# Optimal Electric Arc Furnace Model’s Characteristics Using Genetic Algorithm and Particle Swarm Optimization and Comparison of Various Optimal Characteristics in DIgSILENT and EMTP-RV

Authors: Amir Mohammad Entekhabi Nooshabadi; Shakiba Sadeghi; Hamed Hashemi-Dezaki
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

The stochastic and nonlinear characteristics of electric arc furnaces (EAFs) lead to power quality challenges in the power system. In studying EAF behaviors, having optimized characteristics/models, selecting a suitable and optimum model that adapts to the actual characteristics of EAFs, and investigating simulation software’s capability for implementing EAF models are essential. However, the literature shows a research gap in investigating EAF simulations in various software products based on different models. This paper studies several time-domain models, such as piece-wise linear, modified piece-wise linear, hyperbolic, exponential, and exponential-hyperbolic models, for EAF modeling and simulation. The optimal estimation of parameters for the introduced models is necessary to adapt actual EAF characteristics. Thus, one of the studies taken in this paper is optimizing the EAF model’s characteristics. The proposed optimization problem is solved using the genetic algorithm (GA) and particle swarm optimization (PSO). Moreover, the optimized models are simulated in DIgSILENT and EMTP-RV to investigate different EAF models from the viewpoint of accuracy and efficiency. The optimization of different EAF models’ characteristics and comparison of EMTP-RV and DIgSILENT in simulating EAF behavior are the contributions of this paper. The proposed method is validated based on the actual data of a realistic EAF-based steel company in Iran. The obtained results show that the modified piece-wise linear model has the most accuracy in identifying the EAF behavior. The test results based on DIgSILENT and EMTP-RV simulations imply that the EAF could be simulated with high accuracy using modified piece-wise linear and piece-wise linear models. In general, EMTP-RV has expressed more accuracy in simulating different EAF models, and the simulation execution speed of EMTP-RV is around 2.5 times faster than DIgSILENT. In contrast, DIgSILENT is more suitable to facilitate the power system studies of EAF according to its extensive study tools and library.


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## Body

## 1. Introduction

### 1.1. Motivation and Incitement

The deployment of electric arc furnaces (EAFs) is one of the well-known and efficient approaches in steel industries [1, 2]. The EAF-based technologies have ranked as the second steelmaking process [3]. Due to the stochastic changes in the arc length and other behaviors of electric arcs during the fusion and refining process, EAF’s characteristics would be dynamic, nonlinear, and time-variant [4, 5]. Accordingly, EAF causes power quality problems in the supply network, such as voltage and current imbalances [6], harmonics [7, 8], and voltage flicker [9–11]. To investigate the EAF’s adverse effects in the point of common coupling (PCC) and provide appropriate solutions to deal with the disturbances, using precise EAF mathematical models and implementing the EAF in specialized simulation software is essential [12–14]. Although several studies have been done in the field of EAFs, there is a research gap in selecting an appropriate model incorporating a suitable simulation environment. This article tries to fill such a research gap by studying various EAF models, using the optimized estimated parameters in EMTP-RV and DIgSILENT.


### 1.2. Literature Review

Several studies have proposed various models to identify the EAF behavior and characteristics [15, 16]. Models used to describe EAF have been classified into two categories, e.g., frequency-domain and time-domain models. In the frequency-domain models, the EAF is modeled using a harmonic current source or a harmonic voltage source [17]. On the other hand, in time-domain models, the EAF models are presented based on the nonlinear differential equations of the furnace [18, 19] or the nonlinear approximation of the EAF’s voltage-current characteristic [4, 20]. Also, in [21], the EAF dynamic models have been proposed, considering the stochastic variations in arc length.In [18], the EAF has been modeled using a cascade combination of variable resistance and inductance in parallel with a current source. The variable resistance and inductance have modeled the EAF dynamic and time-varying behavior. Also, the current source has modeled EAF current harmonics. Reference [19] reported a current-controlled voltage source to model the EAF and calculate the voltage flicker.Based on the voltage-current characteristic of an actual EAF presented in [22], references such as [23, 24] have presented mathematical models approximating the EAF voltage-current characteristic. The approximated EAF models’ characteristics have been used in the subsequent investigations for EAF studies. In [25], EAF voltage-current characteristics have been used for flicker studies. EAF power quality has been studied in [26] using voltage-current characteristics. The EAF voltage-current characteristics in [23] have been followed from the flicker studies viewpoint, considering the EAF transformers and cable system. In other articles, such as [27], the EAF voltage-current characteristics have been used for power quality investigations. In [28], the exponential-hyperbolic model has been proposed, and the model on the power system was studied using the PSCAD. Voltage harmonics and other power quality issues have been studied in [29, 30], using EAF voltage-current characteristics.Practical approaches are associated with the conformity of EAF models to the actual characteristics. This accuracy and conformity can be provided by estimating parameters for EAF models. Reference [31] has studied parameter identification of EAF models, which adapted arc measured values to estimate parameters of the EAF models. Despite the importance of representing EAF with an accurate model, parameter estimation and optimization of the EAF model’s characteristics have not received much attention in the literature.The frequency-domain models have been mainly used for harmonic analysis in the EAF upstream networks [32]. On the other hand, time-domain models are suitable for studying EAF in power systems. Thus, several time-domain models derived by approximating the EAF voltage-current characteristic, such as piece-wise linear, modified piece-wise linear, hyperbolic, exponential, and exponential-hyperbolic models, have been introduced in the literature for studying the EAF behaviors [22, 33].The literature review in the field of EAF modeling and simulation is summarized in Table1. As seen, less attention has been paid to optimizing EAF’s various models’ characteristics in the literature. Moreover, according to Table 1, in most recent studies, such as [19, 32, 34], MATLAB has been used for EAF modeling. Despite the many capabilities of MATLAB, this software does not have the conditions and requirements for specialized studies in the power system fields. Several references have studied the EAF models in PSCAD and EMTP-RV. PSCAD is a power system simulation software that has been utilized for EAF simulation in some research works, such as [4, 35]. Also, EMTP-RV is a technically advanced software for transient analysis in power systems. In more recent papers, like [4, 36], EMTP-RV has been used for EAF simulation studies. Moreover, DIgSILENT is a leading power system software that can be used for simulating EAF models. However, less attention has been paid to DIgSILENT to simulate the EAF and corresponding studies.Table 1
Summary of the literature review in the field of EAF modeling.
RefYearEAF modelCharacteristic optimizationSoftwarePiece-wise linearModified piece-wise linearHyperbolicExponentialExponential-hyperbolicOther modelsMATLABPSCADEMTP-RVDIgSILENT[35]2017✓✓✓✓✓Arc resistance✗✓✓✗✗[22]2014✗✗✗✗✗Wavelet-transform and neural-network✗✓✗✗✗[25]2007✓✓✓✗✗✗✗✓✗✗✗[26]2016✗✗✓✓✓Cassie-Mayer✗✓✗✗✗[23]2008✗✗✓✓✓✗✗✗✓✗✗[27]2003✗✗✓✗✗✗✗✗✓✗✗[31]2010✗✗✓✗✗✗GA✗✓✗✗[28]2009✗✗✗✓✓✗✗✗✓✗✗[29]1998✓✓✓✗✗✗✗✗✗✓✗[30]1994✗✗✓✗✗✗✗✗✗✓✗[32]2014✗✗✗✗✗Cassie-Mayer✗✓✗✗✗[34]2017✗✗✗✗✗Wavelet-transform and ANN✗✓✗✗✗[19]2012✗✗✗✗✗Markov✗✓✗✗✗[4]2018✗✗✗✗✗CHAOSGA✗✓✓✗[36]2001✗✗✗✗✗CHAOS✗✗✗✓✗Proposed method✓✓✓✓✓✗GA/PSO✗✗✓✓The literature shows a research gap in investigating EAF simulations in various software programs based on different models with optimized estimated model parameters.


### 1.3. Contributions and Paper Organization

As discussed, different time-domain static models have been developed for identifying the EAF behaviors in the power system. The optimal estimation of parameters for the EAF models is necessary to adapt actual EAF characteristics. One of the studies taken in this paper is optimizing the EAF model’s characteristics by estimating the models’ parameters. The proposed optimization problem is solved using the GA and PSO.The appropriate and effective simulation environment/software for implementing the EAF models is another essential aspect of EAF’s accurate simulation. In recent works, EAF models have been studied in general simulation environments like MATLAB rather than specialized software for power system simulations. To fill this knowledge gap, this research implements various optimized models’ characteristics of EAF in leading power system software, such as EMTP-RV and DIgSILENT. Moreover, EMTP-RV and DIgSILENT performance in simulating EAF models have been compared in terms of accuracy and speed.The most important contributions of this paper in the research field of EAF modeling and simulation are listed as follows:Parameter estimation of EAF different models to adapt measured arc characteristics through optimizationOptimizing the EAF models’ characteristics, using the GA and PSOImplementation of different models for the EAF in EMTP-RV and DIgSILENTComparing EMTP-RV and DIgSILENT performance in simulating EAF models, considering accuracy and speed aspectsApplying the proposed studies to an actual EAF test system located in IranThe rest of this paper is organized as follows. In Section2, the problem statement is drawn. In Section 3, various EAF static models in the time-domain are introduced. In Section 4, optimization of EAF models’ characteristics using GA and PSO is discussed. Section 5 presents the implementation and simulation of EAF models. In Section 6, the optimization and simulation results are proposed. Finally, the conclusion is drawn in Section 7.


## 1.1. Motivation and Incitement

The deployment of electric arc furnaces (EAFs) is one of the well-known and efficient approaches in steel industries [1, 2]. The EAF-based technologies have ranked as the second steelmaking process [3]. Due to the stochastic changes in the arc length and other behaviors of electric arcs during the fusion and refining process, EAF’s characteristics would be dynamic, nonlinear, and time-variant [4, 5]. Accordingly, EAF causes power quality problems in the supply network, such as voltage and current imbalances [6], harmonics [7, 8], and voltage flicker [9–11]. To investigate the EAF’s adverse effects in the point of common coupling (PCC) and provide appropriate solutions to deal with the disturbances, using precise EAF mathematical models and implementing the EAF in specialized simulation software is essential [12–14]. Although several studies have been done in the field of EAFs, there is a research gap in selecting an appropriate model incorporating a suitable simulation environment. This article tries to fill such a research gap by studying various EAF models, using the optimized estimated parameters in EMTP-RV and DIgSILENT.


## 1.2. Literature Review

Several studies have proposed various models to identify the EAF behavior and characteristics [15, 16]. Models used to describe EAF have been classified into two categories, e.g., frequency-domain and time-domain models. In the frequency-domain models, the EAF is modeled using a harmonic current source or a harmonic voltage source [17]. On the other hand, in time-domain models, the EAF models are presented based on the nonlinear differential equations of the furnace [18, 19] or the nonlinear approximation of the EAF’s voltage-current characteristic [4, 20]. Also, in [21], the EAF dynamic models have been proposed, considering the stochastic variations in arc length.In [18], the EAF has been modeled using a cascade combination of variable resistance and inductance in parallel with a current source. The variable resistance and inductance have modeled the EAF dynamic and time-varying behavior. Also, the current source has modeled EAF current harmonics. Reference [19] reported a current-controlled voltage source to model the EAF and calculate the voltage flicker.Based on the voltage-current characteristic of an actual EAF presented in [22], references such as [23, 24] have presented mathematical models approximating the EAF voltage-current characteristic. The approximated EAF models’ characteristics have been used in the subsequent investigations for EAF studies. In [25], EAF voltage-current characteristics have been used for flicker studies. EAF power quality has been studied in [26] using voltage-current characteristics. The EAF voltage-current characteristics in [23] have been followed from the flicker studies viewpoint, considering the EAF transformers and cable system. In other articles, such as [27], the EAF voltage-current characteristics have been used for power quality investigations. In [28], the exponential-hyperbolic model has been proposed, and the model on the power system was studied using the PSCAD. Voltage harmonics and other power quality issues have been studied in [29, 30], using EAF voltage-current characteristics.Practical approaches are associated with the conformity of EAF models to the actual characteristics. This accuracy and conformity can be provided by estimating parameters for EAF models. Reference [31] has studied parameter identification of EAF models, which adapted arc measured values to estimate parameters of the EAF models. Despite the importance of representing EAF with an accurate model, parameter estimation and optimization of the EAF model’s characteristics have not received much attention in the literature.The frequency-domain models have been mainly used for harmonic analysis in the EAF upstream networks [32]. On the other hand, time-domain models are suitable for studying EAF in power systems. Thus, several time-domain models derived by approximating the EAF voltage-current characteristic, such as piece-wise linear, modified piece-wise linear, hyperbolic, exponential, and exponential-hyperbolic models, have been introduced in the literature for studying the EAF behaviors [22, 33].The literature review in the field of EAF modeling and simulation is summarized in Table1. As seen, less attention has been paid to optimizing EAF’s various models’ characteristics in the literature. Moreover, according to Table 1, in most recent studies, such as [19, 32, 34], MATLAB has been used for EAF modeling. Despite the many capabilities of MATLAB, this software does not have the conditions and requirements for specialized studies in the power system fields. Several references have studied the EAF models in PSCAD and EMTP-RV. PSCAD is a power system simulation software that has been utilized for EAF simulation in some research works, such as [4, 35]. Also, EMTP-RV is a technically advanced software for transient analysis in power systems. In more recent papers, like [4, 36], EMTP-RV has been used for EAF simulation studies. Moreover, DIgSILENT is a leading power system software that can be used for simulating EAF models. However, less attention has been paid to DIgSILENT to simulate the EAF and corresponding studies.Table 1
Summary of the literature review in the field of EAF modeling.
RefYearEAF modelCharacteristic optimizationSoftwarePiece-wise linearModified piece-wise linearHyperbolicExponentialExponential-hyperbolicOther modelsMATLABPSCADEMTP-RVDIgSILENT[35]2017✓✓✓✓✓Arc resistance✗✓✓✗✗[22]2014✗✗✗✗✗Wavelet-transform and neural-network✗✓✗✗✗[25]2007✓✓✓✗✗✗✗✓✗✗✗[26]2016✗✗✓✓✓Cassie-Mayer✗✓✗✗✗[23]2008✗✗✓✓✓✗✗✗✓✗✗[27]2003✗✗✓✗✗✗✗✗✓✗✗[31]2010✗✗✓✗✗✗GA✗✓✗✗[28]2009✗✗✗✓✓✗✗✗✓✗✗[29]1998✓✓✓✗✗✗✗✗✗✓✗[30]1994✗✗✓✗✗✗✗✗✗✓✗[32]2014✗✗✗✗✗Cassie-Mayer✗✓✗✗✗[34]2017✗✗✗✗✗Wavelet-transform and ANN✗✓✗✗✗[19]2012✗✗✗✗✗Markov✗✓✗✗✗[4]2018✗✗✗✗✗CHAOSGA✗✓✓✗[36]2001✗✗✗✗✗CHAOS✗✗✗✓✗Proposed method✓✓✓✓✓✗GA/PSO✗✗✓✓The literature shows a research gap in investigating EAF simulations in various software programs based on different models with optimized estimated model parameters.


## 1.3. Contributions and Paper Organization

As discussed, different time-domain static models have been developed for identifying the EAF behaviors in the power system. The optimal estimation of parameters for the EAF models is necessary to adapt actual EAF characteristics. One of the studies taken in this paper is optimizing the EAF model’s characteristics by estimating the models’ parameters. The proposed optimization problem is solved using the GA and PSO.The appropriate and effective simulation environment/software for implementing the EAF models is another essential aspect of EAF’s accurate simulation. In recent works, EAF models have been studied in general simulation environments like MATLAB rather than specialized software for power system simulations. To fill this knowledge gap, this research implements various optimized models’ characteristics of EAF in leading power system software, such as EMTP-RV and DIgSILENT. Moreover, EMTP-RV and DIgSILENT performance in simulating EAF models have been compared in terms of accuracy and speed.The most important contributions of this paper in the research field of EAF modeling and simulation are listed as follows:Parameter estimation of EAF different models to adapt measured arc characteristics through optimizationOptimizing the EAF models’ characteristics, using the GA and PSOImplementation of different models for the EAF in EMTP-RV and DIgSILENTComparing EMTP-RV and DIgSILENT performance in simulating EAF models, considering accuracy and speed aspectsApplying the proposed studies to an actual EAF test system located in IranThe rest of this paper is organized as follows. In Section2, the problem statement is drawn. In Section 3, various EAF static models in the time-domain are introduced. In Section 4, optimization of EAF models’ characteristics using GA and PSO is discussed. Section 5 presents the implementation and simulation of EAF models. In Section 6, the optimization and simulation results are proposed. Finally, the conclusion is drawn in Section 7.


## 2. Problem Statement and Research Overview

The nonlinear and stochastic nature of EAF during its operation causes power quality issues, such as voltage and current imbalances, harmonics, flicker, and other disturbances. To investigate the EAF’s abnormal effects and provide appropriate solutions to deal with them, modeling and simulation of EAF characteristics are essential. Thus, precise mathematical models and specialized simulation software that can simulate the stochastic, time-variant, and nonlinear behavior of EAF are primarily required in EAF studies.Selecting an appropriate model that adapts to the actual characteristics of EAFs is an essential issue. In addition, optimizing the estimated parameters of each model based on the actual measured values of the voltage and current of the EAF is inevitable. Moreover, investigating simulation software’s capability for implementing EAF models is another concern that received less attention in the literature. In this paper, the EMTP-RV and DIgSILENT simulations of EAF based on the optimized estimation of various models are examined in the following steps:Step 1: Estimation of EAF models’ parameters, e.g., piece-wise linear, modified piece-wise linear, hyperbolic, exponential, and exponential-hyperbolic models, by the GA and PSO.Step 2: Simulating the EAF based on various models with the optimized estimation of their parameters in EMTP-RV and DIgSILENT.Step 3: Analyzing the simulation results of EMTP-RV and DIgSILENT based on various models and different aspects, like the conformity index.Step 4: Selecting suitable models for EAF simulation in EMTP-RV and DIgSILENT.


## 3. Time-Domain EAF Static Models

### 3.1. Model 1: Piece-Wise Linear Model

The piece-wise linear model is one of the simplest EAF models in the time-domain based on the linear approximation of the EAF’s voltage-current characteristics. In this model, the relation between arc voltage and arc current is expressed by (1) and (2) [29, 37].(1)va=R2ia+vig1−R2R1,i1≤ia≤i2,R1ia,−i1≤ia≤i1,R2ia−vig1−R2R1,−i2≤ia≤−i1,(2)i1=vigR11,i2=vexR2−vig1R2−1R1.In (1) and (2), va and ia denote the arc voltage and arc current. In addition, vig and vex represent the arc ignition voltage and arc extinction voltage, respectively. Moreover, i1 and i2 are used to express the arc ignition current and the arc extinguishing current. As shown in Figure 1, R1 and R2 would be the slope of OA and AB segments in the voltage-current characteristic of the piece-wise linear model.Figure 1
Voltage-current characteristic of the EAF’s piece-wise linear model [37].


### 3.2. Model 2: Modified Piece-Wise Linear Model

The modified piece-wise linear model is developed based on more precise approximations of the EAF’s characteristics. The mathematical expression of the modified piece-wise linear model has been given in [29](3)va=R1ia,inc:−i3≤ia≤i1,dec:−i1≤ia≤i3,vst+vig−vstexpi1−iaiT,inc:i1<ia≤i2,−vst+vst−vigexpi1+iaiT,dec:−i2≤ia<−i1,vst+ia−i2R2,inc:i2<ia,−vst+ia+i2R2,dec:−i2>ia,−vex+ia+i3R3,inc:ia<−i3,vex+ia−i3R3,dec:i3<ia,(4)i1=vigR1,i2=3i1,i3=vexR1,iT=1.5i1.In (3), the expressions inc and dec indicate increasing and decreasing parts in the arc current, respectively. In addition, vst is a voltage between the arc ignition voltage (vig) and the arc extinction voltage (vex). Also, vst happens due to a sudden voltage drop across the EAF electrodes during the melting stage. This voltage causes the arc voltage to decrease exponentially from vig to vst. Moreover, R1, R2, and R3 are the slope of the OA, CB, and BD segments in the voltage-current characteristic of the modified piece-wise linear model, respectively. In (4), i1, i2, and i3, which are constants, are defined. Figure 2 shows the voltage-current characteristic of the modified piece-wise linear model.Figure 2
Voltage-current characteristic of the modified piece-wise linear model [29].


### 3.3. Model 3: Hyperbolic Model

The hyperbolic model is one of the most widely used time-domain EAF models. In this model, the voltage rising time is neglected and causes sudden arc voltage changes during the zero-crossing arc current. The voltage arc in the hyperbolic model is formulated using [30](5)va=vat+CD+iasignumia.In (5), C andD are constant parameters related to the arc power and arc current. Also, vat is the voltage that depends on the arc length. The voltage-current characteristic of the hyperbolic model is shown in Figure 3.Figure 3
Voltage-current characteristic of the hyperbolic model [30].


### 3.4. Model 4: Exponential Model

In the EAF’s exponential model, as shown in (6), an exponential function is used to approximate the EAF’s voltage-current characteristic [28].(6)va=vat1−e−ia/I0signumia.In (6), the constant I0 denotes the slope of positive and negative currents. Moreover, the voltage-current characteristic of EAF’s exponential model is depicted in Figure 4.Figure 4
Voltage-current characteristic of EAF’s exponential model [28].


### 3.5. Model 5: Hyperbolic-Exponential Model

The hyperbolic-exponential model is a combination of hyperbolic and exponential models for EAFs. The mathematical expressions and formulas of the hyperbolic-exponential model are given in [28](7)va=vat+CD+ia,inc:ia>0,−vat+CD−ia,inc:ia<0,vat1−e−ia/I0,dec:ia>0,−vat1−eia/I0,dec:ia<0.According to (7), the arc voltage during current rise is modeled with a hyperbolic function, and during current descent it is modeled with an exponential function. The voltage-current characteristic of the hyperbolic-exponential model is shown in Figure 5.Figure 5
Voltage-current characteristic of EAF’s hyperbolic-exponential model [28].


## 3.1. Model 1: Piece-Wise Linear Model

The piece-wise linear model is one of the simplest EAF models in the time-domain based on the linear approximation of the EAF’s voltage-current characteristics. In this model, the relation between arc voltage and arc current is expressed by (1) and (2) [29, 37].(1)va=R2ia+vig1−R2R1,i1≤ia≤i2,R1ia,−i1≤ia≤i1,R2ia−vig1−R2R1,−i2≤ia≤−i1,(2)i1=vigR11,i2=vexR2−vig1R2−1R1.In (1) and (2), va and ia denote the arc voltage and arc current. In addition, vig and vex represent the arc ignition voltage and arc extinction voltage, respectively. Moreover, i1 and i2 are used to express the arc ignition current and the arc extinguishing current. As shown in Figure 1, R1 and R2 would be the slope of OA and AB segments in the voltage-current characteristic of the piece-wise linear model.Figure 1
Voltage-current characteristic of the EAF’s piece-wise linear model [37].


## 3.2. Model 2: Modified Piece-Wise Linear Model

The modified piece-wise linear model is developed based on more precise approximations of the EAF’s characteristics. The mathematical expression of the modified piece-wise linear model has been given in [29](3)va=R1ia,inc:−i3≤ia≤i1,dec:−i1≤ia≤i3,vst+vig−vstexpi1−iaiT,inc:i1<ia≤i2,−vst+vst−vigexpi1+iaiT,dec:−i2≤ia<−i1,vst+ia−i2R2,inc:i2<ia,−vst+ia+i2R2,dec:−i2>ia,−vex+ia+i3R3,inc:ia<−i3,vex+ia−i3R3,dec:i3<ia,(4)i1=vigR1,i2=3i1,i3=vexR1,iT=1.5i1.In (3), the expressions inc and dec indicate increasing and decreasing parts in the arc current, respectively. In addition, vst is a voltage between the arc ignition voltage (vig) and the arc extinction voltage (vex). Also, vst happens due to a sudden voltage drop across the EAF electrodes during the melting stage. This voltage causes the arc voltage to decrease exponentially from vig to vst. Moreover, R1, R2, and R3 are the slope of the OA, CB, and BD segments in the voltage-current characteristic of the modified piece-wise linear model, respectively. In (4), i1, i2, and i3, which are constants, are defined. Figure 2 shows the voltage-current characteristic of the modified piece-wise linear model.Figure 2
Voltage-current characteristic of the modified piece-wise linear model [29].


## 3.3. Model 3: Hyperbolic Model

The hyperbolic model is one of the most widely used time-domain EAF models. In this model, the voltage rising time is neglected and causes sudden arc voltage changes during the zero-crossing arc current. The voltage arc in the hyperbolic model is formulated using [30](5)va=vat+CD+iasignumia.In (5), C andD are constant parameters related to the arc power and arc current. Also, vat is the voltage that depends on the arc length. The voltage-current characteristic of the hyperbolic model is shown in Figure 3.Figure 3
Voltage-current characteristic of the hyperbolic model [30].


## 3.4. Model 4: Exponential Model

In the EAF’s exponential model, as shown in (6), an exponential function is used to approximate the EAF’s voltage-current characteristic [28].(6)va=vat1−e−ia/I0signumia.In (6), the constant I0 denotes the slope of positive and negative currents. Moreover, the voltage-current characteristic of EAF’s exponential model is depicted in Figure 4.Figure 4
Voltage-current characteristic of EAF’s exponential model [28].


## 3.5. Model 5: Hyperbolic-Exponential Model

The hyperbolic-exponential model is a combination of hyperbolic and exponential models for EAFs. The mathematical expressions and formulas of the hyperbolic-exponential model are given in [28](7)va=vat+CD+ia,inc:ia>0,−vat+CD−ia,inc:ia<0,vat1−e−ia/I0,dec:ia>0,−vat1−eia/I0,dec:ia<0.According to (7), the arc voltage during current rise is modeled with a hyperbolic function, and during current descent it is modeled with an exponential function. The voltage-current characteristic of the hyperbolic-exponential model is shown in Figure 5.Figure 5
Voltage-current characteristic of EAF’s hyperbolic-exponential model [28].


## 4. Optimization of EAF Models’ Characteristics

### 4.1. Definition of the Proposed Optimization Problem

The main objective of the proposed optimization problem is to identify the parameters of different EAF models. Parameters of each model are defined as decision variables. The proposed objective function (OF) is presented in (8). In (8), OF and N represent the OF of the proposed optimization problem and the total number of data samples for comparing the model with measured values. Also, vamodel, vaactual, iamodel, and iaactual denote the modeled arc voltage, the measured arc voltage, the modeled arc current, and the measured arc current, respectively. The proposed optimization problem aims to minimize the error between the voltage and current waveforms of the model and the measured voltage and current waveforms.(8)OF=∑n=1Nvamodeln−vaactualnmaxvamodel2+iamodeln−iaactualnmaxiamodel2.The optimization problem of the EAF model’s characteristics is a nonconstrained minimization problem. However, specifying the upper and lower boundaries for each model input parameter reduces the problem search space and leads to better convergence to the optimal global value. The parameter boundaries can be assigned using the definition of parameters in the EAF model and based on the results of previous studies.


### 4.2. Proposed Optimization Problem Solving

According to the nonlinear nature of the proposed optimization problem, evolutionary algorithms are more appropriate to solve it. PSO and GA are efficient optimization techniques for solving nonlinear problems in terms of speed, simplified implementation, robustness, and capability of converging to global optima. In this paper, the GA and PSO algorithms are utilized. The statistical analysis will be performed to evaluate the performance of both PSO and GA in solving the proposed optimization problem. The statistical analysis would be helpful for examining whether the optimization algorithms have overcome the local optimization traps around the feasible region or not. Moreover, the convergence and the minor deviation in reaching the global optima in various runs and different cases should be used to investigate the suitability of the algorithms for solving the proposed optimization problem. It should be noted that the proposed optimization problem can be solved by other optimization algorithms. However, the examination of the capability of the selected optimization algorithms to find the global optima and other performance criteria is necessary.The flowcharts of solving the proposed optimization problem to estimate the parameters of the EAF’s models by the GA and PSO algorithms are shown in Figures6 and 7, respectively. As illustrated in Figures 6 and 7, the network data, problem constraints, and EAF measured values should be imported to define the optimization problem. Network data include the resistance and inductance values of flexible cables and the secondary voltage level of the EAF transformer. According to the EAF model, the allowed range for decision variables (model parameters) and the problem constraints are determined. Also, measured arc voltage and arc current in a cycle are imported to the proposed optimization problem as the base waveforms. Finally, the optimization process starts using one of the heuristic algorithms.Figure 6
Optimization of EAF models’ characteristics using GA and investigating DIgSILENT and EMTP-RV simulations.Figure 7
Optimization of EAF models’ characteristics using PSO and investigating DIgSILENT and EMTP-RV simulations.


#### 4.2.1. GA

The GA is a numerical problem-solving technique that is inspired by Darwin’s evolution theory. In GA, each chromosome is a potential solution to the problem, and the genes are related to the problem variables [38]. The GA initializes with a randomly generated population of chromosomes. In each iteration, OF value of each chromosome is evaluated. Then, the selection operator selects some of the best chromosomes using a stochastic process [39]. The crossover and mutation operators are applied to the selected chromosomes, producing a new generation of chromosomes [40, 41]. This process continues until a certain number of iterations or convergence criteria is reached. Eventually, the algorithm converges to the fittest individuals over the iterations.


#### 4.2.2. PSO

The PSO is a metaheuristic optimization algorithm developed based on the social behavior of birds flocking in search of food sources. In the PSO, particles are search agents in the feasible space, and the particles’ position represents valid solutions to the problem. Particles move to new positions to discover better positions by tracking optimal ones [42]. In the beginning, particles are scattered randomly in the feasible region. As shown in (9), xik and vik are respectively the position and velocity of the ith particle in the kth iteration. In each iteration, the OF value corresponding to each particle is evaluated, and the global best position denoted as gbest and the individual best position denoted as Pbesti are updated. The velocity of particles is updated using (9), and they are transferred to the new positions using (10).(9)vik+1=ω×vik+r1×c1×Pbesti−xik+r2×c2×gbest−xik,(10)xik+1=xik+vik+1.According to (9), r1 and r2 are random vectors in the range of [0 1]. Also, c1 and c2 indicate the personal learning coefficient and global learning coefficient, respectively. In (9), ω represents the inertia weight which decreases during iterations using a Damp coefficient through(11)ωk+1=ωk×Damp.This process continues until a certain number of iterations or convergence criteria is reached. Finally, particles would converge to the optimal position over successive movements [43, 44].


## 4.1. Definition of the Proposed Optimization Problem

The main objective of the proposed optimization problem is to identify the parameters of different EAF models. Parameters of each model are defined as decision variables. The proposed objective function (OF) is presented in (8). In (8), OF and N represent the OF of the proposed optimization problem and the total number of data samples for comparing the model with measured values. Also, vamodel, vaactual, iamodel, and iaactual denote the modeled arc voltage, the measured arc voltage, the modeled arc current, and the measured arc current, respectively. The proposed optimization problem aims to minimize the error between the voltage and current waveforms of the model and the measured voltage and current waveforms.(8)OF=∑n=1Nvamodeln−vaactualnmaxvamodel2+iamodeln−iaactualnmaxiamodel2.The optimization problem of the EAF model’s characteristics is a nonconstrained minimization problem. However, specifying the upper and lower boundaries for each model input parameter reduces the problem search space and leads to better convergence to the optimal global value. The parameter boundaries can be assigned using the definition of parameters in the EAF model and based on the results of previous studies.


## 4.2. Proposed Optimization Problem Solving

According to the nonlinear nature of the proposed optimization problem, evolutionary algorithms are more appropriate to solve it. PSO and GA are efficient optimization techniques for solving nonlinear problems in terms of speed, simplified implementation, robustness, and capability of converging to global optima. In this paper, the GA and PSO algorithms are utilized. The statistical analysis will be performed to evaluate the performance of both PSO and GA in solving the proposed optimization problem. The statistical analysis would be helpful for examining whether the optimization algorithms have overcome the local optimization traps around the feasible region or not. Moreover, the convergence and the minor deviation in reaching the global optima in various runs and different cases should be used to investigate the suitability of the algorithms for solving the proposed optimization problem. It should be noted that the proposed optimization problem can be solved by other optimization algorithms. However, the examination of the capability of the selected optimization algorithms to find the global optima and other performance criteria is necessary.The flowcharts of solving the proposed optimization problem to estimate the parameters of the EAF’s models by the GA and PSO algorithms are shown in Figures6 and 7, respectively. As illustrated in Figures 6 and 7, the network data, problem constraints, and EAF measured values should be imported to define the optimization problem. Network data include the resistance and inductance values of flexible cables and the secondary voltage level of the EAF transformer. According to the EAF model, the allowed range for decision variables (model parameters) and the problem constraints are determined. Also, measured arc voltage and arc current in a cycle are imported to the proposed optimization problem as the base waveforms. Finally, the optimization process starts using one of the heuristic algorithms.Figure 6
Optimization of EAF models’ characteristics using GA and investigating DIgSILENT and EMTP-RV simulations.Figure 7
Optimization of EAF models’ characteristics using PSO and investigating DIgSILENT and EMTP-RV simulations.


### 4.2.1. GA

The GA is a numerical problem-solving technique that is inspired by Darwin’s evolution theory. In GA, each chromosome is a potential solution to the problem, and the genes are related to the problem variables [38]. The GA initializes with a randomly generated population of chromosomes. In each iteration, OF value of each chromosome is evaluated. Then, the selection operator selects some of the best chromosomes using a stochastic process [39]. The crossover and mutation operators are applied to the selected chromosomes, producing a new generation of chromosomes [40, 41]. This process continues until a certain number of iterations or convergence criteria is reached. Eventually, the algorithm converges to the fittest individuals over the iterations.


### 4.2.2. PSO

The PSO is a metaheuristic optimization algorithm developed based on the social behavior of birds flocking in search of food sources. In the PSO, particles are search agents in the feasible space, and the particles’ position represents valid solutions to the problem. Particles move to new positions to discover better positions by tracking optimal ones [42]. In the beginning, particles are scattered randomly in the feasible region. As shown in (9), xik and vik are respectively the position and velocity of the ith particle in the kth iteration. In each iteration, the OF value corresponding to each particle is evaluated, and the global best position denoted as gbest and the individual best position denoted as Pbesti are updated. The velocity of particles is updated using (9), and they are transferred to the new positions using (10).(9)vik+1=ω×vik+r1×c1×Pbesti−xik+r2×c2×gbest−xik,(10)xik+1=xik+vik+1.According to (9), r1 and r2 are random vectors in the range of [0 1]. Also, c1 and c2 indicate the personal learning coefficient and global learning coefficient, respectively. In (9), ω represents the inertia weight which decreases during iterations using a Damp coefficient through(11)ωk+1=ωk×Damp.This process continues until a certain number of iterations or convergence criteria is reached. Finally, particles would converge to the optimal position over successive movements [43, 44].


## 4.2.1. GA

The GA is a numerical problem-solving technique that is inspired by Darwin’s evolution theory. In GA, each chromosome is a potential solution to the problem, and the genes are related to the problem variables [38]. The GA initializes with a randomly generated population of chromosomes. In each iteration, OF value of each chromosome is evaluated. Then, the selection operator selects some of the best chromosomes using a stochastic process [39]. The crossover and mutation operators are applied to the selected chromosomes, producing a new generation of chromosomes [40, 41]. This process continues until a certain number of iterations or convergence criteria is reached. Eventually, the algorithm converges to the fittest individuals over the iterations.


## 4.2.2. PSO

The PSO is a metaheuristic optimization algorithm developed based on the social behavior of birds flocking in search of food sources. In the PSO, particles are search agents in the feasible space, and the particles’ position represents valid solutions to the problem. Particles move to new positions to discover better positions by tracking optimal ones [42]. In the beginning, particles are scattered randomly in the feasible region. As shown in (9), xik and vik are respectively the position and velocity of the ith particle in the kth iteration. In each iteration, the OF value corresponding to each particle is evaluated, and the global best position denoted as gbest and the individual best position denoted as Pbesti are updated. The velocity of particles is updated using (9), and they are transferred to the new positions using (10).(9)vik+1=ω×vik+r1×c1×Pbesti−xik+r2×c2×gbest−xik,(10)xik+1=xik+vik+1.According to (9), r1 and r2 are random vectors in the range of [0 1]. Also, c1 and c2 indicate the personal learning coefficient and global learning coefficient, respectively. In (9), ω represents the inertia weight which decreases during iterations using a Damp coefficient through(11)ωk+1=ωk×Damp.This process continues until a certain number of iterations or convergence criteria is reached. Finally, particles would converge to the optimal position over successive movements [43, 44].


## 5. EAF Simulation in DIgSILENT and EMTP-RV

According to the mathematical models of the EAF, the arc voltage at any time step is calculated based on the arc current at that moment. Thus, in EAF models, a new relationship is required to calculate the arc current at any time. According to the EAF power supply circuit illustrated in Figure8, using the KVL relation from the secondary side of the EAF transformer to the electrode ends, (12) is extracted. In (12), Leq and Req are the equivalent resistance and inductance of the network.(12)V−Leqdiadt−Reqia=va⟶V−va=Leqdiadt+Reqia.Figure 8
EAF power supply circuit.Since the EAF models in DIgSILENT and EMTP-RV are implemented discretely, converting (12) to the discrete relation in (13) is necessary.(13)VT−vaT=LeqiaT+1−iaTΔt+ReqiaT.In (13), Δt and T denote the duration of simulation steps and the simulation step number. By arranging (13), the arc current is calculated at any moment based on the arc current and arc voltage at the previous step as given in(14)iaT+1=ΔtLeqVT−vaT+LeqΔtiaT+ReqiaT.Using (14) and assuming an initial value for arc current, the EAF models can be implemented in different software. The EAF mathematical models are simulated in DIgSILENT using the DSL environment and in EMTP-RV using control block diagrams.In DIgSILENT and EMTP-RV libraries, no power element simultaneously controls its voltage and current. Thus, in the EAF element structure, a current source in series with a variable resistor is used, as shown in Figure9.Figure 9
EAF equivalent circuit and structure.As seen in Figure9, if the arc current waveform (ia) is applied to the current source and the arc resistance waveform (va/ia) is applied to the variable resistor, the EAF voltage-current characteristic is created at the PCC (Bus (A)).Indeed, the voltage of Bus (A) and the current passing through this bus are precisely equal to the EAF voltage and current. The current of Bus (A), which is the same as the arc’s current, causes the arc’s voltage in Bus (A) to pass through the arc resistance. The EAF implementation using the mentioned elements in DIgSILENT and EMTP-RV is shown in Figure10.Figure 10
Implementation of EAF element in DIgSILENT and EMTP-RV.


## 6. Test Results

### 6.1. Test System Descriptions and Measurement Results

The proposed method is tested on an existing EAF plant in Iran. The single-line diagram of the understudy EAF’s supply system is shown in Figure11. As depicted, the studied EAF has been connected to the external grid through two sets of transformers, including the 400/63 kV power transformer and the EAF transformer (63/0.9 kV) [15]. A 63 kV cable system with a nominal current of 1100 A with 1320 m length has been installed for the power transmission section. The EAF electrodes are connected to the supply system via flexible cables. The electrodes and flexible cables are modeled through leq and req impedances.Figure 11
Single-line diagram of the understudy EAF’s supply network.The EAF voltage, current waveforms, and harmonic distortion have been measured/recorded to validate the proposed study. The Fluke 437-II Power quality monitor and energy analyzer have been used in this study. In Figure12, the EAF voltage and current measurements are shown. Also, Figure 13 presents the recorded harmonic distortion of the studied EAF [8].Figure 12
Measured voltage and current of the understudy EAF system [15].Figure 13
Recorded harmonic distortion of the understudy EAF system [15].The measured waveforms are used as the reference waveforms of voltage and current in the EAF optimization problem. In Figure14, the measured values and characteristics of understudy EAF are shown.Figure 14
EAF measured values and characteristics of understudy EAF. (a) Voltage. (b) Current. (c) Voltage-current characteristic.
(a)(b)(c)


### 6.2. Optimization Results

The measurement values are imported into the optimization problem. Also, the GA and PSO setting parameters are selected, as given in Tables2 and 3, for solving the proposed optimization problem.Table 2
The GA setting parameters.
ParametersQuantityNumber of decision variablesModel 14Model 26Model 33Model 42Model 54Population size1000Maximum number of iterations100Crossover rate0.65Mutation rate0.25Table 3
The PSO setting parameters.
ParametersQuantityNumber of decision variablesModel 14Model 26Model 33Model 42Model 54Population size1000Maximum number of iterations100ω initial value1Damp0.996c11.496c21.496The convergence diagrams of GA and PSO in solving the optimization problem for different models are shown in Figure15.Figure 15
Convergence diagrams of GA and PSO in solving the characteristic optimization of different EAF models. (a) Piece-wise linear model. (b) Modified piece-wise linear model. (c) Hyperbolic model. (d) Exponential model. (e) Hyperbolic-exponential model.
(a)(b)(c)(d)(e)As revealed by test results shown in Figure15, the convergence of GA is much higher than PSO. In addition, it can be concluded that as the number of decision variables in the optimization problem of the EAF model characteristics increases, the PSO convergence speed decreases.The statistical analysis of solving the proposed optimization problem by the GA and PSO in optimizing the EAF model characteristics is presented in Figure16 and Table 4. The GA would be suitable for solving discrete problems, while the PSO algorithm has a better performance in solving continuous problems. Because the optimization problem of the EAF model’s characteristics is continuous, it is expected that the PSO results in better performance in finding the optimal points. Statistical analysis shows that although the PSO has a relatively slow convergence rate than the GA, it is more robust in finding the global optimal point. For instance, in the statistical analysis diagram of the hyperbolic model, although the obtained optimal values are equal to each other with one-tenth of accuracy, the GA has not been able to find the global optimal point with one hundred-thousandths of accuracy.Figure 16
The GA and PSO optimum objectives in solving the optimization problem of the EAF model’s characteristics under various runs. (a) Piece-wise linear model. (b) Modified piece-wise linear model. (c) Hyperbolic model. (d) Exponential model. (e) Hyperbolic-exponential model.
(a)(b)(c)(d)(e)Table 4
Statistical analysis of GA and PSO in solving the optimization problem of different EAF models.
ParametersModel 1Model 2Model 3Model 4Model 5GAPSOGAPSOGAPSOGAPSOGAPSOMean4.8114.7793.5063.5217.92557.92554.02454.02455.92785.9278Standard deviation0.0420.0040.0560.0771.7 × 10−52.5 × 10−51 × 10−41.3 × 10−54.6 × 10−55.7 × 10−5Worst4.9104.7933.6523.7397.92567.92564.02484.02455.92805.9280Best4.7754.7753.4473.4477.92547.92544.02447.02445.92785.9278Number of runs30303030303030303030Since all models include a few decision variables, both algorithms reach the global optima in almost all cases. Therefore, the standard deviations in optimumOFs and solutions would not be significant in such small-sized problems. However, the smaller the standard deviation for an algorithm in solving an optimization problem, the algorithm is more reliable in finding the global optima.According to the statistical test results and the convergence diagrams in different cases, it is possible to ensure that the GA and PSO techniques have suitable performance in solving the proposed problem. It can be concluded that the converged solutions are global optimums. Since the presented research focuses on optimizing different EAF models’ characteristics rather than the optimization method, it has been confined to introducing GA and PSO to solve the optimization problem. However, solving the optimization problem of EAF models’ characteristics is not restricted to the presented techniques, and other optimization algorithms can be used to solve it.The optimized parameters for EAF models’ characteristics using the GA and PSO are given in Table5. Moreover, the optimum voltage-current characteristics of different EAF models are shown in Figure 17.Table 5
Optimal values of parameters in different EAF models.
Model no.ModelEAF model parametersOF value1Piece-wise linearvig = 477.306695 Vvex = 34.440595 VR1 = 0.052594 ΩR2 = ‒0.001078 Ω4.775482Modified piece-wise linearvig = 234.344244 Vvex = 251.885441 Vvst = 297.241896 ΩR1 = 0.046316 ΩR2 = ‒0.000261 ΩR3 = 0.002301 Ω3.447183Hyperbolicvat = 106.617882 VD = 693646.7778 ΩC = 293081906.4989 Ω7.925484Exponentialvat = 308.121744 VI0 = 4096.036572 A4.024495Hyperbolic-exponentialvat = 339.770057 VI0 = 3765.370477 AC = 6620839.7101 ΩD = 985684344.6322 Ω5.92786Figure 17
EAF voltage-current characteristics based on extracted optimal parameters. (a) Piece-wise linear model. (b) Modified piece-wise linear model. (c) Hyperbolic model. (d) Exponential model. (e) Hyperbolic-exponential model
(a)(b)(c)(d)(e)The harmonics content of voltage and current in EAF models has been compared with measured values in Tables6 and 7. The OF value for different models of the EAF comparing the harmonics contents with the measured values shows that model 2 (modified piece-wise linear) has the atmost accuracy in modeling the EAF behavior.Table 6
Comparison of the harmonics content of arc voltage in EAF models.
Measured/modelsHarmonic orderTHDV%2 (%)3 (%)4 (%)5 (%)7 (%)Measured value0.1520.2980.1210.1950.3550.851Piece-wise linear0.1450.1320.1140.3050.2850.737Modified piece-wise linear0.1460.1510.1150.3300.2320.735Hyperbolic0.1450.3500.1150.4840.4270.905Exponential0.1450.1240.1140.2520.2270.712Hyperbolic-exponential0.1630.4150.0980.3080.2750.989Table 7
Comparison of the harmonics content of arc current in EAF models.
Measured/modelsHarmonic orderTHDI%2 (%)3 (%)4 (%)5 (%)7 (%)Measured value3.2033.4952.1874.2352.2137.781Piece-wise linear3.1941.2530.9023.4572.1855.588Modified piece-wise linear3.1961.2630.8963.7252.2275.781Hyperbolic3.0852.2880.7965.4422.8987.554Exponential3.3001.3900.9653.3952.0485.641Hyperbolic-exponential2.9881.3891.0973.5152.2015.648According to the problem conditions and the intended studies, other models can be used for EAF simulation. For instance, although theOF value of the hyperbolic model is greater than others, it has a much better performance in modeling EAF voltage and current harmonic components. Also, it has been highly considered in harmonic studies. On the other hand, steady-state studies have noticed the exponential model according to the appropriate value of the OF and the model’s simplicity. Nevertheless, the exponential model has weaknesses in modeling the arc voltage and arc current harmonic components.


### 6.3. EMTP-RV and DIgSILENT Simulation Results

Models 1 to 5 are implemented in DIgSILENT and EMTP-RV using DSL and control circuits in this section. It is worth noting that simulated characteristics under different models in both software are accurately similar to the optimized models’ characteristics. The EAF element should be simulated with the same accuracy as the mathematical model in the understudy system using efficient specialized simulation software. The critical point is that the simulated EAF must implement the voltage and current signals generated by the model with the same accuracy and quality in the power grid.In simulation studies, the EAF element is implemented in the power grid indicated in Figure11. The power system parameters and specifications of the actual test system have been presented in Table 8.Table 8
Power system parameters of the understudy actual test system.
Power system elementPower system parameterUpstream networkSsc = 17274 MVAV = 400 kVXth = 10.16 ΩPower transformerSnom = HV/MV/LV: 220/220/110 MVAVnom = HV/MV/LV: 400/63/30.5 kV%UK = 13.81% (HV-MV, 220 MVA base)12.75% (HV-LV, 110 MVA base)4.97% (MV-LV, 110 MVA base)EAF transformerSnom = 125 MVAVnom = 63/0.9 kV%UK = 6%Transmission lineR = 0.012597 ΩX = 0.09 ΩSeries reactorL = 7.2892 mHFlexible cableR = 7.1 μΩL = 4.0459 μHElectrodesR = 33.25 μΩL = 5.69 μHThe EAF element is simulated utilizing various models in DIgSILENT and EMTP-RV. Figure18 shows the voltage waveforms generated by the simulated EAF compared to the measured values and the voltage waveforms of the model. In the EAF modeling, a controlled current source receives the same output current of the mathematical model from the power grid. Thus, the EAF current waveforms in both software are the same as the current waveforms in the respective model. Therefore, the arc voltage waveform of the EAF element, created by passing the arc current through the variable resistance at the furnace bus, is compared with the arc voltage waveform of the respective model. However, a large number of data samples contained in the measured, modeled, and simulated waveforms of the arc voltage complicate this comparison. Therefore, the conformity index and instantaneous error have been used to quantitatively compare the performance of DIgSILENT and EMTP-RV in EAF simulations based on different models.Figure 18
Comparison of EAF voltage based on measurements, mathematical models, and simulation results. (a) Piece-wise linear model. (b) Modified piece-wise linear model. (c) Hyperbolic model. (d) Exponential model. (e) Hyperbolic-exponential model
(a)(b)(c)(d)(e)The conformity index for two parameters of the same material is defined according to the least square method. TheOF of the optimization problem of the EAF model’s characteristics is the sum of the conformance indices between the measured values and the model of arc voltage and arc current in one cycle. For comparing the performance of DIgSILENT and EMTP-RV in EAF simulations based on different models with the optimized estimation of their parameters, the conformity index between the measured values, the mathematical model, and the simulated arc voltage waveform per one cycle is given in Table 9.Table 9
Evaluation of conformity index for comparison of measurements, models, and simulation results.
EAF modelConformity indexMeasured values and EAF model valuesMeasured values and DIgSILENT valuesMeasured values and EMTP-RV valuesEAF model values and DIgSILENT valuesEAF model values and EMTP-RV valuesModel 11.272341.184271.162360.101290.07321Model 20.942631.009270.994480.038000.02685Model 34.776146.832023.714692.482960.99000Model 41.603826.021721.636696.592630.09218Model 53.4118630.84623.1634832.30070.54073As the conformity index considers the conformance of the total cycle in the evaluation, the instantaneous error represents another criterion that observes the conformance of the arc voltage waveform sample by sample. The instantaneous error is helpful in detecting the significant deviations from the reference waveforms. For comparing the performance of DIgSILENT and EMTP-RV, the maximum and minimum instantaneous error between the measured values, the mathematical model, and the simulated arc voltage waveform is given in Table10.Table 10
Evaluation of instantaneous error for comparison of measurements, models, and simulation results.
EAF modelInstantaneous error (V)Measured values and EAF model valuesMeasured values and DIgSILENT valuesMeasured values and EMTP-RV valuesEAF model values and DIgSILENT valuesEAF model values and EMTP-RV valuesModel 1Max = 183.3528Max = 205.6554Max = 201.5437Max = 57.18210Max = 48.40490Min = 0.000183Min = 0.000969Min = 0.000179Min = 0.000014Min = 0.000018Model 2Max = 290.9013Max = 404.1609Max = 390.6713Max = 268.1159Max = 135.6356Min = 0.000394Min = 0.0012Min = 0.000243Min = 0.000113Min = 0.000012Model 3Max = 675.8746Max = 970.3291Max = 673.1311Max = 1222.602Max = 963.5367Min = 0.0015Min = 0.0029Min = 0.0024Min = 0.0041Min = 0.000042Model 4Max = 188.8421Max = 243.8591Max = 205.1512Max = 234.1085Max = 42.2956Min = 0.000028Min = 0.0017Min = 0.000236Min = 0.000913Min = 0.000023Model 5Max = 408.9165Max = 1280.135Max = 596.5685Max = 1290.619Max = 653.6361Min = 0.000145Min = 0.000354Min = 0.000046Min = 0.000261Min = 0.000127Comparing the conformity index between the model values and the simulated values by DIgSILENT, the DIgSILENT has simulated only models 1 and 2 with high accuracy. The significant increase in conformity index for model 5 is due to the instantaneous peaks generated in DIgSILENT, which can be explained using the instantaneous error. According to Table10, the maximum instantaneous error between the model values and the simulated values by DIgSILENT under model 5 is greater than in other models. The conformity index and instantaneous error between the model values and the simulated values by EMTP-RV indicate that this software can implement different EAF models with very high accuracy. In models 1 and 2, the EAF voltage waveform in each software is approximately equal to the voltage waveform of the model. Therefore, these models are efficiently appropriate for simulating EAF in both software programs.Due to the large number of data samples in the understudy waveforms, the instantaneous error could be better when considering the instantaneous peaks generated in both software programs in comparison. The instantaneous error between the model values and the simulated values demonstrates that the instantaneous peaks under simulated models 1, 2, and 4 are less than simulated models 3 and 5.According to the nature of models 3 and 5, in simulation with simulation intervals (0.04μs), these models undergo rapid changes at the moment of passing zero. The rapid changes cause high-frequency oscillations in the DIgSILENT; as in model 5, the oscillation peak increases to 12 kA. In comparison, EMTP-RV is more accurate and can control these rapid changes in the power system.The comparison of simulation execution time shows that the speed of EMTP-RV is 2.5 times faster than DIgSILENT. Also, unlike EMTP-RV, which has been often used for transient studies, DIgSILENT is a suitable platform for implementing power systems according to its powerful user interface, extensive study tools, and comprehensive industrial library. Therefore, to facilitate the power system studies of EAF test systems and steel company electrical grids, it is preferred to implement the EAF load in the DIgSILENT. Since model 2 precisely expresses the EAF behavior and has been implemented with equal accuracy in DIgSILENT and EMTP-RV, it is concluded that model 2 can be suggested as the best alternative for implementing the EAF-based test systems in DIgSILENT.


## 6.1. Test System Descriptions and Measurement Results

The proposed method is tested on an existing EAF plant in Iran. The single-line diagram of the understudy EAF’s supply system is shown in Figure11. As depicted, the studied EAF has been connected to the external grid through two sets of transformers, including the 400/63 kV power transformer and the EAF transformer (63/0.9 kV) [15]. A 63 kV cable system with a nominal current of 1100 A with 1320 m length has been installed for the power transmission section. The EAF electrodes are connected to the supply system via flexible cables. The electrodes and flexible cables are modeled through leq and req impedances.Figure 11
Single-line diagram of the understudy EAF’s supply network.The EAF voltage, current waveforms, and harmonic distortion have been measured/recorded to validate the proposed study. The Fluke 437-II Power quality monitor and energy analyzer have been used in this study. In Figure12, the EAF voltage and current measurements are shown. Also, Figure 13 presents the recorded harmonic distortion of the studied EAF [8].Figure 12
Measured voltage and current of the understudy EAF system [15].Figure 13
Recorded harmonic distortion of the understudy EAF system [15].The measured waveforms are used as the reference waveforms of voltage and current in the EAF optimization problem. In Figure14, the measured values and characteristics of understudy EAF are shown.Figure 14
EAF measured values and characteristics of understudy EAF. (a) Voltage. (b) Current. (c) Voltage-current characteristic.
(a)(b)(c)


## 6.2. Optimization Results

The measurement values are imported into the optimization problem. Also, the GA and PSO setting parameters are selected, as given in Tables2 and 3, for solving the proposed optimization problem.Table 2
The GA setting parameters.
ParametersQuantityNumber of decision variablesModel 14Model 26Model 33Model 42Model 54Population size1000Maximum number of iterations100Crossover rate0.65Mutation rate0.25Table 3
The PSO setting parameters.
ParametersQuantityNumber of decision variablesModel 14Model 26Model 33Model 42Model 54Population size1000Maximum number of iterations100ω initial value1Damp0.996c11.496c21.496The convergence diagrams of GA and PSO in solving the optimization problem for different models are shown in Figure15.Figure 15
Convergence diagrams of GA and PSO in solving the characteristic optimization of different EAF models. (a) Piece-wise linear model. (b) Modified piece-wise linear model. (c) Hyperbolic model. (d) Exponential model. (e) Hyperbolic-exponential model.
(a)(b)(c)(d)(e)As revealed by test results shown in Figure15, the convergence of GA is much higher than PSO. In addition, it can be concluded that as the number of decision variables in the optimization problem of the EAF model characteristics increases, the PSO convergence speed decreases.The statistical analysis of solving the proposed optimization problem by the GA and PSO in optimizing the EAF model characteristics is presented in Figure16 and Table 4. The GA would be suitable for solving discrete problems, while the PSO algorithm has a better performance in solving continuous problems. Because the optimization problem of the EAF model’s characteristics is continuous, it is expected that the PSO results in better performance in finding the optimal points. Statistical analysis shows that although the PSO has a relatively slow convergence rate than the GA, it is more robust in finding the global optimal point. For instance, in the statistical analysis diagram of the hyperbolic model, although the obtained optimal values are equal to each other with one-tenth of accuracy, the GA has not been able to find the global optimal point with one hundred-thousandths of accuracy.Figure 16
The GA and PSO optimum objectives in solving the optimization problem of the EAF model’s characteristics under various runs. (a) Piece-wise linear model. (b) Modified piece-wise linear model. (c) Hyperbolic model. (d) Exponential model. (e) Hyperbolic-exponential model.
(a)(b)(c)(d)(e)Table 4
Statistical analysis of GA and PSO in solving the optimization problem of different EAF models.
ParametersModel 1Model 2Model 3Model 4Model 5GAPSOGAPSOGAPSOGAPSOGAPSOMean4.8114.7793.5063.5217.92557.92554.02454.02455.92785.9278Standard deviation0.0420.0040.0560.0771.7 × 10−52.5 × 10−51 × 10−41.3 × 10−54.6 × 10−55.7 × 10−5Worst4.9104.7933.6523.7397.92567.92564.02484.02455.92805.9280Best4.7754.7753.4473.4477.92547.92544.02447.02445.92785.9278Number of runs30303030303030303030Since all models include a few decision variables, both algorithms reach the global optima in almost all cases. Therefore, the standard deviations in optimumOFs and solutions would not be significant in such small-sized problems. However, the smaller the standard deviation for an algorithm in solving an optimization problem, the algorithm is more reliable in finding the global optima.According to the statistical test results and the convergence diagrams in different cases, it is possible to ensure that the GA and PSO techniques have suitable performance in solving the proposed problem. It can be concluded that the converged solutions are global optimums. Since the presented research focuses on optimizing different EAF models’ characteristics rather than the optimization method, it has been confined to introducing GA and PSO to solve the optimization problem. However, solving the optimization problem of EAF models’ characteristics is not restricted to the presented techniques, and other optimization algorithms can be used to solve it.The optimized parameters for EAF models’ characteristics using the GA and PSO are given in Table5. Moreover, the optimum voltage-current characteristics of different EAF models are shown in Figure 17.Table 5
Optimal values of parameters in different EAF models.
Model no.ModelEAF model parametersOF value1Piece-wise linearvig = 477.306695 Vvex = 34.440595 VR1 = 0.052594 ΩR2 = ‒0.001078 Ω4.775482Modified piece-wise linearvig = 234.344244 Vvex = 251.885441 Vvst = 297.241896 ΩR1 = 0.046316 ΩR2 = ‒0.000261 ΩR3 = 0.002301 Ω3.447183Hyperbolicvat = 106.617882 VD = 693646.7778 ΩC = 293081906.4989 Ω7.925484Exponentialvat = 308.121744 VI0 = 4096.036572 A4.024495Hyperbolic-exponentialvat = 339.770057 VI0 = 3765.370477 AC = 6620839.7101 ΩD = 985684344.6322 Ω5.92786Figure 17
EAF voltage-current characteristics based on extracted optimal parameters. (a) Piece-wise linear model. (b) Modified piece-wise linear model. (c) Hyperbolic model. (d) Exponential model. (e) Hyperbolic-exponential model
(a)(b)(c)(d)(e)The harmonics content of voltage and current in EAF models has been compared with measured values in Tables6 and 7. The OF value for different models of the EAF comparing the harmonics contents with the measured values shows that model 2 (modified piece-wise linear) has the atmost accuracy in modeling the EAF behavior.Table 6
Comparison of the harmonics content of arc voltage in EAF models.
Measured/modelsHarmonic orderTHDV%2 (%)3 (%)4 (%)5 (%)7 (%)Measured value0.1520.2980.1210.1950.3550.851Piece-wise linear0.1450.1320.1140.3050.2850.737Modified piece-wise linear0.1460.1510.1150.3300.2320.735Hyperbolic0.1450.3500.1150.4840.4270.905Exponential0.1450.1240.1140.2520.2270.712Hyperbolic-exponential0.1630.4150.0980.3080.2750.989Table 7
Comparison of the harmonics content of arc current in EAF models.
Measured/modelsHarmonic orderTHDI%2 (%)3 (%)4 (%)5 (%)7 (%)Measured value3.2033.4952.1874.2352.2137.781Piece-wise linear3.1941.2530.9023.4572.1855.588Modified piece-wise linear3.1961.2630.8963.7252.2275.781Hyperbolic3.0852.2880.7965.4422.8987.554Exponential3.3001.3900.9653.3952.0485.641Hyperbolic-exponential2.9881.3891.0973.5152.2015.648According to the problem conditions and the intended studies, other models can be used for EAF simulation. For instance, although theOF value of the hyperbolic model is greater than others, it has a much better performance in modeling EAF voltage and current harmonic components. Also, it has been highly considered in harmonic studies. On the other hand, steady-state studies have noticed the exponential model according to the appropriate value of the OF and the model’s simplicity. Nevertheless, the exponential model has weaknesses in modeling the arc voltage and arc current harmonic components.


## 6.3. EMTP-RV and DIgSILENT Simulation Results

Models 1 to 5 are implemented in DIgSILENT and EMTP-RV using DSL and control circuits in this section. It is worth noting that simulated characteristics under different models in both software are accurately similar to the optimized models’ characteristics. The EAF element should be simulated with the same accuracy as the mathematical model in the understudy system using efficient specialized simulation software. The critical point is that the simulated EAF must implement the voltage and current signals generated by the model with the same accuracy and quality in the power grid.In simulation studies, the EAF element is implemented in the power grid indicated in Figure11. The power system parameters and specifications of the actual test system have been presented in Table 8.Table 8
Power system parameters of the understudy actual test system.
Power system elementPower system parameterUpstream networkSsc = 17274 MVAV = 400 kVXth = 10.16 ΩPower transformerSnom = HV/MV/LV: 220/220/110 MVAVnom = HV/MV/LV: 400/63/30.5 kV%UK = 13.81% (HV-MV, 220 MVA base)12.75% (HV-LV, 110 MVA base)4.97% (MV-LV, 110 MVA base)EAF transformerSnom = 125 MVAVnom = 63/0.9 kV%UK = 6%Transmission lineR = 0.012597 ΩX = 0.09 ΩSeries reactorL = 7.2892 mHFlexible cableR = 7.1 μΩL = 4.0459 μHElectrodesR = 33.25 μΩL = 5.69 μHThe EAF element is simulated utilizing various models in DIgSILENT and EMTP-RV. Figure18 shows the voltage waveforms generated by the simulated EAF compared to the measured values and the voltage waveforms of the model. In the EAF modeling, a controlled current source receives the same output current of the mathematical model from the power grid. Thus, the EAF current waveforms in both software are the same as the current waveforms in the respective model. Therefore, the arc voltage waveform of the EAF element, created by passing the arc current through the variable resistance at the furnace bus, is compared with the arc voltage waveform of the respective model. However, a large number of data samples contained in the measured, modeled, and simulated waveforms of the arc voltage complicate this comparison. Therefore, the conformity index and instantaneous error have been used to quantitatively compare the performance of DIgSILENT and EMTP-RV in EAF simulations based on different models.Figure 18
Comparison of EAF voltage based on measurements, mathematical models, and simulation results. (a) Piece-wise linear model. (b) Modified piece-wise linear model. (c) Hyperbolic model. (d) Exponential model. (e) Hyperbolic-exponential model
(a)(b)(c)(d)(e)The conformity index for two parameters of the same material is defined according to the least square method. TheOF of the optimization problem of the EAF model’s characteristics is the sum of the conformance indices between the measured values and the model of arc voltage and arc current in one cycle. For comparing the performance of DIgSILENT and EMTP-RV in EAF simulations based on different models with the optimized estimation of their parameters, the conformity index between the measured values, the mathematical model, and the simulated arc voltage waveform per one cycle is given in Table 9.Table 9
Evaluation of conformity index for comparison of measurements, models, and simulation results.
EAF modelConformity indexMeasured values and EAF model valuesMeasured values and DIgSILENT valuesMeasured values and EMTP-RV valuesEAF model values and DIgSILENT valuesEAF model values and EMTP-RV valuesModel 11.272341.184271.162360.101290.07321Model 20.942631.009270.994480.038000.02685Model 34.776146.832023.714692.482960.99000Model 41.603826.021721.636696.592630.09218Model 53.4118630.84623.1634832.30070.54073As the conformity index considers the conformance of the total cycle in the evaluation, the instantaneous error represents another criterion that observes the conformance of the arc voltage waveform sample by sample. The instantaneous error is helpful in detecting the significant deviations from the reference waveforms. For comparing the performance of DIgSILENT and EMTP-RV, the maximum and minimum instantaneous error between the measured values, the mathematical model, and the simulated arc voltage waveform is given in Table10.Table 10
Evaluation of instantaneous error for comparison of measurements, models, and simulation results.
EAF modelInstantaneous error (V)Measured values and EAF model valuesMeasured values and DIgSILENT valuesMeasured values and EMTP-RV valuesEAF model values and DIgSILENT valuesEAF model values and EMTP-RV valuesModel 1Max = 183.3528Max = 205.6554Max = 201.5437Max = 57.18210Max = 48.40490Min = 0.000183Min = 0.000969Min = 0.000179Min = 0.000014Min = 0.000018Model 2Max = 290.9013Max = 404.1609Max = 390.6713Max = 268.1159Max = 135.6356Min = 0.000394Min = 0.0012Min = 0.000243Min = 0.000113Min = 0.000012Model 3Max = 675.8746Max = 970.3291Max = 673.1311Max = 1222.602Max = 963.5367Min = 0.0015Min = 0.0029Min = 0.0024Min = 0.0041Min = 0.000042Model 4Max = 188.8421Max = 243.8591Max = 205.1512Max = 234.1085Max = 42.2956Min = 0.000028Min = 0.0017Min = 0.000236Min = 0.000913Min = 0.000023Model 5Max = 408.9165Max = 1280.135Max = 596.5685Max = 1290.619Max = 653.6361Min = 0.000145Min = 0.000354Min = 0.000046Min = 0.000261Min = 0.000127Comparing the conformity index between the model values and the simulated values by DIgSILENT, the DIgSILENT has simulated only models 1 and 2 with high accuracy. The significant increase in conformity index for model 5 is due to the instantaneous peaks generated in DIgSILENT, which can be explained using the instantaneous error. According to Table10, the maximum instantaneous error between the model values and the simulated values by DIgSILENT under model 5 is greater than in other models. The conformity index and instantaneous error between the model values and the simulated values by EMTP-RV indicate that this software can implement different EAF models with very high accuracy. In models 1 and 2, the EAF voltage waveform in each software is approximately equal to the voltage waveform of the model. Therefore, these models are efficiently appropriate for simulating EAF in both software programs.Due to the large number of data samples in the understudy waveforms, the instantaneous error could be better when considering the instantaneous peaks generated in both software programs in comparison. The instantaneous error between the model values and the simulated values demonstrates that the instantaneous peaks under simulated models 1, 2, and 4 are less than simulated models 3 and 5.According to the nature of models 3 and 5, in simulation with simulation intervals (0.04μs), these models undergo rapid changes at the moment of passing zero. The rapid changes cause high-frequency oscillations in the DIgSILENT; as in model 5, the oscillation peak increases to 12 kA. In comparison, EMTP-RV is more accurate and can control these rapid changes in the power system.The comparison of simulation execution time shows that the speed of EMTP-RV is 2.5 times faster than DIgSILENT. Also, unlike EMTP-RV, which has been often used for transient studies, DIgSILENT is a suitable platform for implementing power systems according to its powerful user interface, extensive study tools, and comprehensive industrial library. Therefore, to facilitate the power system studies of EAF test systems and steel company electrical grids, it is preferred to implement the EAF load in the DIgSILENT. Since model 2 precisely expresses the EAF behavior and has been implemented with equal accuracy in DIgSILENT and EMTP-RV, it is concluded that model 2 can be suggested as the best alternative for implementing the EAF-based test systems in DIgSILENT.


## 7. Conclusion

Much attention has been paid to EAF modeling in the literature. Different mathematical time-domain models have been developed to describe the EAF’s time-variant and nonlinear behavior. In this paper, the parameters and characteristics of various time-domain models, such as piece-wise linear, modified piece-wise linear, hyperbolic, exponential, and hybrid models, have been optimized using GA and PSO. The actual measurements of a realistic EAF in Iran have been used to validate the obtained optimum characteristics and models. It has been concluded that the modified piece-wise model is adequately efficient in providing suitable accuracy with the actual EAF characteristics compared to other models.This paper has also addressed the simulation and implementation of different mathematical models in specialized power system software. Investigating the simulation results of DIgSILENT and EMT-RV for various EAF models is another contribution of this research. The test results extracted from DIgSILENT and EMTP-RV indicated that the EAF could be simulated with high accuracy using modified piece-wise linear and piece-wise linear models. However, the simulation of other EAF models in DIgSILENT might not be adequately precise. In contrast, the simulation results in EMTP-RV by different EAF models have been satisfying. Moreover, the simulation execution speed of EMTP-RV is around 2.5 times faster than DIgSILENT. Test results and comparative analysis illustrate this research’s importance in optimizing the EAF models and investigating DIgSILENT and EMTP-RV simulations by various models.



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*Source: 9952315-2022-06-20.xml*