CC BY
21ANALYSIS OF THE DISTORTION OF THE SYMMETRY OF THREE-PHASE CURRENTS AT A
SINGLE-PHASE INDUCTION LOAD
Kinev E.
Ph.D., director of Thermal Electrical Systems LLC,
Krasnoyarsk, Russia Tyapin A.
Postgraduate student, Siberian Federal University,
Krasnoyarsk, Russia
ABSTRACT
The article presents the results of a study of the modes of induction equipment designed for aluminum extrusion. Powerful single-phase inductors connected to a three-phase network cause asymmetry of currents and voltages, worsening the work of the rest of the equipment of the workshop, sharply increasing losses. The study of the mode of a single-phase inductor in a three-phase network was carried out by software using parametric circuit models built using hybrid analysis. Replacing approximate engineering methods with automated computational algorithms and mathematical modeling made it possible to perform a more accurate calculation. To increase control flexibility in practice, automation of balancing using a microcontroller is proposed. In the problem, the advantages and rationality of heating aluminum with electricity are shown, which allow us to proceed to a calculated estimate of the efficiency of electric heating in comparison with gas.
Keywords: balancing device, single-phase load, induction heating, longitudinal magnetic field inductor, steady-state mode, automatic mode control, vector symmetrization diagrams, hybrid analysis.
Introduction. At enterprises equipped with equipment for the extrusion of aluminum ingots, electric and gas heating of aluminum is used. Electric heating of ingots, for a long time seemed difficult to control and un-obvious, despite its simplicity and environmental friendliness. Therefore, induction heating is often inferior to the primary role of gas [1]. This situation turned out to be a reality even in the Siberian regions, where gas is supplied by road. For example, in Krasnoyarsk there are local sources of high-power electric generation. The reason for gas heating can be the high cost of electricity, the high cost of inductors and microcontroller equipment. However, it is likely that the harm caused to the environment by burning fossil fuels is underestimated [2].
At the same time, in the 21st century, transnational companies successfully develop, design, use and sell induction equipment for aluminum extrusion. An example of a powerful inductor (a) of foreign manufacture and a power source (b) is shown in Fig. 1.
Judging by the global economic situation, modern highly efficient induction equipment allows you to fully use the advantages of electric heating of ingots. There is an obvious priority for electromagnetic inductors over gas-fired heaters for aluminum. Apparently, enterprises need a thorough assessment of technical and technological limitations in the use of electric heating. It is necessary to overcome the small controllability of the inductors, increase the load balancing efficiency and ensure a reduction in the associated electric energy losses
a b
Fig. 1. General view of a foreign-made inductor with a power source
The means of redistributing the induction load usually include special balancing devices (BD). Such devices are created on the basis of static batteries of capacitors, electromagnetic chokes and electronic devices. The symmetrizing components for induction heating devices (IHD) are very numerous, diverse. Different schemes are to varying degrees suitable for solving the problem of redistributing power between phases. The most widely used single-phase induction
load as a BD is the Shteinmets circuit with electrical connections. This is due to its simplicity and obviousness. Less commonly, Scott's scheme and other means based on electromagnetic devices are used. Singlephase induction devices operate in the mode of reactive power compensation, therefore it is preferable to consider options for balancing circuits with electrical connections [3, 4]. Only technical aspects of the load balancing problem are investigated in the article.
Formulation of the problem. In the study, it is necessary to develop a methodology for automated calculations and modeling, to create a schematic model of an induction device. Next, you should perform a computer analysis of the modes, obtain the characteristics of the metering when controlling the load and show the effectiveness of controlled electric heating. To solve the problem of increasing the flexibility of controlling the modes of induction equipment, it is advisable to show the components of an automated process control system (ACS TP) based on an industrial controller. The results of mathematical modeling should demonstrate the advantages and effectiveness of electromagnetic inductors. As a result of a numerical experiment, it is necessary to obtain a quantitative assessment of the asymmetry of currents, suitable for studying the comparative economic efficiency of various methods of heating aluminum.
Solution. One of the factors limiting the use of electric heating can be considered the complication of controlling the electrical modes of the power supply system of induction equipment when using technologies built using powerful asymmetric power consumers. The installed capacities of induction devices (ID) can range from hundreds of kilowatts to tens of megawatts [5]. In this case, the power supply of powerful inductors is often performed single-phase, since their three-phase symmetrical design is either not possible for structural or technological reasons, or impractical for technical and economic indicators. The voltages in the windings of the ID do not exceed the level of the standard of the distribution networks up to 1000 volts. Therefore, currents in linear and phase wires can reach tens of kiloamperes. The connection of a powerful single-phase load to a three-phase network results in a long asymmetric mode in currents and voltages, usually characterized by reverse sequence currents. Asymmetry sharply worsens the working conditions of electric machines and other workshop equipment, up to accidents, causes additional losses, reduces wire throughput [6].
Symmetrization of a powerful single-phase load must be considered with reference to serial inductors used in enterprises. In addition to the inductor, the power supply circuit of the induction installation con-
tains numerous switching devices, measuring instruments with sensors, elements of automatic control devices and telecommunications. Therefore, in the calculation of the modes, the power supply circuit in a three-line representation is simplified, limited by the components of the mode balancing. Distribution network equivalent circuit elements are not taken into account [7].
Industrial enterprises operate a large number of induction heaters of periodic and methodological action, among which modifications of inductors of the OKB type, series 894, 944, 970, 1080, etc. can be distinguished. Control devices, as a rule, are combined into a system called an automated control system. The degree of efficiency and accuracy of the automated control system depends on the project budget. Industrial microcontroller ACS, characterized by increased flexibility and reliability, have a high cost. Relay systems are much cheaper, but equally inferior in efficiency [8].
A general view of a single-phase electromagnetic inductor designed for extrusion of aluminum ingots and a simplified diagram of its connection to a three-phase network through a balancing device are shown in Fig. 2. The inductor is used in batch systems, has a short length, is made of a rectangular copper tube and is designed for liquid cooling. To regulate power and current, taps are provided in the winding. An induction coil is fixed between steel shields and has external ferromagnetic elements that act as screens. Inside the inductor there is a steel non-magnetic muffle with a slit and a lining, for insulation of coils.
The symmetry mode of a single-phase inductor was carried out for installation with the following parameters. The resistance of the inductor is equal to ri = 0.039 Ohm; xi = 0.174 Ohm, uncompensated power factor cos 9e = 0.219; the current in the inductor is Ii = 2130 A, the consumed active power is Pi = 177 kW. For the resonance mode, the capacitance of the BC of the compensating device (C = 16183 ^F) was determined at a temperature of T = 20 °C.
A typical circuit for connecting a single-phase inductor to the line voltage CA contains a power capacitors connected in parallel to the inductor coil to create a resonance of currents (Fig. 2, b).
a b
Fig. 2. General view of a single-phase inductor and a scheme for its balancing
The magnitude of the current in the induction wire The traditional balancing circuit contains a powerful can reach several kiloamperes, and the generated reac- battery of capacitors at voltage AB and a power inductive power in the winding can reach several megawars. tor with the ability to control the number of turns, at the
line voltage BC. Effective balancing of the mode in the triangle is ensured provided that the inductor is converted into a resistive load. In this case, as a rule, only one balancing element is subject to regulation, either a capacitor bank or a choke. Joint regulation is very difficult to coordinate, because there are an excessively large number of regime options. If it is necessary to increase the power of the inductor, a voltage-boosting transformer (VBT) is included in the circuit, designed to increase the operating voltage of the winding to the level of 0.5 kV.
An example implementation of an induction device is shown below. First, elements of an automated control system for the induction unit mode are considered.
1. Multifunctional power meter SELEC MFM384-R-C. The general view of the device and the main connection diagrams are shown in Fig. 3.
The device allows you to measure the rms value (RMS) of the current and voltage, power factor, active and reactive component, network frequency. The device supports the protocol for transmitting information via Modbus RTU (RS485). The presented configuration of the ACS allows you to build a simple relay-contactor control system for capacitor banks or use an industrial controller. The diagnostic capabilities of the asymmetric mode incorporated in the MFM384-R-C module are taken into account.
The SELEC MFM384-R-C Multifunction Power Meter monitors the change in reactive component and transfers this value to the PLC. The controller, in turn, compares it with the set values and, when rejected, gives the command to operate the switch the reactive elements of the capacitor bank modules.
6
Fig. 3. Hardware implementation of the primary measuring link
The micro-controller operation algorithm is programmed taking into account the patterns and limits of the change in the IHD mode (Fig. 8). In addition, provide the necessary measures of software-algorithmic and hardware protection.
2. Programmable logic controller (PLC). Practice has shown that with high requirements for the quality of process control, confirmed by the significant project budget, it becomes necessary to use a microcontroller. A general view of the industrial controller complete with analog and digital modules is shown in Fig. 4.
a
Fig. 4. Industrial controller
The practice of using universal industrial PLCs of well-known manufacturers, for example, Phoenix Contact, Allen-Bradley, Siemens, has shown their high efficiency. In the context of restrictions on the supply of foreign equipment, can consider the use of automation equipment of domestic production. The requirement of support in the technology under consideration for stable
data transmission protocols, operation algorithms, as well as modern, well-protected information networks and a convenient system interface seems quite natural.
3. Switching device (contactor, proximity electronic key, solid state relay, etc.). The circuitry and the algorithm for using the switching device should provide switching of high currents in the reactive elements,
especially when switching the regulation stages of induction devices. A description of the interaction order of the ACS modules, as well as the construction of an algorithm for transmitting control signals, may be of interest in the framework of another project, and has not been considered here.
In calculation practice, the estimation of asymmetric IHD modes in a three-phase network is carried out using the asymmetry coefficient of current kN, the asymmetric power factor of the loaded three-phase network kM and the power utilization factor of the symmetric elements ki, which is equal to the ratio of the load power to the total power of the balancing elements.
The asymmetry coefficients (fe) of the currents ki and voltages kau are of paramount importance. They are defined as the ratio of the modules of currents (voltages) of the reverse sequence I2U2) to the module of the component of currents of the direct sequence Ii(Ui):
kal = VI1, kaU = U2 /U1. (1)
In a symmetrical power supply system, there is no reverse sequence of alternating operational parameters in phases, therefore, the asymmetry coefficient is zero. The value of the mode parameters for the reverse sequence I2 and u2 characterizes the degree of deviation of the state of a three-phase electrical installation from a symmetric one [5, 9].
The power factor of an asymmetrically loaded distribution network of an industrial enterprise is determined by the formula:
f
kM = cos •
1+4
-A
-1
V
(2)
J
where cos^i is the power factor of the direct sequence, which in turn is determined by the expression:
f _V1
cos 9i =
1 +
(qab + Qbc + Qca ) (pab + Pbc + Pca )2
2
(3)
The power factor of a three-phase network with unbalanced loading is the product of two factors. The first factor cos^i, shows how many times the power factor km decreases due to the phase shift between voltages and currents [10]. The second factor is a function
of the current asymmetry coefficient and shows how many times the losses in the network increase compared to the losses that would occur when transmitting the same power with a symmetrical load distribution between phases.
The total power of the electrical installation, measured in kVA, is estimated by the expression.
S = J P2 + Q2 + N2 + T2
(4)
where P is the active power (kW), Q is the reactive power (kvar), N is the asymmetry power (kvar), T is the distortion power (kvar).
Only active power serves to do useful work. Taking into account the reactive power Q necessary for generating an alternating magnetic field in electric machines, one should strive to reduce the fraction of power determined by distortion of symmetry N and power T due to distortion of the shape of voltages and currents.
Obviously, with a significant change in the parameters of the induction device, the initially tuned system is outside the resonance of current. In this case, the asymmetry of the currents consumed from the source is extremely significant. To assess the degree of regime change, the concept of the depth of load regulation is introduced
^ = ^max /^min = ^max /rmin , (5)
The degree of influence of a single-phase load on a three-phase distribution network depends on a combination of factors and the most important are listed below: a) the unit power of an asymmetric power consumer, b) the power variation range during the study, c) voltage stability and system power at the connection point, d) co - power factor of an asymmetric load, e) duration of an asymmetric mode, e) number of asymmetric loads.
The study adopted idealized operating conditions for a single-phase induction installation up to 200 kW, when setting currents in resonance, for a continuous switching mode with a load variation range of up to 25%. The expressions for constructing the adjustment characteristics taking into account the depth of the change in the load and its phase are presented below [5, 11]:
IA(X) = lj4X2 - 6Â + 3/^3 , IC(X) = /a(2X- 1)/V3 .
(6)
IA(9) = Ia^ 1 • tg9h + tg29H , IC(9) = Ia^1 -V3 • tg9H + 3 • tg2^/V3 . (7)
Applying the obtained expressions, it is possible to construct the characteristics of changes in the currents of the balancing device when regulating the load
mode. In relative coordinates in fig. 6a shows the dependences of currents Ia, Ib, Ic on the depth of load regulation. In fig. 6b shows the dependences of currents on the load power factor.
a b
Fig. 5. Curves of changes in the currents of the BD when changing the mode of a single-phase ID
Нужно заметить, что представленные характеристики пригодны в использовании на этапе применения инженерных расчетных методик. Вместе с тем в схемотехническом моделировании удобнее использовать характеристики, представленные в абсолютных величинах. Поэтому в дальнейшем применяются схемные модели элементов с размерностью Ом, Гн, Ф, а настройку АСУ ТП выполняют по реальным резонансным характеристикам индукторов и по датчикам режимных характеристик распределительной сети.
It should be noted that the presented characteristics are suitable for use at the stage of application of engineering calculation methods. At the same time, it is more convenient to use the characteristics presented in absolute values in circuit simulation. Therefore, in the future, circuit models of elements with dimensions of Ohm, H, F are used, and the control system is configured according to the real resonant characteristics of the inductors and according to the sensors of the operational characteristics of the distribution network.
In engineering calculations, for the analysis of asymmetric modes, the method of symmetric components (MSG) and the techniques based on it are used [12]. When translating the approach using mathematical modeling, it is also possible to use MSC for a wide class of induction devices [4, 9]. However, the calculation of the mode in symbolic images or by instantaneous values seems more preferable if it is necessary to quantify the operational characteristics, since it allows you to sort through a large number of options with great speed and track the change in the state of the system in dynamics, as a set of established modes.
The use of the method of symmetrical components, as well as other methods for assessing the asymmetry of the ID, is certainly extremely useful [5], since
it gives an idea of the integral parameters of the installations and allows the use of engineering calculation methods. Such an approach is advisable in the analysis of aggregated operational indicators of a group of induction plants on a workshop scale. But for a detailed analysis of the current distribution in the induction installation, it is not possible to use this approach. It is more convenient to apply the analysis of the detailed description of the installation according to the circuit model. In this case, a numerical experiment is carried out in symbolic form, as well as in the time domain by instantaneous values for a given spectrum of harmonics.
The calculation of the electromagnetic modes of inductors is conveniently performed using software environments for mathematical modeling of electrical circuits [13, 14], as well as other computer, hardware and software equipment, including proven high-performance computing environments and developed model libraries suitable for automated analysis [15, 16]. The purpose of the study is to replace the engineering practice of calculations with mathematical modeling of the ID modes using detailed circuit models. The results can be obtained using other computational tools, as well as other models of different components of induction devices [17 - 19].
A schematic representation of the reactances and electrical models of controlled sources (CS) used in the simulation is presented in Fig. 6. Color highlighting of the component is used to set the initial ratio of directions and select characters in the matrix equations of the modified of analysis. The model of a voltage controlled voltage source (VCVS) is shown in Fig. 6 b The model is most widely used as a voltage sensor in arbitrary branches, since the primary branch is devoid of conductivity.
b
а
c
Fig. 6. Elemental basis of mathematical modeling environment
Можно показать пример построения матричного уравнения для емкости, причем во временной области матричное описание модели оказывается третьего порядка. Матрицу параметров формируют по компонентным уравнениям ветви:
1
We can show an example of constructing a matrix equation for capacity, and in the time domain, the matrix description of the model is of the third order. The matrix of parameters is formed according to the component equations of the branch:
Vm(t) - Vn(t) -1JiC(t)dt = 0, - Vm(t) + Vn(t) +1J¿C(t)dt = 0 .
c-
(8)
Then the structure of the model is described by a matrix expression in the time or symbolic domain
m n ¡c
m 0 0 1 iMt )] 0
n 0 0 -1 Vn(t ) = 0
iC -1 1 1J dt ic(t ) 0
(9)
For the schematic image of VCVS shown in Fig. 6, b, the mode of transmission of electricity through a controlled source is determined by the expression:
«2(0 = kui(t) = k [Vg(f) - Vh(t)] = Vr(t) - Vn(t), (10)
where: ui(t), m(t) is the input and output voltage of the VCVS, k = ku is the transfer coefficient.
Often, a similar source is used as a dividing element in circuits where it is required to ensure the conditions of galvanic isolation. For example, we can show
the record of the component equations and the structure of the mathematical model of the current source (Fig. 6, c), controlled by current (CSCC), which have the following form:
Vg (t) - Vh(t) = 0, i2(t)=kI • iit).
i2(t) - kI • i! (t) = 0.
(11)
The system of extended nodal equations for a CSCC is compiled taking into account the controlling and controlled branches.
g h r n i1 i2
g 0 0 0 0 1 0 v )! 0
h 0 0 0 0 -1 0 Vh(t ) 0
r 0 0 0 0 0 1 Vr(t ) 0
n 0 0 0 0 0 -1 Vn(t ) 0
i1 1 -1 0 0 0 0 i1(t ) 0
i2 0 0 0 0 - kl 1 i2(t ) 0
(12)
You may notice that behind each controlled source there is a fifth and higher order model in the described language basis, therefore the order of the resulting system of equations is very high. In computer space, the formation of a matrix description is performed automatically and has no special features. The control module, by default, embeds the necessary stamps in the general description of the model generated before performing the calculations. For machine-oriented computational methods, the dimension of systems of equations is not decisive.
An example of a simplified model of a singlephase ID in a three-phase network, constructed using the described elemental basis, is shown in Fig. 7. A numerical study of the established and transient modes of
an induction installation according to the model of Fig. 7 perform as a multivariate computing project.
In addition to a file with a textual description of the macromodel, in the English-language directory, all the necessary routines, databases, libraries, command and auxiliary modules are grouped. Inside the file, tasks ensure the availability of ordered links to plug-in models and libraries, observing the language syntax and applying the necessary comments. The computational project is programmed for a given number of stages of calculation and computational operations. The calculation results are shipped to tabular files and are quickly monitored during the calculation. When using optimization procedures, a computing project can be completed ahead of schedule, immediately after obtaining the desired result.
Fig. 7. A simplified model of a single-phase IHD with Steinmetz balancing
As a result of the mode simulation and automatic and corrective actions and control algorithm settings
control of the state of the parametric models of the BD, are generated. The corresponding vector diagrams for
families of vector diagrams are obtained by which a the set of modes of the induction device, when balanc-
measure of the efficiency of automation is evaluated, ing the ID, are presented in Fig. 8.
=2100 A"
Fig. 8. VD settings of the ID when symmetrizing by a capacitor and symmetrical components
The symmetry of the battery capacitors, corresponds to the vector diagram shown in Fig. 8 a. The diagram is based on a symmetrical triangle of linear stresses. Current vectors are located in nodes of an equilateral triangle of network voltage. The diagrams of asymmetric consumption currents (Fig. 8, b) and their symmetric components (Fig. 8, c) were separately enlarged. In practice, the balancing is carried out precisely by regulating the capacitance, due to the convenience of parallel connection or disconnection of the required number of capacitors. Capacitive regulators are quite flexible to operate and are characterized by small losses.
The order of regulation of the electric mode can be explained by the vector diagram (VD). By increasing the capacitance (Fig. 8, a), the length of the vector IbC, which moves along the dashed line, is increased. The value of the phase shift A9 becomes equal to zero when moving from the position of asymmetry to symmetrical consumption. With precise regulation, the single-phase
4 + Ia - lCb = 0, " 4 + 4r " 4 =
mode of the inductor goes into resonance at a current of about 2100 amperes.
The balancing device is configured in such a way that the equivalent current is -In according to the designations in fig. 2b is formed by the sum of the resonant current of the inductor IID and the current ICc of the capacitance of the compensating capacitor bank. Since a single-phase induction load is connected between phases A and C, insofar as the electric equilibrium formulas constructed according to the first Kirchhoff law, currents Ia and Ic of the same phases participate together with the load current. The reference node for the resonance currents selected node C. Therefore, the adjustment procedures with the capacitor bank current ICb in the diagram are tied to nodes A and C. If the inductor current Idr remains constant by adjusting the current of the BC capacitance, tuning is performed in which the current vector is moved by a parallel shift along the dashed line in each of nodes A and C.
In this case, the current vectors Ia and Ic change the phase angle, taking a position that is ideally symmetrical for a three-phase system.
0 , 4 - llD - ICc = 0, Icb + 4 - 4r = 0. (8)
A similar state of the VD is typical for the drift of the operating parameters of the IHD during heating of the load in a batch inductor. At the same time, in practice, the magnitude of the phase shift varies in a slightly smaller range than that shown in the VD (Fig. 10). An
analysis of the control parameters during heating is discussed below. For a set of single-phase ID modes, when balanced by a choke, the vector diagrams shown in Fig. 9.
y 'Cb V a<P
Fig. 9. The settings VD of the IHD mode when balancing by the throttle
The node A was chosen as the reference node for configuring the resonant currents. The magnitude of the inductor current during regulation changes, and the vector Idr itself moves along the dashed straight line shown in Fig. 9. The equations of electric equilibrium (8) written down above fully reflect the stepwise change in the controlled quantities reflected in the vector diagram. The extreme positions of the vectors correspond to the dynamics of the IHD mode during heating from start to finish. In this case, the value of A9 noted in the diagram is somewhat larger than the phase shift of the current and voltage at the end of heating. The fact is that the presented vector diagrams are more suitable for displaying a qualitative comparison of regime parameters.
The construction of hodographs can be considered a very obvious way of presenting the results of analysis. A quantitative assessment of the ratio of operational parameters should be carried out using a tabular presentation of the results obtained by modeling. An even more effective means of evaluating the results is to build graphical dependencies.
To compare the technical analysis schemes with the traditional approach to the analytical mode, below are the results of calculating the set of established parameters with symbols of symmetrical components. The calculation is based on a vector diagram of a single-phase inductor, shown in Fig. 8. With a resonant current of about 2.1 kA in the inductor winding with a power of 180 kW, the mode of balancing with the inductor is deliberately violated. In this case, the currents of phases A and B are somewhat different from the current of phase C (Ic = 355 A), which remained without a shift. In this case, phase shifts of A^a and A^b currents turn out to be very significant. Moreover, the equivalent phase currents of the asymmetric mode Ia and Ib, as well as the phase C current, are obtained in
the form of a geometric sum of the same symmetrical components of the forward and reverse sequence:
1 a = Ia1 + Ia2, Ib = Ib\ + Jb2, 1c = hi +Ic2
(9)
Thus, the application of the method of symmetrical components, obviously quantitatively confirms the conclusion about the degree of resulting asymmetry of one steady-state regime of the induction system. The judgment on the totality of intermediate regimes requires an additional in-depth study, under which an updated program of the numerical experiment is formed and massive calculations are performed.
Through simulation, it is easy to estimate the degree of distortion of the power supply mode with erroneous balancing. Parametric modeling tools allow you to control both smooth and stepwise increase in the current asymmetry coefficient. The duration of the heating mode remains unchanged, about 80 s. Judging by the vector diagram (Fig. 8, a), the distortion of the currents in phases B and C is determined by the integral value of the asymmetry coefficient, determined by the formula (1)
kiiI = I2/11 = 72-100/351 = 20,5%.
Obviously, the value of the asymmetry coefficient obtained for the end of heating is more than four times higher than the limits established by the requirements of GOST [20]. When setting the task of analyzing the dynamic characteristics of the asymmetry coefficient, the program of the computational experiment with the model is accordingly changed and tables with calculated values of the current asymmetry coefficient are obtained, according to which a regularity is formed in the form of graphical dependencies. An example of a family of characteristics of the asymmetry coefficient of the currents of an induction installation is shown in Fig. 10.
0 10 20 30 40 50 60 70 t, c 80
Fig. 10. Characteristics of drift the asymmetry coefficient during heating
A significant increase in the current asymmetry coefficient, from the beginning to the completion of heating, is shown for four cases of tuning the microcontroller ACS. The reason for the excessive increase in asymmetry can be considered not quite correct tuning of the control system. For curves 1, 2, 3, 4, the range of variation of the parameters of both balancing devices, which are simultaneously regulated, was set by 3, 6, 9, 12 %, respectively. The result obtained in the graphs shows that the adjustment of the BD taking into account changes in the parameters of the inductor [21] should be much more accurate.
It is noteworthy that only characteristic 5 in Fig. 10 meets the requirements of GOST. It was obtained with an accuracy of tuning of the balancing capasitors to a value of about 2% with a constant state of the throttle. Therefore, we can assume that the marked accuracy of the BD settings in a given corridor of changes in the electrophysical parameters of the inductor is enough to ensure the proper quality of the induction installation mode. You may notice that for other parameters of the IHD, other modeling systems, and other settings of the modes of the balancing devices, the results may differ.
Conclusion. The presented results of modeling the modes of induction equipment demonstrate quite wide possibilities for the analysis of asymmetric induction complexes. No less opportunities are provided by modern means of monitoring and control of technological modes, especially those built on the basis of industrial microcontrollers. The accuracy of the research results is enhanced by the construction of efficient models and computational algorithms that take into account the asymmetry of currents and voltages, causing the generation of distortion power in the system, as well as the influence of higher harmonics. Given the priority of environmental factors, the use of induction heating in the extrusion of aluminum, instead of gas, can be considered uncontested, even taking into account the need, to overcome the consequences of asymmetric modes of inductors and power supply systems.
References
1. TanHH A. A., KnHeB E.C. [Two-phase induction load in a three-phase distribution network]. Dvu-khfaznaya induktsionnaya nagruzka v trekhfaznoy raspredelitel'noy seti. Energy security and energy saving, № 1, 2020, pp. 42-51. ISSN 2071-2219. DOI: 10.18635/2071 -2219-2020-1 -42-51. (In Russ.)
2. Kinev E., Tyapin A. Improving the quality of electricity of a metallurgical enterprise by balancing the induction load. Norwegian Journal of development of
the International Science. Oslo, 2020, No 40. Vol. 1. pp. 33-40. ISSN 3453-9875.
3. Sluhotsky, A.E., Nemkov V.S., Pavlov N.A., Bamuner A.V. [Induction heating plants]. Ustanovki induktsionnogo nagreva. L.: Energy Publishing House, 1981. 328 p. (In Russ.)
4. Aliferov A.I., Lupi S., Forzan M. [Electrotech-nological installations and systems. Induction heating plants]. Elektrotekhnologicheskie ustanovki i sistemy. Ustanovki induktsionnogo nagreva. Tutorial. Novosibirsk: NSTU, 2017. 160 p. (In Russ.)
5. Gitgarts D.A., Mnukhin L.A. [Balancing devices for single-phase electrothermal installations]. Simmetriruyushchie ustroystva dlya odnofaznykh el-ektrotermicheskikh ustanovok. M.: Energy, 1974. 120 p. (In Russ.)
6. Shidlovsky A.K., Moskalenko G.A. [Balancing devices with transformer phase-shifting elements]. Sim-metriruyushchie ustroystva s transformatornymi faz-osdvigayushchimi elementami. Kiev: Naukova Dumka Publishing House, 1981. 204 p. (In Russ.)
7. Danilushkin A.I., Zhivotyagin D.A. [Energy-efficient modes of operation of induction heaters in a metal processing line]. Energoeffektivnye rezhimy raboty induktsionnykh nagrevateley v tekhnolog-icheskoy linii obrabotki metalla. In the book: Prospects for the development of processing technologies and equipment in mechanical engineering. Collection of scientific articles of the 4th All-Russian scientific and technical conference with international participation. Kursk: Company «University Book», 2019. pp. 82-85. (In Russ.)
8. Shidlovsky A.K., Novsky V.A., Kaplychny N.N. [Stabilization of electric energy parameters in electric networks]. Stabilizatsiya parametrov elektrich-eskoy energii v elektricheskikh setyakh. Kiev: Nau-kova Dumka Publishing House, 1989. 312 p. (In Russ.)
9. Fomin N.I., Zatulovsky L.M. [Electric furnaces and induction heating plants]. Elektricheskie pechi i ustanovki induktsionnogo nagreva. M.: Metallurgy, 1979. 247 p. (In Russ.)
10. Tyapin A.A., Kinev E.S. Topological approach to the assessment of electromagnetic asymmetry of the MHD machine. Wschodnioeuropejskie Czasop-ismo Naukowe (East European Scientific Journal). Warsaw, Poland. 2019, No 2(42), part 1, pp. 40-47.
11. Tyapin A.A., Kinev E.S. [The choice of inverter circuit for a linear metallurgical MHD machine]. Vybor skhemy invertora dlya lineynoy metallur-
gicheskoy MGD-mashiny. Electric equipment: operation and repair. № 3, 2020. pp. 23-29. ISSN: 20749635. (In Russ.)
16. Kinev E.S., Tyapin A.A., Yefimov S.N. [Assessment of the asymmetry of the induction machine with the parameters of symmetrical components] Bulletin of the Voronezh State Technical University. 2018. T. 14, No. 6, pp. 68-79. (in Russ.)
13. Chua, L.O. [Machine analysis of electronic circuits: Algorithms and computational methods]. Mash-innyy analiz elektronnykh skhem: Algoritmy i vychislit-el'nye metody. Per. from English / L.O. Chua, Lin Pen-Min. M.: Energy, 1980. 640 p. (in Russ.)
14. Chua L. O, Desoer C., Kuh E. Linear and nonlinear circuits / McGraw-Hill, New York. 1987. -839 p. ISBN 0-07-010898-6.
15. Aliferov A.I., Bikeev R.A., Goreva L.P. [A software package for calculating inductive resistances of rectilinear conductor systems arbitrarily located in space]. Programmnyy kompleks dlya rascheta in-duktivnykh soprotivleniy sistem pryamolineynykh provodnikov, proizvol'no raspolozhennykh v pros-transtve. In the collection: Electrical Engineering, Elec-tromechanics and Electrotechnology. EEE-2009 materials IV scientific and technical conf. with international participation. Novosibirsk: NSTU, 2009. pp. 206-212. (in Russ.)
16. Bazarov A.A., Danilushkin A.I., Danilushkin V.A., Vasiliev I.V. [Modeling of electromagnetic processes in a multilayer three-phase induction cylindrical system]. Modelirovanie elektromagnitnykh protsessov v mnogosloynoy trekhfaznoy induktsionnoy tsilindricheskoy sisteme. Bulletin of the Samara State
Technical University. Series: Engineering. 2017. № 3 (55). pp. 50-60. (in Russ.)
17. Aliferov A.I., Bikeev R.A., Goreva L.P., Blank A.V. [Study of electrical parameters of circular ferromagnetic conductors]. Issledovanie elektricheskikh parametrov ferromagnitnykh provodnikov kruglogo secheniya. News of higher educational institutions. Electromechanics. 2012. № 6. pp. 59-61. (in Russ.)
18. Lupi S., Forzan M., Aliferov A.I., Meleshko A.A. [The active resistance of the inductor when in-auction heating of the inner round surfaces]. Aktivnoe soprotivlenie induktora pri induktsionnom nagreve vnutrennikh tsilindricheskikh poverkhnostey. Electrical Engineering. 2010. № 5. pp. 43-47. (in Russ.)
19. Kuvaldin A.B. [Features of the calculation of the parameters of the electromagnetic field in ferromagnetic steel]. Osobennosti rascheta parametrov el-ektromagnitnogo polya v ferromagnitnoy stali. Industrial electric heating and electric heating. 2014. № 2. pp. 26-30. (in Russ.)
20. GOST 13109-67. Electric Energy. Electromagnetic compatibility of technical equipment. Quality standards for electric energy in general-purpose power supply systems. M.: IPK publishing house of standards, 1999. Srandartinform, 2006. 32 p. (in Russ.)
21. Bazarov A.A., Pavlova O.E. [Electrotechnical complex for heating ingots with uniform energy consumption]. Elektrotekhnicheskiy kompleks dlya nagreva slitkov s ravnomernym energopotrebleniem. In the book: Modern Russian equipment to improve the reliability of efficiency and safety of the energy complex of Russia. Collection of scientific papers. Samara: Samara State Technical University, 2019. pp. 6-9. (in Russ.)
