is still significant. On the contrary, the dynamic behavior of v-i characteristic can be accurately constructed with the RBFM.

TABLE I CONTENTS OF CURRENT AND VOLTAGE FOR HCIM AND HVSM

HCIM HVSM Harmonics Current (%) Harmonics Voltage (%)

2 2.5% 2 0.5% 3 6.5% 3 18% 4 0.8% 4 0.5% 5 8% 5 2.5% 6 1% 6 1% 7 3.5% 7 2% 8 0.5% 8 0.5% 9 2% 9 1.5%

V. CONCLUSIONS

In this paper, the classic and advanced modeling methods for the AC electric arc furnace based on the actual recorded data are compared. Field measurements taken from an actual steel plant power system is used to show the accuracy of the compared models based on computer simulations. Results show that HCIM and HVSM present an insufficient accuracy when modeling the V-I characteristic of the EAF load, due to insufficient harmonic components are included in models. Another limitation of adopting HCIM and HVSM is that FFT is implemented in the steady state and cannot be used to describe the dynamic behavior of the EAF load. To overcome these drawbacks, two different curve-fitting methods for EAF modeling are proposed. Although, CSIM can model the dynamic behavior of EAF, but the precision of the model may be affected by inadequate sampling parameters. Finally, the structure of RBFNN is developed to accurately model the EAF. The RBFM not only models the EAF load accurately but also includes the abilities to describe and predict the dynamic characteristic of EAF loads.

VI. REFERENCES

[1] T. Zheng, E. B. Makram and A. A. Girgis, “Effect of Different Arc Furnace Models on Voltage Distortion,” in Proceedings of IEEE Int’l Conf. Harmonics Quality of Power (ICHQP), Greece, pp. 1079-1085, Oct. 1998.

[2] GIGRE Working Group 36-05, “Harmonics, characteristic parameters, methods of study, estimates of existing values in the network,” Electra, no. 77, pp.35-54, 1981.

[3] S. Varadan, E. B. Makram, and A. A. Girgis, “A new time domain voltage source model for an arc furnace using EMTP,” IEEE Trans. on Power Delivery, vol. 11 no. 3, pp. 1685-1691, July 1996.

[4] M. A. P. Alonso and M. P. Donsion, “An improved time domain arc furnace model for harmonic analysis,” IEEE Trans. on Power Delivery, vol. 19, no. 1, pp. 367-373, Jan. 2004.

[5] G. C. Montanari, M. Loggini, A. Cavallini, L. Pitti, and D. Zaninelli, “Arc-furnace model for the study of flicker compensation in electrical networks,” IEEE Trans. on Power Delivery, vol. 9, no. 4, pp. 2026-2036, Oct. 1994.

[6] L. Tang, S. Kolluri, and M. F. McGranaghan, “Voltage flicker prediction for two simultaneously operated AC arc furnaces,” IEEE Trans. on Power Delivery, vol. 12, no. 2, pp. 985-992, April 1997.

[7] K. J. Tseng, Y. Wang, and D. M. Vilathgamuwa, “An experimentally verified hybrid Cassie-Mayr electric arc model for power electronics simulations,” IEEE Trans. on Power Electronics, vol. 12, no. 3, pp. 429-436, May 1997.

[8] H. Mokhtari and M. Hejri, ”A new three phase time-domain model for electric arc furnace using MATLAB,” in Proc. IEEE Asia-Pacific T&D Conf., vol.3, pp. 2078-2083, Oct. 2002.

[9] E. Acha, A. Semlyen, and N. Rajakovic, “A harmonic domain computational package for nonlinear problems and its application to electric arcs,” IEEE Trans. on Power Delivery, vol. 5, no. 3, pp. 1390-1397, July 1990.

[10] O. Ozgun and A. Abur, “Flicker study using a novel arc furnace model,” IEEE Trans. on Power Delivery, vol. 17, no. 4, pp. 1158- 1163, Oct. 2002.

[11] T. Zheng and E. B. Makram, “An Adaptive Arc Furnace Model,” IEEE Trans. on Power Delivery, Vol. 15, No. 3, July 2000, pp. 931-939.

[12] J. G. Mayordomo, L. F. Beites, R. Asensi, etc al., “A new frequency domain arc furnace model for iterative harmonic analysis,” IEEE Trans. on Power Delivery, Vol. 12, Oct. 1997, pp. 1771-1778.

[13] E. M. Gisela and U. Frank, Numerical Algorithms with Fortran, Springer, 1996.

[14] J. Zhang, G. G. Walter, Y. Miao, and W. N. W. Lee, “Wavelet Neural Networks for Function Learning,” IEEE Trans. on Signal Processing, Vol. 43, No. 6, June 1995, pp. 1485-1497.

[15] W. S. Vilcheck and D. A. Gonzalez, “Measurements and simulations - for state-of-the-art harmonic analysis,” in Proceedings of the 1988 IEEE Industry Applications Society Annual Meeting, vol. 2, pp. 1530-1534, Oct. 1988.

VII. BIOGRAPHIES

Gary W. Chang (M’94-SM’01) received his Ph.D. degree from the University of Texas at Austin in 1994. He is a professor of the Department of Electrical Engineering at National Chung Cheng University, Taiwan. His areas of research interest include power systems optimization, harmonics, and power quality. Dr. Chang is a member of Tau Beta Pi and a registered professional engineer in the state of Minnesota. He chairs the IEEE Task Force on Harmonics Modeling & Simulation.

Yu-Jen Liu (S’07) received his BSEE from National Kaohsiung University of Applied and Sciences at Kaohsiung, Taiwan and MSEE from National Chung Cheng University at Chia-Yi, Taiwan, in 2003 and in 2005, respectively. He is currently working toward his Ph.D. at National Chung Cheng University. His areas of research interests include power system modeling, harmonics and real-time simulation.

Cheng-I Chen (S’07) received his BSEE and MSEE from National Chung Cheng University at Chia-Yi, Taiwan, in 2004 and in 2006, respectively. He is currently working toward his Ph.D. at the same institute. His areas of research interests include power quality and signal processing.