Research on the transmission of electric power via the increased frequency coaxial line to a remote low-power load (consumer)

. The article gives the analyses of the existing limitations in the transmission of electric power to a remote low-power load (consumer) and the alternative system of an increased frequency electric power supply is offered. The primary parameters of an increased frequency coaxial line are examined at different operating modes. It is shown that in the study progress the influence of the equipotential surface of the Earth’s field on the primary parameters of an increased frequency coaxial line was identified. The calculation methodology of the primary parameters of an increased frequency coaxial line is given taking into consideration the influence of the equipotential surface of the Earth’s field . The obtained results can be used at the designing of increased frequency electric power supply systems for remote low-power loads (consumers).


Introduction
The transmission of electric power to a remote low-power load (consumer) is an actual research problem.A remote low-power load (consumer) is the electric power consumer for whom the building of the electricity supply system of the utility frequency (50/60 Hz) from the centralized, decentralized or combined zone of electric power supply is not profitable economically.A remote low-power load (consumer), for example: a living house, stations and posts of hydro meteorological services, cellular towers, exploration and mining facilities, hunting facilities, tourist complexes, some military facilities etc., as a rule do not exceed 30 kW of installed capacity, but are located far away from the centralized, decentralized and combined power supply zone and have to use local autonomous power sources (gasoline and diesel generators, accumulator batteries, renewable energy sources [1]).This forced measure significantly increases the economic costs of the owner of a lowpower load (consumer), for the purchase of fuel, delivery of fuel to the location, as well as the maintenance of DC systems and etc.Besides all these facts many public and private projects with low-power loads (consumers), which are located far away from the above mentioned power supply zones, simply cannot be launched because of the same economic reasons.Investment profitability of the project goes beyond the limits and this fact leads to the project closure.The owner of the low-power load (consumer) just does not know other alternative power supply systems.
Basically, the increase of the investment in the project with remote low-power loads (consumers) is caused by the construction of the overhead power lines (overhead lines).The maximum length of the overhead power line is defined by such criteria as: 1) voltage loss at the end of the line; 2) minimum cross section by economic density; 3) minimum cross section by mechanical strength; 4) minimum cross section by thermal resistance; 5) installed capacity.It is possible to increase the length of the overhead lines observing the above criteria at the stage of design by increasing the cross section of conductors or by increasing the voltage class of the overhead lines.But the construction of high-voltage (for example, 10 kV) overhead lines with a small load factor is impractical in economic aspects.Because of the same economic reasons, it is also not practical to increase the cross section of the conductors.
For many remote low-power loads (consumers) the way out of this situation can be the power supply system at an increased frequency.Industrial frequencies of 50 and 60 Hz are the most common in power supply systems, but there are power supply systems at other frequencies.The scientists of the Russian Federation and the USSR were and are engaged in the problems of the power supply systems at an increased frequency: V.S. Kulebakin, A.P. Lvov, T.B. Leshchinskaya, N.I.Sokolov, R.N. Sokolova, A.M. Sokolov, O.A. Roshchin, T.E.Shadrikov, A. Tankoy, A.Yu. Filozhenko, as well as such foreign scientists as F.P. Emery, Darius Irani, Roderick Javier, Garcia Montoya, Akrem Mohamed Elrajoubi, Simon S. Ang, and others [2][3].
At the present moment, it is possible to increase the industrial frequency of an alternating current with the help of frequency converters, which are divided into two classes: 1) rotating electric machines; 2) semiconductor static frequency converters.The change in electromagnetic induction at the AC frequency increase gives the opportunity to design smaller electrical machines.In that context the increased frequency power supply systems have found their application in the situations where the mass and size of electric machines reduce capital and operating costs to a minimum.The increased frequency electrical installations have found the most extensive application in the aircraft industry, shipbuilding, oil and oil-extracting industry and induction heating of materials [4][5].

Materials and methods
The development of electrical engineering makes progress [6], new equipment on a constant and alternating electric current appears every year.In power supply systems each type of the current has its own positive and negative sides.For example, parallel DC electric machines do not require synchronization, there is no reactive electric power in the DC circuits.But the DC power supply system is not widely used.This system is economically suitable for the transmission of electricity for export at switching from an industrial frequency of 50 Hz to an industrial frequency of 60 Hz, as well as at the highpower transmission of electricity over very long distances.The AC power supply system has opened the possibility to transmit the electricity to the consumers at different distances economically advantageous.But AC has its drawbacks connected with the capacitance between the wires and the inductive component, which increase the voltage drop at the end of the overhead and cable lines, as well as significantly reduce the carrying capacity of power lines.
As a rule, a low-power electric load (consumer) consists of a set of electric receivers such as household electrical appliances, electric heating devices, a lighting system, etc.Let us find the maximum length of the overhead line construction from the electric load according to the nominal voltage standard in the Russian Federation.Let us express the electric load by the active power only, then the maximum length of the overhead line can be found by the formula: where γ is the conductivity of the wire material, m/Ω•mm 2 ; Fwire cross-section, mm 2 ; Unomnominal voltage of overhead lines, kV; ∆Umaxmaximum voltage deviation, kV; Pnomnominal power of the electric load, kW.
According to the 'Regulations for Electrical Installation', GOST 32144-2013 and GOST 33073-2014 the voltage deviation for power supply networks at the end of the overhead lines from the nominal value should be ± 5%.To make calculations we will use the most common non-insulated wire with a steel core and aluminum wires.Let us summarize the received data in Table 1.We can observe from Table 1 that the maximum length of the overhead line construction increases at the larger cross-section of the conductor section.It is impossible to transmit electricity for several kilometers for low-power electric power units (5-80 kW) at a voltage of 0.4 kV.The construction of the overhead lines of another 6 or 10 kV voltage class leads to an increase in capital costs.Also, the usage of a larger cross-section of the conductor section must be justified by the economic current density.According to the analysis made for the majority of low-power electric loads located far away from the centralized, decentralized and combined power supply area, the construction of the overhead lines is not economically convenient.

Results
Previously we spoke about the usage of increased frequency electrical installations in such industries as shipbuilding, aircraft, oil and oil-extracting industry.But all the abovementioned application of increased frequency electrical installations can be reduced to a local objectan aircraft, a ship, a drilling derrick etc.So there appears a question about the E3S Web of Conferences 463, 03008 (2023) EESTE2023 https://doi.org/10.1051/e3sconf/202346303008possibility to transmit increased frequency electricity to a low-power load (consumer) for a long distance, for example, three kilometers, through an overhead power line.Having asked this question, it was decided to make a complex research on the transmission of increased frequency electricity to a low-power load consumer (EC) for a long distance.
In the course of studies, it was found out that the distribution of an increased frequency current over the cross section of the conductor depends on the location of the conductors relative to each other.In case the increased frequency current in the conductors flows in one direction it is distributed over their cross section as far from each other as it is possible.In case the increased frequency current in the conductors flows in different directions it is distributed as close to each other as it is possible fig. 1.This phenomenon of redistribution of the increased frequency current across the cross-section of the conductor is called the proximity effect [7][8][9].Regardless of the shape of the cross section and the configuration of the conductors, the proximity effect will be present when the increased frequency current flows.It is necessary to distinguish the coaxial design of the cable from all the configurations.Coaxial cable consists of two concentric conductors situated coaxially and isolated from each other.The increased frequency current passing through the conductors is focused multidirectionally on the surfaces of the conductor which are facing each other.Thus the distribution of the current over the cross-section area of the conductor occurs peripherally according to the surface effect fig. 2 [10].The inductive resistance of the coaxial cable is much smaller than of other conductor configurations, this is due to the minimum distance between the conductors and the surface effect.It is this type of cable that is the most optimal for further research.An overhead transmission line with an increased frequency coaxial line was built in the case of the research.Let us give a method for calculating the primary parameters of the increased frequency coaxial line.To calculate the electrical properties of the increased frequency coaxial line we will use the well-known circuit method with distributed parameters of long lines, but with  At the calculation by the method of a circuit with distributed parameters of long lines a simplification is introduced that a voltage drop occurs in both wires, but we can consider that this voltage drop is concentrated only on the upper half of each element of the equivalent replacement scheme.This kind of simplification is also true for the increased frequency coaxial line, but requires a more detailed calculation of the primary parameters of the increased frequency coaxial line depending on the coaxial configuration of the conductors and the transmission of electric energy by an increased frequency.During the construction of a refined equivalent substitution scheme, it is necessary to take into account the resistance of the direct R1 and the reverse R2 conductors, the inductance of the direct L1 and the reverse L2 conductors, inductance between the conductors LB, capacity between the conductors C0 and active conductivity of insulation between the conductors G0 fig. 4. The resistance of the length unit, Ω/m, the straight R1 and the reverse R2 of copper conductors is found by the formula: where d is the diameter of the inner conductor, mm; D is the inner diameter of the outer conductor, mm; f is the AC frequency, Hz.The inductance of the increased frequency coaxial line loop referred to the length unit length, H/m, is composed of the internal inductance of the conductors L1 and L2, as well as the inductance between the conductors LB.Thus the inductance of the increased frequency coaxial line loop with copper conductors can be found by the formula: The parameters C0 and G0 can be calculated from the complex conductivity of the coaxial circuit insulation.Capacity between the conductors per the length unit, F/m, is calculated by the formula: where ε is the relative permittivity of the ambient.
The active conductivity of the insulation between the increased frequency coaxial line conductors, S/m, is calculated by the formula: ) where tan δ is the tangent of dielectric loss angle of the insulation material.
The graph of the dependence of the primary parameters of the increased frequency coaxial line on the frequency is presented in fig. 5.It is shown in the graph that as the AC frequency increases, the resistance and conductivity of the insulation of the increased frequency coaxial line increases and the inductance decreases.The capacity between the conductors is independent of the frequency.The results of calculations of primary parameters of the increased frequency coaxial line at different frequencies are summarized in Table 2.
Two modes of operation of electric power transmission via the increased frequency coaxial line were considered in the course of our research.In the first mode of operation, two conductors of the increased frequency coaxial line were connected to the outputs of the secondary winding of the increased frequency transformer (IFT1) of the power supply source (PS), one of the terminals of the transformer was grounded.At the other end of the increased frequency coaxial line only one conductor was connected to the secondary winding of the increased frequency transformer (IFT2), the second conductor remained isolated.One of the outputs of the increased frequency transformer (IFT2) was grounded fig.6.The first mode involves a single-wire power transmission line, where the ground serves as the return conductor.In this case, the external conductor of the increased frequency coaxial line is grounded and represents a protective conductor against the electric shock.The advantage of the first mode is the stable primary parameters regardless of the height of the line.The increased frequency coaxial line consists of two conductors, in the first mode of the operation the external conductor shields the internal conductor, therefore, such a system of conductors, strictly speaking, forms a capacitor.The capacity of the capacitor is the capacity between its conductors, i.e. the capacity between the conductors per length unit C0.The capacity between the conductors C0 is independent of the presence of any other conductors located outside the shielding conductor.The disadvantage of this mode is the reduced efficiency of electric power transmission through the increased frequency coaxial line due to the high resistance of the return conductor.The results of the research of the first mode are presented in Table 3.
In the second mode two conductors of the increased frequency coaxial line were connected to the outputs of the secondary winding of the increased frequency transformer (IFT1) power source, one of the ends of the transformer was grounded.At the other end of the increased frequency coaxial line, two conductors were also connected to the secondary winding of the increased frequency transformer (IFT2) electric consumer, one of the transformer ends was grounded fig. 7.
The second mode implies a two-wire power transmission line, while the external conductor of the increased frequency coaxial line is considered the neutral conductor.The advantage of the second mode of operation is the increased efficiency of electric power transmission via the increased frequency coaxial line due to the low resistance of the reverse conductor compared to the first mode of operation.The disadvantage of this mode is the change in the distributed parameters of the increased frequency coaxial line depending on the height of the line.The results of the study of the second mode are presented in Table 4.

Discussion
In the course of studies of the second operating mode, it was found out that the change in the distributed parameters of the increased frequency coaxial line depending on the height of the line is caused by the equipotential surface.Due to the grounding of one of the ends of the secondary windings of the IFT1 and IFT2, a negligible low electric current of increased frequency, which is not taken into account in the calculations, flows through the ground.
Accordingly, charges are formed on the external conductor of the increased frequency coaxial line when an increased frequency current flows, and charges of the opposite sign are formed on the ground surface.The surface of the earth is the equipotential surface of the field as the ground is a conductor.According to the method of images (or method of mirror images), charges can be taken into account by considering an opposite conductor placed at a depth equal to the height of its suspension fig.8.In the second mode of operation, the capacity between the external conductor and the ground in the wire-ground system CWG is equal to the double capacity between the external conductor and its mirror image because the capacity that is equivalent to two identical serially connected capacities is equal to the half of each capacity fig.9.The capacity of the wire-ground system is calculated with the help of the following formula: where ε is the relative permittivity of the ambient; ε0 ≈ 8.86×10 -12electrical constant, F/m; hheight of the coaxial line, m; R = D/2radius of external conductor, m.The dependence of the capacity of the wire-ground system of the coaxial line on the height of the line is shown in fig.10.Thus, the equipotential surface makes an adjustment at the calculation of the primary parameters of the increased frequency coaxial line, in addition to the capacity between the conductors C0 it is necessary to take into account the capacity of the wire-ground system CWG fig.11.The capacity of the increased frequency coaxial line consists of the capacity between the C0 conductors and the capacity of the wire-ground system CWG, F/m, and is found by the formula: 12.The obtained results of the designed and actual voltage at the beginning and at the end of the increased frequency coaxial line as well as the efficiency of power transmission in the second mode are presented in Table 5. Taking into account the additions mentioned above and the more detailed calculation of the circuit with distributed parameters, the error of the designed and actual efficiency of power transmission through the increased frequency coaxial line was 4%.

Conclusion
The results of the study of power transmission to a remote low-power load (consumer) via an increased frequency coaxial overhead line are given.The procedure for calculating the primary parameters of the coaxial overhead line in the electromagnetic field of the increased frequency is given.Calculation of the dependence of the capacitance of the wireearth system of the coaxial overhead line on the equipotential surface of the ground field is given.The developed calculation method can be used for the preliminary analysis of the primary parameters of the coaxial overhead line connected to an increased frequency power supply.

Fig. 1 .
Fig. 1.Distribution of the increased frequency AC over the cross-section of the conductors: a) the current flows in different directions, b) the current flows in one direction.

Fig. 2 .
Fig. 2. Distribution of the increased frequency current over the cross section of the coaxial cable conductors.

E3S
Web of Conferences 463, 03008 (2023) EESTE2023 https://doi.org/10.1051/e3sconf/202346303008some additions.The additions and more detailed calculations of the distributed parameters circuit are due to the specific operation of the increased frequency coaxial line.Distributed parameters of the increased frequency coaxial line are characterized by the primary parameters:  R0the resistance of the unit of the length of the direct and reverse conductors;  L0line loop inductance, also related to the length unit;  C0capacity between the conductors per unit length;  G0active conductivity of the insulation between the conductors per unit length of the conductors.An equivalent scheme for replacing of the part of the increased frequency coaxial line is shown in fig.3.

Fig. 3 .
Fig. 3. Equivalent scheme for replacing of the part of the increased frequency coaxial line.

Fig. 4 .
Fig. 4. Specified scheme for replacing of the part of the increased frequency coaxial line.

Fig. 5 .
Fig. 5. Dependence of the primary parameters of the increased frequency coaxial line on the AC frequency.

Fig. 6 .
Fig. 6.The first mode of the work of the increased frequency coaxial line.

Fig. 7 .
Fig. 7.The second mode of the work of the increased frequency coaxial line.

Fig. 8 .
Fig. 8. Placement of the outside conductor of the coaxial line and its mirror image in the wire-ground system.

Fig. 9 .
Fig. 9. Capacity between the external conductor and the ground in the wire-ground system.

Fig. 10 .
Fig. 10.Dependence of the capacity of the wire-ground system of the coaxial line on the height of the line.

Fig. 11 .
Fig. 11.Equivalent scheme for the replacement of the part of the coaxial line considering the capacity of the wire-ground system.

E3S
Web of Conferences 463, 03008 (2023) EESTE2023 https://doi.org/10.1051/e3sconf/202346303008dependence of the capacity of the increased frequency coaxial line on the height is shown in fig.

Fig. 12 .Table 5 .
Fig. 12. Dependence of the capacity of the coaxial overhead line from the height.Table 5.The results of the calculated and actual parameters of the power transmission via the coaxial overhead line in the second working mode.Voltage at the beginning of the coaxial overhead line, actual, V Voltage at the end of the coaxial overhead line, actual, V Efficiency, actual, % 1176 1054 76 Voltage at the beginning of the coaxial overhead line, calculated, V Voltage at the end of the coaxial overhead line, calculated, V

Table 1 .
The maximum length of the overhead power line.

Table 2 .
Primary parameters of the increased frequency coaxial line at different AC frequencies.

Table 3 .
The results of the research of the first mode of the increased frequency coaxial line.

Table 4 .
The results of the research of the second mode of the increased frequency coaxial line.