Cooperation of an Electric Arc Device with a Power Supply System Equipped with a Superconducting Element

Abstract

This paper analyses the validity of using superconductors in the power supply system of arc devices. Two cases were analysed: when an additional superconducting element was included in the conventional power supply system and when the total power supply system was made of a superconductor. The analysis was carried out by simulating the cooperation of the arc receiver with its simplified power supply system in Matlab Simulink software. The characteristics of the changes of the arc current, its conductance and voltage as a function of the arc length changes for selected superconductor parameters, i.e., different values of the critical current and different values of the resistance in the resistive state, are given. The time courses of these quantities as well as the courses of resistance changes in the superconductor at randomly varying arc lengths are presented. The analysis showed that by selecting the critical current and resistance in the resistive state of the superconductor, arc parameters such as arc current drawn and arc conductance can be influenced. By making the entire power system from a superconductor, the arc current can be increased by 1.8% for a 1 cm arc and by 1% for a 1 cm arc. The ability of the superconductor to lose its superconducting state and return to that state can be used to limit the value of the current drawn by the arc over certain ranges of arc length. The range of these lengths can be controlled by selecting the value of the critical current of the superconductor. By selecting the resistance of the superconductor in the resistive state, the value of the limited current can be influenced. In the case studied, for a 1 cm arc length, an arc current 45% lower was obtained when the superconductor was in the resistive state.

1. Introduction

At present, about 30% of the world’s steel is produced annually using electric arc furnaces and the problems of their operation and the operation of their power supply systems are well known. Observing the technologies of steel melting, it seems that no significant development of steel arc furnaces and their feeding systems has been recorded for 40 years. In order to obtain the required performance indicators, while reducing production costs [1], attempts are made to improve the organisation of the technological process by designing buildings and the layout of electrical steelworks installations so as to ensure the most favourable transport and logistics layout. Attempts are made to optimise the process [2,3,4] using computer simulations with genetic algorithms and artificial intelligence [5,6].

It appears that further development of the production of steel in arc furnaces would be possible through the implementation of superconductors in their direct power supply system, especially superconducting furnace transformers, but also superconducting high-current paths and superconducting current limiters. For economic reasons, the proportion of electrical energy converted into heat should be as high as possible, since only this part of the energy is used for the metallurgical process and to cover the heat losses accompanying this process. The amount of electrical energy that is converted into heat in the arc furnace depends on the arc parameters and on the electrical parameters of the power supply system [7]. The use of the properties of superconductors, such as zero resistance in the superconducting state and the ability to switch between superconducting and resistive states in the power circuits of arc receivers may influence their operating characteristics. At the current scale of arc furnace use, even a small improvement in process efficiency can bring significant economic benefits on an annual basis [8].

The phenomenon of superconductivity has been known since 1911, but it was not until the discovery of high-temperature superconductivity in 1986 that interest in the use of superconductors in technology increased. The devices in which the use of superconductors is most promising are transformers, cables, and current limiters.

The most useful property of superconductors is their ability to conduct high currents with very low energy losses. The small cross-sections of superconducting wires allow the construction of transformer windings with small radial and axial dimensions, which makes it possible to reduce the size of the transformer by 30–40%. The use of superconductors makes it possible to reduce the total losses in a superconducting transformer, in comparison with a transformer of the same power but with copper windings, by approximately 0.3% in the fully loaded condition [9]. Superconducting transformer windings also have some ability to limit short circuit current [10,11,12,13]. This property of superconductors is used more widely in superconducting current limiters.

Several types of superconducting current limiters have been developed [14,15,16], of which the resistive current limiter is the simplest in design [17]. A resistive limiter is an element made of a superconductor, which is directly and in series connected to the protected circuit. The current limitation in the circuit occurs due to the transition of the superconductor to the resistive state from the superconducting state when the critical current of the superconductor is exceeded [18,19]. This transition is accompanied by a sharp increase in the resistance of the superconducting element. The critical current is the characteristic quantity for a superconductor and depends on the type of superconducting material, its geometry, temperature, and the value of the external magnetic field [20,21]. The resistance of a superconductor in the resistive state also depends on the superconducting material, its geometry, and temperature.

The already-mentioned characteristics of superconductors predispose them for use in the construction of power cables [22,23,24,25]. It is possible to construct superconducting cables for currents greater than 2 kA while maintaining their functional small cross-sections and simultaneously reducing transmission losses relative to conventional cables.

To date, there has been no extensive research on the potential application of superconductors in arc furnace power systems. There is a lack of information on the problems of superconducting devices interfacing with arc-furnace loads and the effects of superconducting devices on the performance of arc-furnace devices. The aim of this study is to recognize the effects of superconductor properties, such as zero resistance and the ability to transition from superconducting to resistive state, on the basic arc parameters, i.e., arc current, arc conductance and arc voltage.

In this paper, the possibility of using superconductors in electric arc furnace (EAF) power systems is analysed. The arc furnace described by Cassie-Mayer mathematical model was the subject of theoretical analysis. The furnace parameters necessary for the calculations were taken from the literature. The receiver model was supplied by a simplified equivalent diagram of the arc furnace power supply system with an additional superconducting element included. The superconducting element was simulated as an ideal superconductor with simplified characteristics of the transition from superconducting to resistive state. Two cases were analysed: when an additional superconducting element was included in the conventional power supply circuit and when the whole power supply circuit was made of superconductor. The simulation was carried out in Matlab Simulink.

The study shows that the use of superconductors, in place of conventional conductive materials, in the arc furnace power circuit can provide some benefits. Nearly zero resistance of superconductors in the superconducting state and much lower losses in superconducting wires, compared to conventional copper wires, make it possible to increase the value of the arc power while keeping the power drawn from the mains unchanged. The ability of superconductors to transition from the superconducting state to the superconducting state can be used to limit the value of currents in an arcing circuit. It appears that by selecting the critical value of the superconductor current, it is possible to limit the short circuit currents. It also appears that with an efficient superconductor cooling system, it becomes possible to shape the current-voltage characteristics of the arc by selecting the critical value of the superconductor current. Thus, it seems that the replacement of conventional materials with superconducting materials in the arc furnace power circuit should improve the performance of such a furnace and improve the economics of the process.

2. Arc Receiver Power Supply System

The mathematical model of the EAF must take into account the interaction between the electric arcs and their power supply system. The simplified arc receiver power supply system adopted for the calculations is shown in Figure 1. The arc is supplied in a single-phase circuit from a sinusoidal voltage source e through a resistance R_s and an impedance L_s representing the current path parameters and a resistance R_hts representing the superconductor.

For simulation, it is necessary to determine the parameters of this circuit. In [26], it is stated that the resistance of the high-current circuit is equal to 2.1 mΩ, while the reactance is equal to 4 mΩ. More detailed data from measurements of the power circuit are contained in the work [27] where it is reported that the resistances and inductances of the flexible conductors, bus conductors and coal electrodes of the furnace are 7.1 μΩ, 4.0959 μH, 3.55 μΩ, 3.32 μH, 29.7 μΩ, 2.37 μH, respectively. The paper [28] states that the resistance of the connection between the transformer and the furnace is 0.669 mΩ and the inductance of this connection is 2.5 μH. For the calculations, the resistance R_s of the circuit from Figure 1 was assumed equal to 0.38 mΩ and the inductance L_s equal to 8.589 μH following the values given in the work [29]. The maximum value of the supply voltage e equal to 500 V was assumed. The resistance of the superconductor R_hts in the superconducting state was assumed to be zero while in the resistive operating state it was assumed to be equal to 1 mΩ or 100 mΩ. More information about the superconductor model is given in Section 2.2. The circuit parameters from Figure 1 are given in

Table 1. Electrical circuit parameters adopted for calculations.

Parameter	Value Used for Calculation
Rs	0.38 mΩ
Ls	8.589 μH
Em	500 V, 50 Hz
Rhts	0 Ω in a superconducting state 1 mΩ or 100 mΩ in resistive condition

.

The circuit in Figure 1 is described by the equation:

$$e=\left(R_s+R_{hts}\right)i+L_s\frac{di}{dt}+u$$

(1)

where:

$$e=E_{m}sin\left(\omega t+\phi \right)$$

(2)

The voltage u from Equation (1) is given by the equation:

$$u=\frac{i}{g}$$

(3)

2.1. Arc Receiver Model

The conductance g from Equation (3) is described by the Cassie-Mayer mathematical model [30]:

$$g=g_{min}+\left[1-exp\left(-\frac{i^2}{I_0^2}\right)\right]\frac{ui}{E_0^2}+exp\left(-\frac{i^2}{I_0^2}\right)\frac{i^2}{P_0}-\Theta \frac{dg}{dt}$$

(4)

where:

$$\Theta ={\Theta }_0+{\Theta }_{1}exp\left(-\alpha ·\left|i\right|\right)$$

(5)

$$E_0=A+Bl$$

(6)

Typical values of the parameters P₀, I₀, Θ₀, Θ₁, α, A, B, g_min appearing in Equations (4)–(6) are given in

Table 2. Arc parameters adopted for calculations.

Parameter Description	Parameter	Value Used for Calculation
Constant defining the momentarily constant steady state arc voltage E0	A	40 V
Constant defining the momentarily constant steady state arc voltage E0	B	10 V/cm
Arc length	l	0–20 cm
Transition current	I0	10 A
Minimum arc conductance	gmin	0.008 S
Momentary power loss	P0	110 W
Time Constant	Θ0	110 μs
Time Constant	Θ1	100 μs
Constant	α	0.0005 A

[28,29].

An important parameter is the arc length and the frequency of change of this length, i.e., how long the arc burns maintaining a certain length. In the work [28] it was reported that the arc length during charge melting changes within the range of 0 ÷ 40 cm, while during refining within the range of 25 ÷ 30 cm. In [26] it is stated that during charge melting the arc length may vary within 0 ÷ 20 cm, while during refining it may vary from 2 ÷ 8 cm. For the numerical calculations carried out, the arc length was assumed to vary in the range of 0 ÷ 20 cm (Table 2).

2.2. Superconductor Model

A simplified model of a superconductor was used in the study. It was assumed that a superconductor is a resistive binary [31,32], whose resistance during the transition from the superconducting state to the resistive state and vice versa changes in steps [33] within the limits from zero to some fixed value of R_r (Figure 2a). In reality, this transition is not stepwise but occurs smoothly but very quickly [34,35,36]. It is assumed that the loss of superconducting state takes place when the current in the circuit exceeds the critical value for a superconductor I_c, while the return to the superconducting state takes place when the value of the current decreases below the value of I_c. Figure 2b shows the variation of the superconductor resistance R_hts during the sinusoidal current i, whose maximum value exceeds the superconductor critical current I_c. The influence of temperature [37] and magnetic field is completely ignored in the superconductor model adopted.

2.3. Calculation Model

The operation of the arc receiver power supply system with a superconducting element included in the current path was modelled in Matlab using the Simulink graphical environment. The block diagram of the system is shown in Figure 3. The superconducting element (frame “R_hts”) was realised by means of the block “Switch” realising resistance switching for a fixed value of the signal given to its input, whose value corresponds to the value of the critical current of the superconductor. The blocks in the “R_hts” frame implement the superconductor model given in Figure 2. The switching takes place between the block “Constant RHTS0”, representing the resistance of the superconductor in the superconducting state, and the block “Constant RHTS1”, representing the resistance of the superconductor in the resistive state. Box “i” defines Equation (1) and box “g” defines Equation (3).

3. Numerical Analysis

The operation of the arc receiver power supply system was analysed for three cases: (1) when the arc is supplied from a circuit having a constant resistance greater than zero, (2) when the arc is supplied from a circuit with zero resistance, and (3) when the arc is supplied from a circuit with variable resistance. The first case simulates the operation of a normal power circuit used to power metallurgical arc furnaces with the parameters given in Table 1 (R_s = 0.38 mΩ). The second case corresponds to the theoretical situation in which all elements of the power supply system are made of an ideal superconductor which is permanently in a state of superconductivity (R_s = 0 i R_hts = 0). In the third case, a power system with a superconductor that periodically leaves and then returns to the superconducting state (R_hts = 0 lub R_hts > 0) is analysed. In all cases, the parameters of the sinusoidal power source and circuit inductance given in Table 1 and the parameters of the arc receiver given in Table 2 were assumed.

3.1. Power System with Constant Resistance

The arc receiver was supplied through a resistance of R_s = 0.38 mΩ (Table 1) while the superconductor resistance (R_hts = 0) was ignored. This approach makes it possible to estimate the performance of a typical arc receiver power supply system. The simulation was carried out for two arc lengths of 1 cm (Figure 4) and 20 cm (Figure 5).

The simulation shows that the maximum value of the current for the arc with the length l = 1 cm is I_m = 177.9 kA and is higher by 64.6 kA than the maximum value of the current for the arc with the length of 20 cm. The voltage of an arc with a length of 1 cm in the moment of time when the current reaches its maximum value is U = 50 V and is 190 V lower than for an arc with a length of 20 cm. The conductance of the 1 cm arc reaches its maximum value G = 3.5 kS and is 3.1 kS higher than for the 20 cm arc.

3.2. Power Supply System with Zero Resistance

For the calculations, it was assumed that the resistance of the arc receiver power supply system was zero R_s = 0 Ω. This approach made it possible to determine the operating conditions of the system if it were made entirely of superconducting elements that were constantly in a superconducting state (R_hts = 0). The simulation was performed for two arc lengths of l = 1 cm (Figure 6) and 20 cm (Figure 7).

The simulation shows that the maximum current for the l = 1 cm arc is I_m = 181.1 kA and is 53.9 kA higher than the maximum current for the 20 cm arc. The voltage of the 1 cm arc at the instant of time when the current reaches its maximum value is U = 50 V and is 190 V lower than for the 20 cm arc. The conductance of the 1 cm arc reaches a maximum value of G = 3.6 kS and is 3.1 kS higher than for the 20 cm arc.

Comparing the results obtained for a circuit with a resistance of R_s = 0.38 mΩ with the results obtained for a circuit with a resistance of R_s = 0 Ω, it is apparent that making the power circuit from a superconductor which is constantly in a superconducting state increases the arc current by 3.2 kA for an arc of 1 cm length, which is a 1.8% increase in value. There is also an increase in the conductance of such an arc by 74 S, which is a 2% increase. For a 20 cm arc, the use of a superconductor in the power supply system increases the arc current by 13.9 kA, a 1% increase, and the conductance of the arc by 58 S, a 11% increase. The arc voltage remains unchanged. The results obtained from the simulations are summarised in

Table 3. Simulation results for different circuit resistances.

Rs, [mΩ]	l, [cm]	Im, [A]	G, [S]	U, [V]
0.38	1	177,889	3557	50
	20	113,254	471	240
0	1	181,144	3631	50
	20	127,187	529	240

.

3.3. Power System with a Superconductor Losing its Superconducting State

A superconductor plugged into the direct power circuit of an arc furnace can, as it were, act as a resistance switch for that circuit. Such a circuit may be in a low resistance state when the superconductor is in a superconducting state, or a high resistance state when the superconductor is in a resistive state. The parameters that determine the switching are the value of the current taken by the arc and the current critical value of the superconductor. By selecting the ratio of the critical current of the superconductor to the current taken by the arc, it is possible to decide when the superconducting state is lost.

The circuit in Figure 1 was analysed assuming that the resistance R_s is equal to 0.38 mΩ. It was also assumed that the superconducting resistance R_hts varies stepwise as shown in Figure 2. The lower value of this resistance, i.e., in the superconducting state, was assumed to be R_hts = 0 Ω, while for the upper value, i.e., in the resistive state, R_hts = 1 mΩ was assumed for the first case under consideration or R_hts = 100 mΩ for the second case under consideration. The adoption of the two values of the superconductor in the resistive state for the calculations made it possible to determine the degree of influence of the superconductor resistance value on the operation of the circuit with an electric arc. The analysis also took into account different values of the critical current of the superconductor. The effects on the arc and electrical circuit of a superconductor with several values of the critical current adopted from the range from I_c = 60% I_m to I_c > 100% I_m, where I_m = 177,889 A (Table 3) were analysed.

For a critical current greater than 100% I_m, in no case does the superconductor come out of superconductivity and the circuit operation is identical to that discussed in Chapter 3A. For a superconductor critical current lower than 100% I_m, the value of the critical current affects the values of arc currents, voltages and conductances measured for different arc lengths. The characteristics of changes in the maximum values of these parameters as a function of changes in the arc length and at different values of the critical current are presented in Figure 8, Figure 9 and Figure 10.

Figure 8 shows that when the maximum value of the arc current exceeds the critical value of the superconducting current, 3 areas can be distinguished on the characteristic curve I = f(l) at a change of arc length, as shown in Figure 11. In the first area (a), the arc current is damped [38,39] while maintaining its shape close to sinusoidal (Figure 12-a). In area two (b), the current is damped by flattening (cutting off) its peak (Figure 12-b). This flattening is clearly visible for large resistances of the superconductor in the resistive state. In the third region (c), current damping does not occur, and the current waveform is sinusoidal because the superconductor does not exit the superconducting state (Figure 12-c).

The effect of the critical current value is less noticeable in the case of the arc conductance (Figure 9). The greatest differences in conductance values, for different values of the superconductor critical current, are observed for small arc lengths.

The arc voltage does not depend on the value of the critical current of the superconductor and reaches the same value for any of its values at a given arc length (Figure 10).

In Figure 13, in addition to the waveforms of current (i), arc voltage (u) and its conductance (g), the changes in circuit resistance (r) are shown. The time intervals during which the superconductor is in the resistive state are visible as step changes in the circuit resistance from a value of 0.38 mΩ to a value of 1.38 mΩ. Figure 13a shows the case when the operating point of the circuit is selected on the part (a) of the characteristic curve shown in Figure 11. The case when the critical current of the superconductor I_c is 80% of the value of the maximum current I_m (I_c = 80% I_m) that flows in the circuit with the grid resistance equal to R_s = 0.38 mΩ and the arc length l = 1 cm (I_m = 177,889 A, Table 3) is analysed. The characteristics in Figure 13b were obtained by moving the operating point of the circuit to the line (b) of the characteristics in Figure 11. This was done by changing the arc length to l = 13 cm.

The superconductor is in the resistive state for the longest time (Figure 13a) when the circuit operates on the curve (a) of the characteristic from Figure 11. On this curve, the current is limited while maintaining its shape close to sinusoidal. When the working point of the circuit passes to the linear part (b) of the characteristic from Figure 11, the time in which the superconductor is in the resistive state is shortened. In the intervals in which the superconductor is in the resistive state, the value of the current is reduced by cutting off (flattening) the peaks of its waveform to the value equal to the value of the critical current of the superconductor. A similar cut-off of the peaks is also observed in the arc conductance waveforms. Figure 13c shows the case when the superconductor does not leave the superconducting state and the resistance of the circuit is constant in time and equal to 0.38 mΩ, which corresponds to the operation of the circuit on the curve (c) of the characteristic from Figure 11. The length of the arc is then l = 17 cm.

With increasing resistance of the superconductor in the resistive state, the current damping becomes stronger and the range of the curve (a) in Figure 11 becomes narrower until it disappears completely for large values of superconducting resistance (Figure 14).

The real current waveforms obtained from numerical simulations are shown in Figure 15. Increasing the resistance of a superconductor in the resistive state to R_hts = 100 mΩ causes significantly larger changes in the arc conductance (Figure 16) than in the case of R_hts = 1 mΩ (Figure 9). The characteristics of the arc voltage variation as a function of changes in its length are the same for both considered superconductor resistances (Figure 10 and Figure 17).

Figure 18 shows the current, arc conductance, arc voltage and circuit resistance waveforms for three selected operating points on the characteristic of Figure 15. At a superconductor resistance in the resistive state of 100 mΩ there is a clear flattening (cut-off) of the current and arc conductance waveforms at times when the superconductor loses its superconducting state (Figure 18a,b). Figure 18c shows the case when the arc is long enough that the superconductor does not exit the superconducting state and the circuit resistance is constant in time and equal to 0.38 mΩ.

Relating these simulation results to the results presented in Figure 13, it is found that there are clear differences in the current waveforms in the arc length range from 1 cm to 12 cm. In this length range, at time instants when the superconductor exits the superconducting state and for R_hts = 100 mΩ, the maximum current value is constant and maintained at a value equal to the critical current value of the superconductor. At R_hts = 1 mΩ in the arc length range of 1 cm to 12 cm, during the time instants when the superconductor exits the superconducting state, the current value was variable and greater than the critical current of the superconductor. After the arc length exceeded 12 cm, the differences in current variation disappeared.

The analysis shows that by selecting the value of resistance of the superconductor in the resistive state and by selecting the value of the critical current, it is possible to influence the operating characteristics of the arc receiver. The value of the arc current and the value of the arc conductance can be influenced by shaping the course of variation of these quantities as a function of the arc length changes.

3.4. Variable-Length Electric Arc

The analysis of the cooperation of the power system equipped with a superconducting element with an arc of changing length was carried out in a modified system from Figure 3. The modification consisted in replacing the “Constant Larc” block with the system of blocks shown in Figure 19. This system generates an arc length varying randomly in time in the range from 0 to 20 cm as shown in Figure 20-l and Figure 21-l.

The analysis was performed for I_c = 80%I_m, a circuit resistance of R_s = 0.38 mΩ and a resistance of the superconductor in the resistive state equal to R_hts = 100 mΩ.

As can be seen from Figure 20, the current in the circuit maintains a constant value of 142.4 kA in the range of arc lengths where the superconductor exits the superconducting state, i.e., for arc lengths from 0 to 15 cm. When the arc exceeds 15 cm in length, the maximum current values fall below the critical value for the superconductor. The minimum peak current value in this range is 119.2 kA.

Making the total circuit from a superconductor, i.e., R_s = 0 Ω, allows the maximum current value to be maintained at 142.4 kA in the arc length range from 1 cm to 16 cm (Figure 21). Thus, there is an extension of the arc length range in which the superconductor exits the superconducting state. In the arc length range from 16 cm to 20 cm, the superconductor is constantly in the superconducting state and the lowest peak current value is 136.1 kA, which is 16.9 kA higher than when R_s = 0.38 mΩ.

4. Conclusions

The numerical analysis carried out was aimed at analysing the cooperation of an arc receiver with a power supply circuit in which a superconducting element was connected in series in the current path.

The installation of a superconductor as an additional element in a typical arc power supply system, assuming a critical current value for the superconductor lower than the maximum value of the discharge current, has a significant effect on the operating characteristics of the system. The magnitude of this influence depends on the value of the ratio of the critical value of the superconductor current to the maximum value of the current with which the arc is supplied in a normal power system made of resistive materials. The smaller this ratio is than unity, the stronger the effect.

The value of the resistance of the superconductor in the resistive state of its operation is also significant. The strength of the interaction increases with the increase in the ratio of the value of the resistance of the superconductor in the resistive state to the resistance of the system. In the case analysed, for the resistance of the supply system equal to 0.38 mΩ, the resistance of the superconductor in the resistive state 100 mΩ, and the critical value of the superconductor current equal to 80% of the maximum value of the current attainable in the circuit without a superconductor, the maximum value of the current in the supply circuit of the arc receiver was limited and kept constant at 142.4 kA in the arc length range from 0 to 15 cm.

The elimination of the resistance of the power circuit by making the entire circuit from a superconductor widens the arc length range for which the maximum current value is kept at 142.4 kA. The maximum current for the arc length at which the superconductor does not leave the superconducting state is also increased. Making the complete power system from a superconductor while constantly maintaining it in a superconducting state leads to a 2% increase in the value of the arc current with a length of 1 cm and more than a 12% increase in the current with an arc length of 20 cm. Analogous increases occur when comparing the conductance of the arc.

The analysis shows that by selecting the critical value of the superconductor current and its resistance in the resistive operating state, the operating characteristics of the arc receiver can be influenced. In the range of small arc lengths, the superconductor acts as a current limiter by inserting additional resistance into the circuit. In the large arc length range, making the power circuit from a superconductor allows higher current values to be obtained by eliminating the resistance of the power circuit.