Selection of C-Type Filters for Reactive Power Compensation and Filtration of Higher Harmonics Injected into the Transmission System by Arc Furnaces

: This article presents a method for selecting the elements of a C-type ﬁlter working with a conventional LC-type ﬁlter for compensating reactive power and ﬁltering out higher harmonics generated by arc furnaces and ladle furnaces. The study was conducted in a steel mill supplied by a 110 kV transmission system, where higher harmonic currents and nonlinear loads were measured. A series of computer simulations were performed under various operating conditions, and an algorithm for selecting the parameters of a third-order C-type ﬁlter (for suppressing the second harmonic) and two second-order LC-type ﬁlters (for suppressing the third harmonic) was proposed. The ﬁltering system was tested in an arc furnace with the highest rated power, and harmonics in the current spectrum were evaluated. The results of the measurements were used to analyze the e ﬀ ectiveness of the compensation system comprising two passive C-type and LC-type ﬁlters at di ﬀ erent system conﬁgurations. C-type ﬁlters signiﬁcantly inﬂuenced current harmonics. The inﬂuence of the changes in the number of arc furnace transformers on the true Root Mean Square (RMS) of the currents injected into the 110 kV transmission system and on the voltages of the 110 kV busbars was discussed. with an inductive transmission system, the result can be parallel resonance and a higher second harmonic in the transmission system. The results of the analysis indicate that resistor


Introduction
The growing use of electronic systems in power receivers decreases energy consumption in numerous production processes [1][2][3][4], but also creates new risks. The associated dangers include higher harmonic components that lead to the unnecessary activation of protective devices, such as residual-current protection devices [5][6][7], harmonic components that disrupt the operation of automatic synchronizers [8], as well as harmonic components that occur in non-stationary signals in various interdisciplinary applications, for example related to acoustics [9], electric vehicles [10], or even seemingly distant biomedical engineering [11,12].
Arc furnaces and ladle furnaces are the largest power receivers in steel mills. They operate with low power factors [13][14][15][16][17][18] and inject higher harmonic currents into the transmission system. Due to the operating characteristics of an electric arc between two carbon electrodes, the current is asymmetrical relative to ordinates and abscissas, which produces odd (third, fifth, seventh, etc.) as well as even (second, fourth, sixth, etc.) harmonics in the current spectrum [16][17][18][19]. Even and odd harmonic currents have to be reduced in industrial plants [20]. Industrial furnaces are high-power systems that require harmonic and for filtering out higher harmonics. A compensation system should be equipped with a C-type filter to reduce second harmonic generation. In most studies dedicated to the selection of C-type filters, the presence of other filters in the system is disregarded, and the proposed selection methods are narrowed down to C-type filters and the impedance of the transmission system [21,22]. The second harmonic current can be increased when the presence of LC-type filters is disregarded during the design of a C-type filter for reducing out the second harmonic current or when third harmonic filters are installed without a second harmonic filter. It is worth noting that active filters can be used for reactive power compensation and filtration of higher harmonics, but due to the very high power of arc furnaces and the high cost of active filters it is usually unreasonable and therefore passive filters or less often hybrid filters are used [23][24][25]. Static Var Compensator (SVC) systems are also applied to compensate arc furnaces [26][27][28].
Various algorithms for selecting C-type filters have been proposed in the literature [29][30][31][32][33][34]. However, the developed algorithms were verified mainly during computer simulations, and the selected parameters have never been tested in a real-world setting. The operation of an electric arc furnace is a dynamic process, and computer simulations do not always accurately predict the behavior of a compensation system in a steel mill. Computer simulations do not always account for the system's operation in different configurations, including with/without a C-type filter in systems equipped with an LC-type filter, or the system's response to changes in the operating parameters of an arc furnace. A compensation system has been designed and tested in a real-world setting in only one study [35]. However, the choice of resistance parameter, RT, which is the most important step during the selection of C-type filters, was not discussed.

Specification of the Transmission System
LC-type passive filters can have different topologies, which are presented in Figure 1. The filter selection algorithm for a system with arc furnaces was tested in an industrial plant whose power supply system is presented in Figure 2a, and the supply system parameters are described in Table 1. Third and higher harmonics (3, 4, 5, etc.) are usually reduced with second-order LC-type filters (Figure 2b,c). Due to the proximity of the second and first harmonics, a third-order C-type filter is generally applied to limit the second harmonic. The quality of electric power was measured in the plant's substation in the transformer field at 110 kV. Measurement results are discussed in Section 5 for the different transformers.    (c) Diagram of a transmission system with reactive power compensation where higher harmonics are filtered by LC-type and C-type filters when arc furnace 1 is supplied by transformer Tr 3.

Theoretical Analysis
Reactive power consumption was calculated at 60 MVAr, based on the measurements of reactive power of the first harmonic. A 20 MVAr third-order C-type filter was used first due to the presence of the second harmonic, and the remaining harmonics were filtered out with two identical 20 MVAr second-order LC-type filters. The LC-type filter should be selected first, and the third-order C-type filter should be chosen accordingly.

Selection of an LC-Type Filter
The effectiveness of an LC-type filter is determined mainly by the self-resonant frequency (Figure 3), which is defined by the following formula: where X C -represents the capacitive reactance of the bank for the first harmonic; and X DO -the inductive reactance of the choke for the first harmonic.
Energies 2020, 13, x FOR PEER REVIEW 5 of 21 The percent filtration (enhancement) coefficient can have positive or negative values. A positive value ("+") denotes the percent suppression of a given harmonic, whereas a negative value ("−") denotes the percent enhancement of a given harmonic. Suppression efficiency is also determined by the QDO-factor of the choke applied in the system with an LC-type filter ( Figure 4). The goodness-of-fit of a filter choke is determined as follows:   The suppression of the nth harmonic in the system was defined as where U nf -phase voltage of the nth harmonic after compensation with k capacitor units, in V; and U nf -phase voltage of the nth harmonic before compensation, in V. Coefficient k fn denotes the degree to which the harmonic component of the supply voltage is reduced by the compensation system or a passive filter. A given harmonic is suppressed when coefficient k fn is less than 1, and it is enhanced when coefficient k fn is greater than 1. The below formula can be used to express coefficient k fn in percentage terms: The percent filtration (enhancement) coefficient can have positive or negative values. A positive value ("+") denotes the percent suppression of a given harmonic, whereas a negative value ("−") denotes the percent enhancement of a given harmonic.
Suppression efficiency is also determined by the Q DO -factor of the choke applied in the system with an LC-type filter ( Figure 4). The goodness-of-fit of a filter choke is determined as follows: where X DO -choke reactance (phase) for the first harmonic, in Ω; and R DO -choke resistance (phase) for the first harmonic, in Ω.
denotes the percent enhancement of a given harmonic. Suppression efficiency is also determined by the QDO-factor of the choke applied in the system with an LC-type filter ( Figure 4). The goodness-of-fit of a filter choke is determined as follows: where DO X -choke reactance (phase) for the first harmonic, in Ω; and DO R -choke resistance (phase) for the first harmonic, in Ω. The resonant frequency nr should be lower than 3 if the LC-type filters are designed for reducing the third, fourth, fifth, and successive harmonics. In view of the manufacturing tolerance (choke and capacitors) and aging of the system components, the filter was detuned and nr = 2.95 was applied. Phase reactance of the capacitor bank was determined with the use of the below formula:  The resonant frequency n r should be lower than 3 if the LC-type filters are designed for reducing the third, fourth, fifth, and successive harmonics. In view of the manufacturing tolerance (choke and capacitors) and aging of the system components, the filter was detuned and n r = 2.95 was applied. Phase reactance of the capacitor bank was determined with the use of the below formula: where U NS -represents the network rated voltage, the RMS value of the interphase voltage of the first harmonic; and Q Cr -reactive power of the first harmonic injected into the system by the filter. Capacitive reactance of the bank was used to calculate the first harmonic voltage U C1 on the bank terminal: The rated voltage of capacitors should be higher than the first harmonic voltage due to the presence of higher harmonics. The RMS value of voltage on the bank terminal is the algebraic sum of the individual harmonic voltages [37,38]: where U C -RMS value of the voltage on the bank terminal; U C1 -RMS value of the 50 Hz first harmonic voltage on the bank terminal; U Cn -RMS value of the nth harmonic voltage on the bank terminal; n-successive harmonic in the transmission system (second, third, fourth, fifth, etc.); and m-the number of higher harmonics in the transmission system. The above prediction stems from the IEC 60871-1:2014 standard [37] and its European equivalents EN 60871-1:2014 (for capacitors with rated voltage above 1 kV) and IEC 60831-1:1996 [38], as well as its European equivalent EN 60831-1:1996 (for capacitors up to 1 kV). In the literature, the RMS value of voltage on the bank terminal is often confused with the RMS value of the transmission system voltage [13,21,[39][40][41][42], which is defined as the geometric sum of the individual harmonics in accordance with standard EN 50160:1994 [43]. The rated voltage of a capacitor bank should closely correspond to the RMS value of the voltage on the bank terminal. To reduce costs, the difference between the installed reactive power and the reactive power of the first harmonic injected into the system should be minimized. The proportionality of the rated voltage to rated power of a capacitor bank is given by the following formula: Choke reactance was calculated with the use of the following formula:

Selection of a C-Type Filter
C-type filters should be selected after choosing the appropriate LC-type filters. The capacitive reactance of the auxiliary capacitor bank and the inductive reactance of the choke for the first harmonic are equal: where X CP -represents the capacitive reactance of the auxiliary capacitor bank for the first harmonic; and X DO -inductive reactance of the choke for the first harmonic. When the resistance of the choke, auxiliary capacitors, and current circuits (cables, terminals, etc.) is disregarded, equivalent impedance for the first harmonic approximates zero. The main capacitor bank is supplied with a rated voltage of the network powering the entire filter. The selected capacitor bank should inject actual reactive power, calculated based on the below formula, into the transmission system: where X CG -represents the capacitive reactance of the main capacitor bank for the first harmonic. A C-type filter will have self-resonant frequency of Formula (10) can be modified to calculate the value of X CP or X DO with the use of the below equation: The most difficult task in the process of designing a C-type filter is the selection of the damping resistor R T , which should • guarantee that the entire system, with a C-type filter only and with LC-filters, has an inductive character for higher harmonics and a capacitive character for the first harmonic (which was not taken into account in [21,42] as well as [35,[44][45][46][47][48][49]); • filter out higher harmonic currents and reduce higher harmonics voltages; • compensate the reactive power of the first harmonic in view of the manufacturing tolerance and aging of components R, L, and C [50]; • reduce overvoltage in a system with a C-type filter.
Based on the above dependencies, an algorithm was developed for selecting a C-type filter for the required higher harmonic after compensation. The algorithm is presented in a diagram in Figure 5, and additionally, the algorithm equations are given in Appendix A.
System parameters have to be specified in the first step of the proposed algorithm ( Figure 5). Depending on the number of the applied filters, the parameters of the LC-type filters for eliminating higher harmonic components (higher than second) are selected in Steps 2 to 4. In systems with three LC-type filters, the third, fourth, and fifth harmonics are calculated in three loops (s = 3). The parameters of a C-type filter are set as beginning from Step 5. Parameters such as X CG , X CP , X DO , and R DO are selected in Steps 5 and 6. In Step 8, the resistance (R T ) is set to ensure that the equivalent reactance has a positive value X zas > 0 to prevent parallel resonance. The filtering system's ability to reduce a given harmonic to a desirable value is verified in Step 9. If the above requirement is not met, the self-resonant frequency of a C-type filter is increased, and Steps 6 to 9 are repeated. The resonant frequency should not exceed n r = 2, and the manufacturing tolerance of a C-type filter should be taken into consideration. increases with the number of incorporated filters. Resistor RT also discharges the second harmonic current in a C-type filter. Its resistance has to ensure that more than 90% of the active power of the second harmonic is discharged on the resistor and not on the other components (receivers and sources) of the transmission system (Figure 7).

Figure 5.
Algorithm steps for calculating the parameters of a C-type filter for the required Un' as in Equations (A1) to (A18) Appendix A.
The compensation system can be expanded to include a fourth harmonic filter; therefore, a 300 Ω damping resistor was used. The parameters of the C-type filter are presented in Table 2.

Simulation
A number of computer simulations were run for the presented transmission system, including with the use of the proposed algorithm for selecting C-type filters. Different parameters of the transmission system and passive filters were used in the simulations. The choice of the damping resistor was the most important task. The parameters of an LC-type filter were selected first (Table 2), and they were used to analyze the operating requirements of a C-type filter. The highest self-resonant frequency that was possible in view of the manufacturing tolerance of the filter's elements was n r = 1.95. The above value was adopted in Step 5 of the presented algorithm to maximally filtrate the second harmonic. The condition does not have to be verified after Step 9 (see Figure 5 and Appendix A). The main focus of the analysis was to select the parameters for resistor R T , which had been disregarded in the literature [35] or whose selection and influence on the filtering system had been presented incorrectly [21,46]. Resistor parameters have to ensure that the equivalent reactance of the second and third harmonic filters has an inductive character at the frequency of 100 Hz (Figure 6a). If inductive reactance is combined with an inductive transmission system, the result can be parallel resonance and a higher second harmonic in the transmission system. The results of the analysis indicate that resistor Energies 2020, 13, 2330 9 of 19 R T should have minimum resistance of 236 Ω. If a fourth harmonic filter were incorporated into the system, resistance would equal 298 Ω (Figure 6b). All system filters should be taken into account because the minimum resistance increases with the number of incorporated filters. Resistor R T also discharges the second harmonic current in a C-type filter. Its resistance has to ensure that more than 90% of the active power of the second harmonic is discharged on the resistor and not on the other components (receivers and sources) of the transmission system (Figure 7).     Energies 2020, 13, x FOR PEER REVIEW 10 of 21 Figure 7. Active power of a higher harmonic discharged on the damping resistor RT and in the transmission system as a function of the resistance of the damping resistor RT.
In practice, a C-type filter tuned to the third harmonic can be installed in two configurations:  the capacitor bank is arranged in double-star connection ( Figure 8a);  the capacitor bank is arranged in an H-bridge (Figure 8b).
The selection of a specific solution (Figure 8a,b) has significant implications for the overvoltage conditions of the compensation system. In a system where the capacitor bank is arranged in a double-star connection, when a short circuit occurs in the bank terminals, voltage will not be reduced to a safe level by surge protectors because  for the periodic component of fault current (50 Hz), the reactance of the auxiliary bank with a resonant choke has practically zero value, which implies that the periodic component will flow through the auxiliary bank with the same magnitude that is noted during busbar overvoltage;  the flow of high fault current through the auxiliary bank causes very high voltage losses in different elements of the system (XD, XCP); in the analyzed steel mill, overvoltage will exceed 300 kV.
(a)  for the 2 harmonic in for the 3 rd harmonic in for the 4 th harmonic in for the 2 n d harmonic in the transmission system for the 3 rd harmonic in the transmission system for the 4 th harmonic in the transmission system Figure 7. Active power of a higher harmonic discharged on the damping resistor R T and in the transmission system as a function of the resistance of the damping resistor R T .
The compensation system can be expanded to include a fourth harmonic filter; therefore, a 300 Ω damping resistor was used. The parameters of the C-type filter are presented in Table 2.
In practice, a C-type filter tuned to the third harmonic can be installed in two configurations: • the capacitor bank is arranged in double-star connection ( Figure 8a); • the capacitor bank is arranged in an H-bridge (Figure 8b). Figure 7. Active power of a higher harmonic discharged on the damping resistor RT and in the transmission system as a function of the resistance of the damping resistor RT.
In practice, a C-type filter tuned to the third harmonic can be installed in two configurations: the capacitor bank is arranged in double-star connection (Figure 8a); the capacitor bank is arranged in an H-bridge (Figure 8b).
The selection of a specific solution (Figure 8a,b) has significant implications for the overvoltage ditions of the compensation system. In a system where the capacitor bank is arranged in a uble-star connection, when a short circuit occurs in the bank terminals, voltage will not be uced to a safe level by surge protectors because for the periodic component of fault current (50 Hz), the reactance of the auxiliary bank with a resonant choke has practically zero value, which implies that the periodic component will flow through the auxiliary bank with the same magnitude that is noted during busbar overvoltage; the flow of high fault current through the auxiliary bank causes very high voltage losses in different elements of the system (XD, XCP); in the analyzed steel mill, overvoltage will exceed 300 kV. When the bank is arranged in an H-bridge, a short circuit in the auxiliary bank will have value than the current rating of the filter and compensator system. Effective protection auxiliary bank and the damping resistor poses an additional problem in both configuration required protection has to respond to the  first harmonic in a 50 Hz system to protect the auxiliary bank;  all harmonics to protect the damping resistor.

Validating a Designed Filter in a Real-World System
A system of filters (Figures 2b,c, 8a, and 9a,b) was developed for the parameters prese Table 2, and the higher harmonics In and Un were measured. Measurements were conducted transformer field at 110 kV (measuring point of the analyzed steel mill), and the results are pre in Figures 10 and 11. Measurements were carried out for two system configurations: wi The selection of a specific solution (Figure 8a,b) has significant implications for the overvoltage conditions of the compensation system. In a system where the capacitor bank is arranged in a double-star connection, when a short circuit occurs in the bank terminals, voltage will not be reduced to a safe level by surge protectors because

•
for the periodic component of fault current (50 Hz), the reactance of the auxiliary bank with a resonant choke has practically zero value, which implies that the periodic component will flow through the auxiliary bank with the same magnitude that is noted during busbar overvoltage; • the flow of high fault current through the auxiliary bank causes very high voltage losses in different elements of the system (X D , X CP ); in the analyzed steel mill, overvoltage will exceed 300 kV.
When the bank is arranged in an H-bridge, a short circuit in the auxiliary bank will have a lower value than the current rating of the filter and compensator system. Effective protection for the auxiliary bank and the damping resistor poses an additional problem in both configurations. The required protection has to respond to the • first harmonic in a 50 Hz system to protect the auxiliary bank; • all harmonics to protect the damping resistor.

Validating a Designed Filter in a Real-World System
A system of filters (Figure 2b,c, Figures 8a and 9a,b) was developed for the parameters presented in Table 2, and the higher harmonics I n and U n were measured. Measurements were conducted in the transformer field at 110 kV (measuring point of the analyzed steel mill), and the results are presented in Figures 10 and 11. Measurements were carried out for two system configurations: with two parallel transformers T1 and T2 (Figure 2b), and with one transformer T3 (Figure 2c). Measurements were also performed when the higher harmonic filters were turned off in both configurations. The effectiveness of the C-type second harmonic filter and its influence on the transmission system were evaluated with the C-type filter turned off. The variations in the RMS values of the second, third, and fourth harmonic voltages during several operating cycles of an arc furnace are presented in Figure 11.

Conclusions
The following conclusions can be formulated based on the computer simulations performed in this study:


The effectiveness of a passive LC-type filter is strongly influenced by the degree to which the order of self-resonant frequency departs from the order of the filtered harmonic and from the order of the harmonic. Lower-order harmonics (third, fourth, and fifth) are significantly more influenced than the higher-order harmonics. In higher-order harmonics, the effectiveness of an LC-type filter will be reduced by only 10% when the value of nr decreases from 11 to 10. Therefore, a single filter can be used to eliminate two harmonics. For example, a filter with nr = 10.5 will suppress the 11th harmonic by around 95% and the 13th harmonic by 90% (Figure 3).

Conclusions
The following conclusions can be formulated based on the computer simulations performed in this study: • The effectiveness of a passive LC-type filter is strongly influenced by the degree to which the order of self-resonant frequency departs from the order of the filtered harmonic and from the order of the harmonic. Lower-order harmonics (third, fourth, and fifth) are significantly more influenced than the higher-order harmonics. In higher-order harmonics, the effectiveness of an LC-type filter will be reduced by only 10% when the value of n r decreases from 11 to 10. Therefore, a single filter can be used to eliminate two harmonics. For example, a filter with n r = 10.5 will suppress the 11th harmonic by around 95% and the 13th harmonic by 90% (Figure 3).
• A filter's effectiveness is also influenced by choke resistance. However, this parameter has a negligent effect on filters where the order of self-resonant frequency considerably departs from the filtered harmonic. The protective choke influences a filter's effectiveness when the value of n r approximates or exceeds the filtered harmonic, which could lead to parallel resonance (n r = 2.9-3.3 in Figure 4).

•
The option of installing other higher-order filters in the system must be taken into account in the process of selecting the resistance R T of a damping resistor in a C-type filter. In a system equipped with a C-type filter only, the minimum resistance of the damping resistor is 100 Ω. In a system equipped with a C-type filter and a third-order LC-type filter, the minimum resistance of the damping resistor is 236 Ω (Figure 6a). The minimum resistance of the damping resistor increases to 298 Ω when a fourth-order LC-type filter is installed in the system (Figure 6b). In this study, the resistance of the damping resistor was set at R T = 300 Ω.

•
The true power generated by a damping resistor peaks at 40 kW for the second harmonic when resistance approximates 80 Ω. The true power generated by the resistor and the transmission system decreases at higher resistance values (Figure 7).

•
In practice, a C-type filter can be installed in two configurations: a double-star connection (Figure 8a) or an H-bridge (Figure 8b). In this study, a double-star configuration was selected due to the ease of detecting short circuits in systems with a C-type filter.
The results of the analysis performed in a real-world system equipped with C-type passive filters for second harmonic generation and LC-type passive filters for third harmonic generation revealed the following: • The activation of C-type and LC-type filters increased the RMS value of the second harmonic current injected into the transmission system by 10.8% when one transformer was used and by 7.2% when two transformers were used (Figure 10b), which increased the RMS value of higher harmonic voltage on the 110 kV busbars (Figures 10a and 11a) by 10.1% when one transformer was used and by 7.1% when two transformers were used.

•
The absence of a C-type filter for compensating second harmonic generation and the activation of only one LC-type filter for compensating third and higher harmonic generation increased the RMS value of second harmonic current injected into the transmission system by 62% (Figure 10a). The absence of a second harmonic filter increased the RMS value of second harmonic voltage by 68% (Figures 10a and 11a), which indicates that a C-type filter should be used for compensating harmonic generation in arc furnaces.

•
The increase in equivalent impedance between an arc furnace and a measuring point resulting from a reduction in the number of transformers (from two to one) decreased the RMS value of second harmonic current injected into the transmission system by 7.1% when the capacitor bank was turned off and by 3.4% when the capacitor bank was turned on (Figure 10b). The above decreased the RMS value of second harmonic voltage by 7.6% when the capacitor bank was turned off and by 4.6% when the capacitor bank was turned on (Figure 10a).

•
The activation of higher harmonic filters decreased the RMS value of third harmonic current by 94% when one transformer was used and by 53% when two transformers were used (Figure 10d). The RMS value of higher harmonic voltage on 110 kV busbars decreased by 88% and 52%, respectively (Figures 10c and 11b).

•
A single transformer should be used instead of two parallel transformers with similar power. The above solution increases impedance on the 110 kV side and decreases the RMS value of the third harmonic injected into the transmission system by 8.8% when the capacitor bank is turned off and by 28% when the capacitor bank is activated (Figure 10d). Lower generation of third harmonic current decreases the RMS value of third harmonic voltage on 110 kV busbars by 9.1% and 27%, respectively (Figure 10c).

•
The application of a C-type filter only for the second harmonic decreases the RMS value of the third harmonic current injected into the transmission system by 43.8% and decreases the RMS value of third harmonic voltage by 44% on 110 kV busbars.

•
The use of only one transformer instead of two also decreases the RMS value of fourth and fifth harmonic voltages and currents injected into the transmission system (Figures 10e-h and 11c).

•
When the second and third harmonic filter is replaced with the third harmonic filter only, the RMS values of the second and third harmonic currents and voltages are decreased, which results from a drop in the equivalent impedance of the filters for harmonics higher than the third harmonic (Figure 10e-h). Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.
Abbreviation R Sn transmission system resistance (phase): in Ω X Sn transmission system reactance (phase) for the nth harmonic, in Ω X CG reactance (phase) of the main capacitor bank for the first harmonic, in Ω X CP reactance (phase) of the auxiliary capacitor bank for the first harmonic, in Ω X DO choke reactance (phase) for the first harmonic, in Ω I n nth harmonic current, in A R DO choke resistance (phase) for the first harmonic, in Ω R T resistance (phase) of the damping resistor for the first harmonic, in Ω R f p resistance (phase) of the pth LC-type filter for the nth harmonic, in Ω X fn p reactance (phase) of the pth LC-type filter for the nth harmonic, in Ω R zas n (1,p+1) equivalent resistance (phase) of the first to the pth LC-type filter for the nth harmonic, in Ω X zas n (1,p+1) equivalent reactance (phase) of the first to the pth LC-type filter for the nth harmonic, in Ω R LC n equivalent resistance of all LC-type filters (phase) for the nth harmonic, in Ω X LC n equivalent reactance all LC-type filters (phase) for the nth harmonic, in Ω R C n equivalent resistance of a C-type filter (phase) for the nth harmonic, in Ω X C n equivalent reactance of a C-type filter (phase) for the nth harmonic, in Ω R zas equivalent resistance of all filters (phase) for the nth harmonic, in Ω X zas equivalent reactance of all filters (phase) for the nth harmonic, in Ω Q Cr actual reactive power supplied to the system, in VAr n harmonic distortions in the transmission system U" nf phase voltage of the nth harmonic before compensation, in V U" nmf interphase voltage of the nth harmonic before compensation, in V U nf phase voltage of the nth harmonic after compensation with k capacitor units in V U nmf interphase voltage of the nth harmonic after compensation with k capacitor banks, in V U nW required phase voltage of the nth harmonic after compensation, in V U NS RMS value of interphase voltage of the first harmonic, in V Q DO choke Q DO -factor n r self-resonant frequency of the choke-capacitor bank s maximum number of LC-type higher harmonic filters m successive LC-type higher harmonic filter