An Arc Furnace as a Source of Voltage Disturbances—A Statistical Evaluation of Propagation in the Supply Network

: This article presents the results of measuring P st indicators at three points of a power system supplying a large source of voltage disturbances—an arc furnace. Measurements were made at three voltage levels: 30, 110, and 400 kV. Recorded values of P st at each point were subjected to statistical analysis, the probability distributions were adjusted to their histograms, and the nature of changes in the basic parameters of these distributions with the distance from the source of disturbances was indicated. The adjustments of the distributions were made using a modiﬁed ﬁreﬂy algorithm.


Introduction
Interest in the issue of power quality specified in the EN 50160 standard [1] is growing along with the development of the energy market. The rapid increase in the number of loads of different power and a different nature of work has highlighted their significant impact on the supply network, including the voltage itself. Voltage fluctuations can have economic and ergonomic effects. Therefore, there is a need to determine the impact of the load for voltage changes and the need to search for and indicate the load causing such changes [2][3][4][5][6][7][8][9]; thus, it is necessary to assess how voltage disturbances propagate through the energy system [10][11][12][13]. The most obvious point on which to focus is the registration of the voltage fluctuation index P st and correlating it with the parameters of the energy system. One should also remember what this measure is and how it is determined; it estimates the level of frustration of a statistical person with statistically healthy eyesight for changes in the brightness of the glow source caused by voltage changes [14]. There is no doubt, however, that the basic step to improve the power quality parameters is to find the source of the disturbances [2][3][4][5][6][7][8][9] and the secondary issue is to use some numerical measures that define voltage fluctuations. The basic indicator of voltage disturbances is P st , which has its drawbacks and so, often attempts are made to replace it or its estimation with other quantities that characterize the power system [15][16][17][18][19][20][21][22][23][24][25][26].
This article presents a statistical analysis of the propagation of voltage disturbances in a power system, the source of which is a large arc furnace. In the literature, the electric arc furnace is often cited as an example of the load causing voltage fluctuations [3,16,20,21,27] and other phenomena [28][29][30]. In this example, the arc furnace under examination is not the only source of disturbance, as there are more ladle and arc furnaces in the immediate vicinity, but this one furnace is clearly larger than the others and its influence is, therefore, clearly visible. In the arc furnace supply system, P st values were measured at three points and at different voltage levels. Technical documentation of the system was also available which made it possible to determine the equivalent impedance of the power system components, i.e., power lines, chokes, transformers, etc. Information on the short-circuit power of 400 kV was also available. This parameter is sometimes difficult to estimate due to the unavailability of information and configuration changes occurring in the energy system itself, made during the use of the system [31].
The continuous development of science and technology creates new problems, the analytical solution for which becomes increasingly difficult or even impossible to establish due to mathematical knowledge or the time needed to determine the solution. Then, modern numerical optimization methods included in the group of artificial intelligence methods are involved, e.g., genetic algorithms, bees, fireflies, ants, particle swarms, cuckoos, gray wolves, and many others. Examples of their use in technical issues are already numerous [32][33][34][35][36][37][38][39] and for this article, a modified firefly algorithm was used [36,38,39].

Analyzed Power Supply System
The arc furnace is one of the most restless loads that deteriorates the power quality [3,16,20,21,27]. It is a source of reactive power, higher harmonics, and significant voltage fluctuation in the supply network, as measured by the P st indicator.
In one Polish steel mill, the arc furnace was modernized, increasing its smelting capacity, i.e., its power was increased. This modernization was followed by the necessity to adapt the power supply system by adding a C-type second harmonic filter to the alreadyexisting third harmonic filters [28]. This resulted in a compensation of increased reactive power and reduction of harmonics below the limits indicated in EN 50160. Unfortunately, it was not possible to reduce the increased influence of the arc furnace on voltage changes in the distribution network. Therefore, it was decided to carry out further modernization of the power supply system by replacing some elements and connecting an SVC system (Static Var Compensator) [31]. In order to be able to assess the effectiveness of these changes, the quality parameters of the electricity were measured and the impact of the arc furnace on the supply network and the propagation of voltage disturbances was assessed by analyzing the P st indicator before making the necessary changes. The arc furnace supply system is shown in Figure 1.  Measurements were made at the same time at three points of the power system ( Figure 1) using Fluke1760 recorders: • 400 kV-C point, above transformer TR3; • 110 kV-B point, above parallel connected transformers TR4 and TR6; • 30 kV-A point, above choke DL.
The main factor influencing the propagation of voltage disturbances in the supply system is the short-circuit power at individual points in the system, i.e., the impedance of the elements in the supply system. Changes in the value of the supply voltage are caused by changes in voltage drops on the elements of the energy system above the measuring point, i.e., changes in the current flowing through these elements. However, the same changes in current cause smaller changes in the supply voltage when the impedance of the supply circuit is lower, that is, when the short-circuit power at this point is higher [40,41].
The impedances of the system elements and the short-circuit powers were determined based on the rated data and other available information (Appendix A). Table 1 lists the impedances and short-circuit powers behind a given element and the percentage influence of the element on the short-circuit impedance of the entire supply track. The values presented show that about 77% of the furnace supply path impedance is in the choke, furnace transformer, and high-current track. Long-term measurements of basic power quality indicators were made in the described power supply system at three selected points of the system (A, B, and C) ( Figure 1). Figure 2 shows a fragment of the current (1 s RMS values at point A). There are two complete smelting processes with three stages each. It is expected that when the current has a high variability there will be high values of the Pst indicator, whereas when the current has low variability, the values of Pst will be smaller, and when the furnace does not melt, possible values of Pst result from the energy background.  There are generally three states in the current: (a) No current-the furnace does not melt; (b) Highly variable current around 2500 A-the initial melting phase with frequent arc flashover between the electrodes and unmelted elements of the charge; (c) Current with low variability around 2500 A-metal in a liquid state.
It is expected that when the current has a high variability there will be high values of the P st indicator, whereas when the current has low variability, the values of P st will be smaller, and when the furnace does not melt, possible values of P st result from the energy background.

Statistical Analysis of Recorded Values of P st at Point A
In the arc furnace supply system (Figure 1  It is expected that when the current has a high variability there will be high v the Pst indicator, whereas when the current has low variability, the values of Pst smaller, and when the furnace does not melt, possible values of Pst result from the background.

Statistical Analysis of Recorded Values of Pst at Point A
In the arc furnace supply system (Figure 1   The first maximum on the histogram corresponds to the furnace being off. The second and third maxima occur when the furnace melts the charge. The third maximum corresponds to the initial stage of the furnace operation, when the electric arc jumps between the unmelted parts. On the other hand, the second maximum occurs at the liquid phase of the metal. For this article, only the operating range of the furnace is analyzed, i.e., the area of the second and third maxima on the histogram. Figure 4 shows a histogram of the value P st dedicated to the operation of the furnace. The criterion of sample separation was to determine the value of P st (10 min) in the range wherein the maximum value of the current (20 ms) did not exceed 200 A. We consider these P st samples when the furnace has been in operation for a while. The graph also shows the distribution of the histogram. responds to the initial stage of the furnace operation, when the electric arc jumps b the unmelted parts. On the other hand, the second maximum occurs at the liqui of the metal. For this article, only the operating range of the furnace is analyzed, area of the second and third maxima on the histogram. Figure 4 shows a histogra value Pst dedicated to the operation of the furnace. The criterion of sample separation was to determine the value of Pst (10 min range wherein the maximum value of the current (20 ms) did not exceed 200 A. sider these Pst samples when the furnace has been in operation for a while. The gra shows the distribution of the histogram.
The histogram in Figure 4 is a combination of two statistical distributions: th alized distribution of the extreme value (1) and the normal distribution (2). The c tion of these distributions is a weighted sum of both processes (3).
x-random variable k-shape parameter b-location parameter (depending on the average value) c-scale parameter (depending on the standard deviation) where: x-random variable μ-average value s-standard deviation where a is the standard distribution weighting factor. Based on the registered values of Pst, the function distribution (3) was adju minimizing the maximum difference on the graphs of the distribution of the re The histogram in Figure 4 is a combination of two statistical distributions: the generalized distribution of the extreme value (1) and the normal distribution (2). The composition of these distributions is a weighted sum of both processes (3). where: x-random variable k-shape parameter b-location parameter (depending on the average value) c-scale parameter (depending on the standard deviation) where: x-random variable µ-average value s-standard deviation where a is the standard distribution weighting factor. Based on the registered values of P st , the function distribution (3) was adjusted by minimizing the maximum difference on the graphs of the distribution of the registered values of P st and the desired distribution by using the Kolmogorov-Smirnov test ( Figure 5). Weight values were sought: a weighted sum, distribution function (3), and parameters of both distributions (1) and (2), i.e., k, b, and c, and µ and s. The firefly algorithm searched for these values of the decision variables for which the Kolmogorov-Smirnov criterion would be the smallest. It is the task of minimizing the maximum value.
values of Pst and the desired distribution by using the Kolmogorov-Smirnov test ( Fig  5). Weight values were sought: a weighted sum, distribution function (3), and parame of both distributions (1) and (2), i.e., k, b, and c, and μ and s. The firefly algorithm searc for these values of the decision variables for which the Kolmogorov-Smirnov crite would be the smallest. It is the task of minimizing the maximum value. Optimization was performed with a modified firefly algorithm [36,38,39]. The rameters used in the firefly algorithm follow: The same parameters of the firefly algorithm were used in all optimization case this article, only the allowed ranges of variability of the searched parameters w changed. Figure 6 shows the optimization result. The indicator Pst histogram is shown with designated probability distribution (3) (Figure 6a). Figure 6b shows the difference in tribution of the measured Pst with the matched function (3). Optimization was performed with a modified firefly algorithm [36,38,39]. The parameters used in the firefly algorithm follow: The same parameters of the firefly algorithm were used in all optimization cases in this article, only the allowed ranges of variability of the searched parameters were changed. Figure 6 shows the optimization result. The indicator P st histogram is shown with the designated probability distribution (3) (Figure 6a). Figure 6b shows the difference in distribution of the measured P st with the matched function (3).      The maximum difference between the distribution of the measured P st and the fitted distribution is 0.0082; however, it should be remembered that voltage disturbances measured by the P st indicator are caused by the operation of the furnace as well as other loads propagating through the system and so, the measured values P st at the measuring point are the result of all these causes. The tested arc furnace is both characteristic of and visible in the energy system due to the size of the generated disturbances.
Factor a informs of the division of the measured values P st between the distributions (1) and (2); 70.72% of the measurements were included in the normal distribution. The positions of the maxima of the distributions (1) and (2) are described by the values b = 3.8444 and µ = 18.4773, respectively.

Statistical Analysis of Recorded Values P st at Point B
In the arc furnace supply system (Figure 1), measurements were also taken at point B-level 110 kV, synchronously with the measurements taken at point A-30 kV. The short-circuit power calculated based on the rated data of the transformer and the cable line at this point is 1191 MVA.
It should also be remembered that the arc furnace tested is not the only source of voltage disturbances in this system. Other ladle and arc furnaces are supplied from the 110 kV level, but are of lower power, together with other accompanying devices. Figure 7a shows a histogram of the value P st at point B. Similarly to point A, the probability distribution was fitted using a modified firefly algorithm. The optimization criterion was to minimize the Kolmogorov-Smirnov test, i.e., to minimize the maximum difference between the distribution function of the measured P st and the matched distribution. The distribution difference diagram is shown in Figure 7b. The maximum difference in distribution of the measured Pst and the fitted distribution is 0.0046.
The histograms in Figure 6a and Figure 7a (for points A and B) are similar in shape, differing mainly in the location of the maxima. The histogram in Figure 7a is narrower (compressed in the Y direction) compared to the histogram in Figure 6a. Table 3 summarizes the parameters characterizing the density distributions for points A and B. The second maximum (value μ) shifted from 18.4773 to 10.9085, which is 59% of the initial value. The first maximum (value b) shifted from 3.8444 to 2.3243, which is 60.46% of the initial value. The values decrease in a similar way for c and s. The maximum difference in distribution of the measured P st and the fitted distribution is 0.0046.
The histograms in Figures 6a and 7a (for points A and B) are similar in shape, differing mainly in the location of the maxima. The histogram in Figure 7a is narrower (compressed in the Y direction) compared to the histogram in Figure 6a.  10.9085, which is 59% of the initial value. The first maximum (value b) shifted from 3.8444 to 2.3243, which is 60.46% of the initial value. The values decrease in a similar way for c and s. It is also worth noting that some values in the "B/A" row of Table 3 correspond to the ratio of the calculated short-circuit power (Table 1) at the measurement points (5).
This value results from the impedance of the supply track. Indicator d AB (voltage disturbance propagation indicator) is the quotient of the short-circuit impedance at point B to the short-circuit impedance at point A (6).
As already mentioned, the value of d AB can be seen in Table 3 in row "B/A" for value b, c, µ, and s. Additional information about the nature of the histogram changes can be formulated based on the dependence graph of the recorded values of P st at the level of 110 kV from the value of P st registered at the 30 kV level (Figure 8). Approximations of measurements of P st can be made with a straight line with an inclination of 0.5978. The same value was obtained by calculating the mean value of the quotient P stB /P stA . The data presented shows that it is possible to estimate the short-circuit impedance at the measuring point (and thus the short-circuit power) based on the value Pst at both points and the impedance between the points ZAB (7).
Therefore, the short-circuit power at the test point can be estimated from (8).
On the basis of Formulas (7) and (8), the impedance between points A and B (ZAB = The data presented shows that it is possible to estimate the short-circuit impedance at the measuring point (and thus the short-circuit power) based on the value P st at both points and the impedance between the points Z AB (7).
Energies 2021, 14, 1076 9 of 14 Therefore, the short-circuit power at the test point can be estimated from (8).
On the basis of Formulas (7) and (8), the impedance between points A and B (Z AB = 0.499 Ω) and the determined propagation coefficient of voltage disturbances d AB = 0.5904 (value µ from Table 3), short-circuit power was calculated in various measuring points of the power supply system (S A = 739 MVA, S B = 1251 MVA) and short circuit impedance at these points (Z A = 1.218 Ω, Z B = 0.719 Ω). The calculated values are close to the values in Table 1.

Statistical Analysis of Recorded Values P st at Point C
In the arc furnace supply system (Figure 1), measurements were taken at point Clevel 400 kV, synchronously with the measurements at points A-30 kV and B-110 kV. The short-circuit power of point C is 4750 MVA (according to the technical documentation). Figure 9a shows the relationship between the recorded values of P st at point C and the value of P st at point B, and in Figure 9b, the relationship between the quotient P stC /P stB and the value of P st from point B. Using the modified firefly algorithm, the probability distributions for both furnace operation cases were determined. Figure 10 shows the histograms and distribution differences for these cases.
Searching for the matching of the distributions, the values of the Kolmogorov-Smirnov test were obtained for two furnace operation cases: 0.0071 and 0.0047, respectively. Table 4 Figure 9 shows that the short-circuit impedance at point C, and thus the short-circuit power, changed during the measurement process. Two groups of points distributed along two approximating lines are clearly visible. At points A and B, this difference was not visible due to the low value of the short-circuit impedance of the network in relation to the short-circuit impedance at points A and B. It can be presumed that during the operation of the furnace there were configuration changes in the supply network. Therefore, it is necessary to separate the two cases of furnace operation in order to be able to identify both short-circuit impedances. The mean value of the points of the lower group (Figure 9b) and the inclination of the lower approximating line (Figure 9a) is 0.1885; for the higher group, it is 0.2575.
Using the modified firefly algorithm, the probability distributions for both furnace operation cases were determined. Figure 10 shows the histograms and distribution differences for these cases.
Searching for the matching of the distributions, the values of the Kolmogorov-Smirnov test were obtained for two furnace operation cases: 0.0071 and 0.0047, respectively. Table 4 presents the parameters of the fitted distributions and the ratios of these values to the values obtained for the measurement point B and A.
Using the modified firefly algorithm, the probability distributions for both furnace operation cases were determined. Figure 10 shows the histograms and distribution differences for these cases.
Searching for the matching of the distributions, the values of the Kolmogorov-Smirnov test were obtained for two furnace operation cases: 0.0071 and 0.0047, respectively. Table 4 presents the parameters of the fitted distributions and the ratios of these values to the values obtained for the measurement point B and A.   Based on the values in Table 4, the value of the voltage disturbance propagation factor d BC was assumed to be 0.2669 and 0.1826 (column µ), and the resulting impedance and short-circuit power values (7) and (8) for the impedance between points B and C (Z BC = 0.565 Ω) are 0.2057 and 0.1262 Ω, and 4375 and 7131 MVA, respectively. During operation, the furnace was powered for 75% of the time from the network, with a short-circuit power of 7131 MVA. Table 4 also presents the quotients of the distribution parameters for individual points. It is not surprising that the coefficient of voltage disturbance propagation between points A and C is the product of the coefficient of voltage disturbance propagation between points A and B, and B and C (9).
At point C, the values of P st reach 15.75% or 10.78% of the value P st at point A, depending on the configuration of the supply network. This means that there will be less fluctuation in the supply network when the impedance of the supply track from point C to the furnace is greater than the impedance above that point. The degree of damping of fluctuations is determined by the ratio of the impedance above the supply point to the track impedance below the point considered.
The reduction of voltage fluctuations between two points of the supply network, d AB , depends on the equivalent impedance of this grid fragment (10).

Conclusions
Using the example of the arc furnace, which is a large, turbulent load significantly different from other loads in the supply network, it was possible to observe the propagation of voltage disturbances in the supply network. The measure of these disorders is the indicator P st , the values of which decrease with the distance from the source of disturbances. The degree of reduction of these values is determined by the voltage disturbance propagation indicator, d, determined for a given section of the supply network. This value depends on the equivalent impedance of a given section (10) or the ratio of impedances (6), or the ratio of short-circuit power (5).
The short-circuit impedance or the short-circuit power at another point in the power system can be determined from the relationships (7) and (8), knowing the voltage disturbance propagation coefficient. The propagation factor of voltage disturbances between two points in the network can also be estimated based on the measured values P st at these points by determining the angle of inclination of the line approximating the relationship P stB /P stA . Another possibility for estimating the propagation value of voltage disturbances is provided by the analysis of statistical processes and estimating the characteristic parameters of the probability distributions, i.e., the location of the distribution maxima and the width of these distributions; in the example presented, these are the values b, c, µ, and s.
The influence of a given load on the supply network at a given point decreases with the increase of the short-circuit power at that point (i.e., with the decrease of the shortcircuit impedance). The greater short-circuit power at the measuring point, the smaller the P st values. This means that the reduction of the P st value is greater when the quotient of the short-circuit impedance at the load connection point to the short-circuit impedance at the measuring point (the measuring point is above the load connection point) is greater.