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10КОРРЕКЦИЯ ХАРАКТЕРИСТИК ТРЕХФАЗНОГО ИНДУКТОРА С ИСПОЛЬЗОВАНИЕМ
ЛОКАЛЬНЫХ РЕЗОНАНСОВ
Кинев Е.С.
к.т.н., директор,
ООО Тепловые электрические системы, г. Красноярск, Россия
Тяпин А.А. аспирант,
ФГОУ ВО Сибирский федеральный университет, г. Красноярск, Россия
THREE-PHASE INDUCTOR PERFORMANCE CORRECTION USING LOCAL RESONANCES
Kinev E.
Ph.D., director of Thermal Electrical Systems LLC,
Krasnoyarsk, Russia Tyapin A.
Postgraduate student, Siberian Federal University,
Krasnoyarsk, Russia
Аннотация
В статье рассмотрены особенности расчета несимметричного установившегося режима трехфазного индукционного нагревателя алюминиевых слитков, по результатам которого предложена коррекция схемы индуктора. Посредством специального схемного решения удается обеспечить требуемую неравномерность распределения токов и мощности по трем секциям трехфазного индуктора при симметричном электропотреблении индукционного комплекса по фазам. Предложенная схема включения секций индуктора при условии создания локальных резонансных режимов, позволяет избежать применения вольтодобавоч-ных трансформаторов. Достигнутые по итогам коррекции режимных характеристик результаты представлены на векторных диаграммах токов и напряжений.
Abstract
The article discusses the features of calculating the asymmetric steady state of a three-phase induction heater of aluminum ingots, according to the results of which a correction of the inductor circuit is proposed. By means of a special circuit solution, it is possible to ensure the required uneven distribution of currents and power over three sections of a three-phase inductor with symmetric phase-state power consumption of the induction complex. The proposed circuit for switching on sections of the inductor, provided that local resonant modes are created, avoids the use of boost boost transformers. The results achieved by the correction of operational characteristics are presented on the vector diagrams of currents and voltages.
Ключевые слова: несимметричная трехфазная нагрузка, индукционный нагрев, установившийся режим, векторная диаграмма токов, коэффициент несимметрии, гибридный анализ.
Keywords: asymmetric three-phase load, induction heating, steady state, vector diagram of currents, asymmetry coefficient, hybrid analysis.
Introduction. Three-phase inductors of a longitudinal magnetic field of industrial frequency are widely used for heating cylindrical aluminum ingots in the press industry [1]. Under conditions of mass production, the three-phase switching on of induction units provides the resulting efficiency of converting electricity to heat, since unit capacities of inductors can reach values of 1 MW or more [2]. The operating modes of inductors and technological requirements are diverse. The greatest power, as a rule, is concentrated in the output section of the induction heater. Therefore, despite the three-phase nature, the consumption currents in phases are not the same, and the load is asymmetric [3]. A certain contribution to the resulting asymmetry of power consumption is made by the drift of the parameters of the induction system during heating [4]. Changing the parameters of some installations can reach 1520% [5]. Such modes can cause an increase in losses, as well as a deterioration in the operation of electrical machines and workshop equipment, up to the emergency state [6]. With limited power of the distribution network, phase asymmetry sharply worsens the quality
of electricity, which may require reconstruction of the power supply system.
To quantify the asymmetry of currents and voltages, measurements are used, the results of which develop measures to overcome the inappropriate condition of the power supply systems of induction equipment [7]. In addition to measurements, mathematical modeling is used as a modern means of studying electromagnetic modes, which is highly efficient and safe when creating extreme equipment operating conditions [8]. Conducting a numerical experiment as an alternative to the physical one, for numerous reasons, seems extremely beneficial [9]. However, for its implementation, it is necessary to equip the hardware and software modeling tools with adequate mathematical models that can reliably calculate the magnetohydrodynamic, electrical, electrothermal and electromagnetic modes of modern equipment [10].
Formulation of the problem. From the condition of uneven distribution of power among sections of a three-phase inductor, the result inevitably follows -asymmetric power consumption in phases. To create
extreme currents in the sections of the inductors, resonant modes are applied, which are tuned by regulating powerful capacitor banks. In addition to capacitor banks, typical schemes of induction plants often contain transformer voltage regulators, so the power supply circuit of the device is complicated. At the same time, the circuitry of induction equipment can be modified in order to create symmetrical power consumption while observing the given inequality of the winding currents and the extreme current of the output section [11]. To achieve this, a combined connection of sections should be applied, redistributing the local resonant modes between the power windings. The power factor of a three-phase induction device (ID), in the course of improvement, should be provided with the greatest possible current resonance conditions.
Solution. Given the complexity of calculating the asymmetric modes of three-phase installations, when solving the problem of modifying the circuitry of induction equipment, machine-oriented automated tools should be used that allow you to quickly monitor the result of changing the connection circuit of windings and compensating devices [12]. The choice of a modeling system can be crucial to obtain a simple and effective result [13]. In addition, it is necessary to have a complete understanding of the circuitry, operating and technological parameters of the entire complex of equipment of the induction installation to be modernized [14].
A general view of a three-phase induction heater for the extrusion of cylindrical aluminum ingots is shown in Fig. 1.
Fig. 1. General view of a three-phase induction heater
The presented three-section inductor has water cooling and is intended for heating large-sized cylindrical aluminum ingots. As part of the technological equipment, the inductor is equipped with an automatic control system (ACS), which provides a rhythmic and uninterrupted mode of operation of a complex of mechanisms and devices involved in the methodical heating and operation of the press. The duration of the heating cycle of the aluminum load with one inductor is 100 -250 seconds and is built into the cycle of the press. The
inductor is designed for three-phase connection and is used to heat large ingots before pressing [15]. It is distinguished by a very significant power consumption, about 100 kW. Using voltage boosting, the total power of the induction installation can reach 0.3 MW.
A typical scheme for connecting a three-section three-phase induction heater (Inductor 1, Inductor 2, Inductor 3) to the distribution network of an industrial enterprise is shown in Fig. 2.
Fig. 2. Simplified connection diagram of a three-phase inductor
The size of a computer model can be quite large, so it's very convenient to build a description using links to individual modules declared as macro models. The circuit designs declared in the description are read into the buffer in the order of their mention in the control module and added to the global description generated in the main memory [11]. The description of nonlinear models has features, and their iterative calculation can be lengthy. The order of the systems of equations for
linear problems is limited only by common sense, as well as by the possibility and convenience of viewing the circuit image, and does not affect the calculation time, even when the optimization module is connected [1].
The circuit model of the inductor for the analysis of the electromagnetic mode is shown in Fig. 3. The model description is generated in the header file in ASCII code.
38 The scientific heritage No 48 (2020)
-Ç)-
Fig. 3. Structure of the circuit model of a three-phase induction device
The model of a three-phase induction heating installation (Fig. 3) is constructed using bipolar and multipole circuits used in most circuit simulators. The image of the elements adopted in the modeling system corresponds to domestic traditions and differs somewhat from models with three connection nodes from foreign programs [16]. The accepted interpretation of the distribution within the model of currents and potentials seems more convenient and fairly obvious.
" Vm(t) + Vn(t) + Rk ■ iR(t) = 0
m
HOT
In the software environment for mathematical modeling of induction devices, the method of modified nodal equations is programmed [1]. The conjugation in a single computational space of traditional topological equations for the regular part of the circuit and component equations for ideal branches and controlled sources allows us to obtain a universal tool suitable for matrix implementation and practical programming [11]. The software environment describes in detail the extremely useful standard four-pole components of the basis of circuit theory, which are called controlled sources. Using standard elements programmed in the shell, it is possible to create multipole models of high
When forming a system of modified nodal equations, typical matrix stamps of elements from the standard library of author software are used. For example, for a resistive element connected to arbitrary nodes m and n, the matrix stamp may be second or third order. It is formed according to the component equations of the branch-conduction or branch-resistance, in the temporal or symbolic domain.
m " 1/R k - 1/R k ' ~Vm (t ) 0
n - 1/R k 1/Rk _ Vn (t)_ 0
m n 'R
m 0 0 1 |>m(t)l 0
n 0 0 -1 Vn(t) = 0
'R -1 1 R k 'R(t ) 0
(1)
(2)
complexity. Therefore, the creation of circuit models of induction devices, if there is information about the structure of the circuit and the algorithms for the operation of the links, is not very difficult.
The structure of circuit models for controlled elements is shown in Fig. 4. The color highlighting of the components is used to set the initial ratio of directions and the choice of signs in the matrix equations of the modified nodal stamp. The model of a voltage controlled current source (VSIC) is shown in Fig. 4 a. Such a model is most widely used as a current sensor in arbitrary branches, since the primary source branch is ideal and does not have resistance.
n
b c
Fig. 4. Elemental basis for the environment of mathematical modeling
The transmission mode for the managed source VSIC is determined by the expression:
u2(t) = k il(t) = Vr(t) - Vn(t), (4)
where: ii(t) is the input current of the controlled source, u2(t) is the output voltage, k = kR is the transfer coefficient equivalent to the transition resistance.
When forming a system of matrix equations, a stamp of a controlled EMF source obtained by component equations is used.
V (t) - Vn (t)+kR • 1 (t)=0, Vg (t) - Vh (t) = 0.
The matrix system of equations for a controlled source of VSIC is shown below
g h r n '1 iz
g 0 0 0 0 1 0 Vg(t )! 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
'1 -1 1 0 0 0 0 i1(t ) 0
iz 0 0 -1 1 kR 0 iz (t) 0
The presented system is of the sixth order and is formed taking into account the regime of the control and controlled branches. For all sources according to fig. 4, matrix equations are automatically generated that are embedded in the global model description [17]. For non-standard elements, such as triples, quadrupoles, multipoles, for example transformers and mutual inductances, macromodels are formed manually or use ready-made ones from the standard library. The structure of macromodels can have a different degree of idealization, different complexity, and, accordingly, a very different order of the matrix structure. Moreover, each model certainly fulfills the prototype device algorithm [18].
When using a voltage boost in the transformer model (Fig. 3), the transformation coefficients are initially taken equal. Having performed a computational experiment, a steady-state symmetric mode is obtained. Then, model tuning is performed by adjusting the transformation coefficients and performing a second count. At the command of the control module of the numerical experiment, the transition to the calculation of the asymmetric mode is performed. The simulation is programmed for a cyclic iterative calculation, carried out after replacing the models of booster transformers of the middle and output phase, on the model with a high transformation ratio. Thus, a new calculation task is quickly generated and a group of files is recalculated for a new task configuration.
The calculation results of the asymmetric electromagnetic mode of a three-phase induction heater in the form of a vector diagram (VD) are presented in Fig. 5.
To display the resonant currents of each parallel circuit, the traditional method of placing vectors at the vertices of a symmetrical triangle of linear stresses is applied. Judging by the vector diagram, the known properties of a parallel LC circuit in each phase are reflected by a corresponding increase in currents in adjacent branches. The magnitude of the current and the temperature intensity is set by the value of the equivalent voltage on each section of the inductor created with the participation of a boost transformer.
In this case, there is an additional possibility of changing the resonance quality by regulating local capacitor banks [19]. However, the mutual influence of the phases does not allow fully independent regulation, due to the immediate onset of the consequences of interference in the form of a distortion of the regime of other sections.
Obviously, for uneven distribution of power between sections of the induction device, one has to resort to the regulation of transformation coefficients for voltage boosting in all phases. The diagram shows that the consequence of such regulation is a sharp discrepancy between the phase current vectors Ia = 474 A, Ib = 332 A, Ic = 540 A consumed by the inductor and the asymmetry of the power supply mode, with a resonant current of the output section of about 1.8 kA. A numerical study of asymmetry using the considered models allows us to obtain a rigorous quantitative assessment of the deterioration of operational parameters at all stages. The measure of this deterioration, formalized in the al-
d
a
gorithm for evaluating comparative economic efficiency, serves as the basis for a decision, for example, on the reorganization of production.
Fig. 5. Vector current diagram for unbalanced mode
The task of analyzing the modes of the induction device has many components, since in addition to ensuring energy efficiency, there is a need for the proper quality and rate of heating of the load. The total duration of the extrusion cycle of the batch of ingots, as well as the absence of accidents during accidental melting of the ingot or other unproductive downtime of the press, provide the final economic efficiency of production.
It can be assumed that the study of the electromagnetic modes of the inductor and the characteristics of the temperature field created by heating in the load can be performed separately and alternately [1, 11]. In this project, the analysis and synthesis of components that provide the necessary electromagnetic mode of the induction installation, by upgrading the design of the inductor and selecting loaded operating modes [20]. Modeling the nature of the electromagnetic and thermal fields in the load can be performed according to recommendations from the literature [21]. At the same time, it is advisable to separate the study of the temperature distribution in the ingot and the analysis of the resulting
economic efficiency of the equipment into separate projects.
There is a technical solution to ensure the required distribution of currents between sections with symmetrical power consumption of the ID in phases [3, 22]. The proposed circuitry solution provides the modernization of the heating installation by combining sections, as well as creating local resonances in the windings. The switching circuit of the inductor coils, using local resonant modes, is shown in Fig. 6.
The presented scheme, with a sequential axisym-metric arrangement of coils, is implemented in the methodical method ID. The direction of the currents, represented by a cross and a dot on the conductor cross-section, corresponds to the inversion of the current phase in the middle section C-B, due to which it is possible to elementarily reduce the phase shift of the currents of adjacent sections of the three-phase system to n/3.
A*M Bftfe Cêic
Fig. 6. Diagram of the inclusion of the inductor when upgrading the induction device
It should be noted that the actual phase shift of the currents is somewhat different, due to the mutual influence of the phases [3] and this complicates the picture of the magnetic field at the junction of the sections. Nevertheless, in the course of modernization, one should strive to further reduce the phase shifts between the currents of adjacent sections, since this serves to level the picture of the thermal field in the aluminum ingot [11]. The circuitry of a three-phase connection of the windings in a triangle, more suitable for the analysis of switching sections and capacitor banks is shown in
A
Fig. 7. It is more visual, since the diagram in Fig. 8, insufficiently clearly reflects the essence of the technical solution, based on the combined connection of the inductor and BC sections using resonances.
The impedances of the branches containing the windings 1, 2, 3 of the induction device for through heating of ingots are different. Therefore, to create the symmetry of currents in phases, the number of turns (w) and the cross-section of wires (S) in sections 1, 2, 3 are selected according to the conditions w1>w2>w3 and
s,<s2<s3.
Fig. 7. Scheme of a modified three-phase induction devic
The first ID coil connected to line voltage AB. It is connected to the first capacitor bank (xC4) in series. The middle coil is connected to the line voltage BC. It is connected to two capacitor banks - in series and parallel (x с5, X C 5). The output section of the inductor carrying the main load is connected to the interfacial voltage CA. It has a parallel branch in the form of an increased voltage resonant capacitor bank (xc6). Typically, for the third stage, capacitor banks with a rated voltage of at least 660 volts are used, which make it possible to increase the voltage by 1.73 times.
The complex current vector Iab coincides with the line voltage vector Uab, since the reactive impedance of the branch is zero. Thus, the current in the coil of section 1 has a phase of 91 = 30°. The current vector
Ica = I'ca + I"ca also coincides in phase with the linear voltage Uca, while the current complex of the coil of section 3 (I'ca) is behind the voltage complex Uca by the angle 9ca- Consequently, the phase of the current is equal to 93 = 150° - 9ca. The current vector of the second section Ibc = I'bc + I"bc also coincides in phase with the voltage Ubc, while the current of the coil 2 (I'bc) lags the voltage Ubc by the angle 9bc = 9ca/2. In addition, winding 2 is inverted, so the current phase has a value: 92 = 270° - 180° - 9bc. Thus, the phase shift between currents in neighboring sections of the ID turns out to be less than 60°:
912 =92 - 91 = 60° " 9bc < 60°, 923 =93 "92 = 60° " 9ca + 9bc = 60° " 9bc < 60°
(10)
Judging by the scheme (Fig. 7), when the inductor sections are switched on combined, not only the current resonance is used locally, but also the voltage resonance, the use of which leads to the required resulting current distribution. However, in the practical design of the circuit diagram, one should remember the properties of series resonance, which are especially harmful
in relation to higher voltage harmonics, and provide for appropriate protective devices.
The circuit model of a three-phase induction heater with a combined inclusion of windings is shown in Fig. 8.
Fig. 8. Circuit computer model of the upgraded induction device
The circuit model generated for the presented technical solution (Fig. 8) is much simpler than the model shown in Fig. 3. This is due to the exclusion of multipolar macromodels used to account for transformer links for nonlinear objects. Dynamic branch currents are controlled using sources EI51 - EI54, EI58, EI59. The electrical distribution point is represented by non-ideal sinusoidal sources ES1 - ES3, connected in a star. The logic of constructing a computational model with this approach seems more obvious, although the calculation of the asymmetric mode, as before, proceeds in machine implementation. The analytical solution of systems of equations for a model above the seventh order is irrational [17]. In addition to current sensors built using VSIC controlled sources, the model diagram contains voltage sensors implemented on VSVC sources EU55 - EU61. When generating a model, it is advantageous to take care in advance of introducing elements suitable for different modes of machine analysis. At different stages, in addition to studying the steady state, a frequency analysis, calculation of dynamics by instantaneous values, or the formation of coefficient matrices for equations of state variables may be needed.
Multiple model corrections during a numerical experiment adversely affect the archiving of large arrays of regime information, which have to be dynamically renamed, and then converted to the necessary formats when accessing third-party graphical media. The task of managing the flow of information is extremely urgent here, so the planning of a numerical experiment should be performed as carefully as possible.
The results of numerical simulation of a modernized induction heater with combined switching on of windings and local resonances of sections according to the model of Fig. 8 are shown in the vector diagram, fig. 9. Analysis of the distribution of currents in the vector diagram presented in Fig. 9 allows us to conclude that the requirement formulated for the formulation of the problem to ensure various currents in sections of a three-phase inductor is fulfilled. According to the scheme (Fig. 8), the currents of the inductor windings are different in magnitude. The electric current Iab of the first section is smaller than the others. The current of the second section is larger in magnitude, has the name I 'bc, and a minus sign, according to the phase inversion condition. The current value of the third section of I 'ca is larger than the others, and its value is set by a capacitor bank in parallel resonance.
The obvious arrangement of the currents of adjacent sections with a phase shift of about tc/12 (912 « 923) allows us to state that the desired positive effect was obtained - the phase shift between the currents was reduced. This effect helps to improve the uniformity of heating of the load at the joints of the sections [3]. However, the quantitative characteristics of the uneven distribution of temperatures in the load require clarification in solving the thermal problem. It is noteworthy that the vectors of phase currents Ia, Ib, Ic, consumed from the network, in the presented circuit solution are completely symmetrical.
Fig. 9. VD for the mode of the modernized induction installation
Thus, the requirement specified at the stage of the statement of the problem and the subsequent optimization of the induction device mode has been met to ensure the resistive nature of the load, since there are no phase shifts between the same linear voltages and currents.
Conclusion. The result of the modernization project for the three-phase installation of induction heating of aluminum ingots, was the development of a circuit for combining sections, with series-parallel connection of capacitor banks to create local resonant modes. The application of a software environment for automated modeling of the set of electromagnetic modes of induction equipment is proposed. The article provides a solution to the problem of machine analysis of electromagnetic modes of a modernized induction installation. Using vector diagrams, it is shown that the required asymmetric mode of sections can be provided, with symmetrical current consumption from the distribution network, under conditions of effective reactive power compensation. The application of the new solution eliminates the use of booster transformers, and the regulation is performed by a resonant capacitor bank.
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