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12(рис. 7) символами представлено изменение фазы: «.» - фаза не менялась; «+» - увеличение фазы на Т / 2; «-» - уменьшение фазы на Т / 2; «г» - изменение фазы на ТТ . Величина изменения фазы однозначно определяет битовое состояние двух подканалов: синхрогруппы (СГ) и кодовой комбинации (КК). Непрерывный анализ последовательности восьми бит, по каждому подканалу, на корректность правил модифицированного кода Бауэра дает результат: СГ=11; КК=7.
Заключение Задача по вычислению основных параметров модулированных сигналов может быть решена стандартными методами обработки вещественных сигналов, без преобразования их в комплексный вид. В некоторых случаях целесообразно работать с вещественными сигналами, в условия ограничения вычислительных ресурсов, так как математические операции с комплексными переменными более трудоемкие. Но так как существующие методы цифровой обработки более эффективны при обработке комплексных сигналов (например, мнимые компоненты не обнуляются при БПФ) и современ-
ные вычислители постоянно наращивают вычислительный потенциал, то обработка сигналов в комплексном виде имеет перспективу.
Определяющим фактором в рассмотренной выше процедуре является применение дискретного преобразование Гильберта к входным вещественным данным и, соответственно, переход к аналитическому представлению сигнала. Данная операция не является ресурсоемкой, применяя микроконтроллерные функции с фиксированной точкой и с рассчитанными коэффициентами в математических пакетах, аналогичных Matlab. И как следствие, рассмотренная процедура декодирования рельсовых сигналов с фазоразностной модуляцией не представляется сложной, благодаря переходу к комплексному (аналитическому) виду измерительного сигнала.
Список литературы:
1. Understanding digital signal processing / Richard G. Lyons. - 3rd ed. 2011. ISBN 0-13-702741-9
2. Сергиенко А. Б. Цифровая обработка сигналов: учеб. пособие. - 3-е изд. - СПб.: БХВ-Петербург, 2011. — 768 с.: ил.
Vlasenko T.S.1
Candidate of sciences (physics and mathematical), manager of department
Hvalin D.I.12 engineer Mystetskyi V.A.1,2
engineer
institute for Safety Problems of Nuclear Power Plants, NAS of Ukraine 2Institute of Electrodynamics, NAS of Ukraine
COMPLEX MATHEMATICAL SIMULATION OF PHYSICAL PROCESSES IN POWERFUL
GENERATOR
Abstract. Algorithm of stage solve a quasi-three-dimensional model for determination of electromagnetic field and distribution temperature in the stator core end zone of powerful generator using a numerical method is given. A model is the intermediate version between two-dimensional and three-dimensional solutions and is based on the numeral calculations in transversal and longitudinal sections of electrical machine, interconnected by a complex of boundary conditions. The mathematical model differs from are known more complete account of the physical and technical factors and the high accuracy of the calculation results with a relatively simple the software implementation. In particular, developed a method for electromagnetic field, allows description the currents of stator winding frontal part with consider the geometric shape of frontal connections and, most importantly, the load condition of turbogenerator.
Keywords: mathematical model, end zone, stator core, winding frontal part, electromagnetic field, temperature.
At present developed a lot of different convenient simulation software for computer calculations and design of electromechanical energy converters, in particular powerful turbogenerators. The most well-known complexes are FEMLAB, ANSYS, MAXWEL and COMSOL Multiphysics.
The COMSOL Multiphysics software [12] deserves special attention, because used today in many universities of the world, as well as the professional researchers of various fields.
The most significant advantage of Comsol Mul-tiphysics software is the ability to solve multiphysical problems allows creating the complex (interrelated)
mathematical models. Also an important advantage is the simplicity of interface in a coefficient form, that enables the researcher to simulate of physical processes without focusing directly on the solving of differential equations, while paying serious consideration to variation calculations (experiments) for solving of set tasks.
As an example, in this paper a complex simulation the electromagnetic and temperature fields in end zone of a powerful turbogenerator using the Comsol Multiphysics 3.5a software is performed. As known, the construction of mathematical models for analysis the physical processes in this zone of turbogenerator is very problematic [4 - 7, 9, 13, 14], because the difficult of
this problem makes researcher to apply a series of simplifications in the formulation of a task, that suitable in some cases and unsuitable for others, resulting obtained solutions often too approximate describe the nature of electromagnetic field.
A three-dimensional model have not found wide application because of complexity the end zones design of powerful generators requires very much computing trouble and even with modern computer technology needs a series of simplifications, also because the formation of meaningful and sufficiently detailed conclusions concerning parameters and characteristics of the object requires a large number of variation calculations [4]. Therefore used an approach for the analysis of heat processes in the stator core end zone of turbogenerator with help a consecutive logic transition from the solution of a simple calculation model the electromagnetic field in active part machine to more difficult calculations models in end zone using the previous results in the following, that allows to obtain a solution for determination of temperature distribution in difficult areas. The use of specialized software for development of mathematical model allowed creating a one that is sufficiently flexible in terms of modification the individual components and allows the researcher to concentrate on achieving of set goal, rather than solve of utilitarian tasks.
As a research object, a serial turbogenerator the TGV-500 type of 500 MW capacity (by Kharkov State Enterprise plant "Electrowazhmash") is considered. That due to the presence of experimental data [7 - 9], that allows to correct the reliability of electromagnetic and thermal calculations and to simplify heat calculation due to known heat transfer coefficients.
A staged construction algorithm of a mathematical model is as following.
On the first stage, a two-dimensional electromagnetic field model in transversal section of central (active) zone of a turbogenerator is considered (Fig. 1). In general case the equation of a two-dimensional electromagnetic field for the axial component of the magnetic vector potentials Az will be:
jroaA + V x (RoVV x a) = Jezez, A = A2e2 (1)
where ra is the angular frequency; c is the electric conductivity; V is the Hamilton operator; is the permeability of vacuum; ^ is the relative permeability; A is the magnetic vector potential; Jz is the external current density; ez is the unit vector.
Since the stator core is composed of sheets the high-alloy cold-rolled electrical steel with a thickness 0.5 mm, eddy current from the radial field can be neglected. The calculation of additional losses in the stator end packets of turbogenerator with acceptable accuracy can be made only for the axial component of the magnetic flux density. So, the electromagnetic field of central part of a turbogenerator in the Cartesian coordinate system satisfies the Poisson equation and, in this case, the well-known model [1] is considered:
cX2
C2 Az
+ —^ = -J*
cy
(2)
Equation (2) is supplemented by next boundary condition on the outside of solving region:
AAg = 0
(3)
and continuity condition for the magnetic flux on the inside of solving region.
In the stator winding taken a symmetric system of phase currents [4, 5] :
*a = Im cos(at + p)
u =
ic =
Im cos(ot + P- 120o) Im cos (ot + P+120° )
(4)
where Im is the magnitude of stator current; p is the angular displacement of axis along which operates the motive-magnetic force of three-phase stator winding in relation to the longitudinal axis of rotor d.
The angle p is determined by the formula:
p = 90o +6 + 9;
(5)
where 0 is the load angle of machine; 9 - phase displacement between voltage and stator current.
The load angle 0 can be found from the equation
[7]:
A Y
Oi
>
tge=-
I cos 9
(6)
(Ujxd)±Is sin where Is , Us are the phases stator current and voltage, respectively; xd is the main inductive resistance; the «plus» sign corresponds to overexcitation mode of
turbogenerator, and the «minus» is underexcitation mode.
The magnitude of currents in the stator and rotor windings and the angle p are given in accordance with the load conditions of turbogenerator.
The model takes into account the rotor movement relative to the stator. The rotation is simulated using a Moving Mesh (ALE). The displacement of the finite-element mesh is given by the transformation equation:
\dx = cos(2rcnt) • X - sin(2 nnt) • Y - X \dy = sin(2rcnt)• X + cos(2rcnt)• Y-Y '
(7)
where dx, dy are the displacement along x and y coordinates, respectively; X, Y are the coordinates of initial points, n is the speed of rotation.
The expression in brackets is the certain angle (in radians), and time t is the parameter by which the rotor does movement on angle (2nnt) relative to the initial position. A static calculation is performed, and one fixed position of rotor and fixed magnitude of currents in the windings (at t =0) are considered, the results of which then used as initial conditions for the dynamic effect (more precisely, the parametric calculation). In this case, the Poisson equation does not depend on time, its solution is determined by the instantaneous values of currents in the windings and the geometry of solving region, which varies because of displacement of rotor coordinates.
After calculated the electromagnetic problem individually solves the problem of eddy currents distribution from the main radial field for each massive element of a turbogenerator (press plate, press fingers, copper shield), and it's the average magnitude is used for calculation the thermal problem.
In the second stage, taking into account the field distribution in central part, the electromagnetic field in longitudinal section is simulated (Fig. 2). Taking into account the symmetry of the machine along axial and radial directions, the calculation area of end zone is considered as the rotor section along its axis and the section of the stator core tooth in the tangential direction (circumferentially) [13, 14]. This section coincides with the rectangular coordinate plane XY and is the symmetry plane of rotor (in Fig. 1 it passes along the radius OO1).
The equation of a two-dimensional magnetic field for the component Az (is tangential component in the solving region) has the form (1). The required structure lines of force for electromagnetic field is formed by means of boundary conditions complex on lines abcdef for the magnetic vector potentials and magnetic field [2, 3] (Fig. 2).
On line bc installed the magnitude of magnetic field in the point O1 (Fig. 1):
H\c = H 01, (8)
and on the line af - condition of symmetry the magnetic field concerning rotor axis:
n x Hf = 0. (9)
Along the line ab installed the distribution of magnetic field, a similar to the distribution along the same line in transversal section (lines OO1, Fig. 1):
Hab = H(r)looi. (10)
It is assumed that for remote lines cd, de Ta ef the magnetic field fails, so on the lines installed:
AXdef =0. (11)
At the same time installed the distribution of magnetic permeability along the radius OO1 - ^|oo1, which varies due to different saturation of stator magnetic core with the radial field.
Thus, the interconnection of electromagnetic fields in central and end zones of turbogenerator is installed. The effect of load condition is taken into account by the magnitude of currents in the stator and rotor windings, angle p and boundary conditions.
Fig. 2
An important stage in the calculation of the electromagnetic field in the end part is the description of currents in the winding frontal part. The proposed model of end zone turbogenerator allows description the currents of stator winding frontal part with consider the geometric shape of frontal connections and, most importantly, the load condition [2, 3]. The direction of phase instantaneous currents iA, iB, iC of stator winding frontal part and the part of its scheme (together with section OOi) is shown in Fig. 3.
The currents in the winding frontal part (Fig. 3)
less than taken in the system (4) at V2 time, since the front parts are bent an average on 45°. Since the longitudinal section of turbogenerator passes along the longitudinal axis of rotor d (Fig. 1), the magnitude of current density of rotor winding frontal part corresponds to the given excitation current.
The electromagnetic calculation is performed for the whole of solving region.
Individually solves the problem of eddy currents distribution from the axial magnetic fluxes of windings frontal parts. Since the rotor moves synchronously with the field, a non-zero electric conductivity is given in the stator core packets, press finger, copper shield and press plate.
Taking the obtained results of electromagnetic calculations as a source of heat losses, the problem of distribution temperature is solved.
The equation of a two-dimensional stationary temperature field is given by equations:
V(-kVT) = Q , (12)
where k is the thermal conductivity; T is the temperature; Q is the heat source.
The heat calculation is performed for only in end zone stator core.
The turbogenerator under consideration has a radial ventilation system. Cold hydrogen is fed into the radial channels between the stator packets and between the press fingers and plate from side of the gap with the further movement towards the stator yoke.
Boundary conditions are installed. Heat transfer to the environment is carried into effect by means of con-vective heat exchange between a heated surface and a stream of cooling hydrogen. The general equation has the form:
-.n-q=q0+h(TM-T) (13)
where h is the heat transfer coefficient, Tinf is the external temperature.
Installed the next boundary condition for the water circulating in the cooling channels of the press plate and stator winding frontal part:
T =T,. (14)
Installed the boundary condition of symmetry on line passes in the middle of the last ninth packet of solving region:
n-(kVT) = 0 .(15)
The finite element mesh is the same for both the electromagnetic and thermal problems.
In order to verify the reliability of developed algorithm was performed a control calculation and comparison of the obtained values of the temperature for end zone of turbogenerator type TGV-500 under rated load conditions in accordance with experimental data for temperature field of stator core end packet and heat transfer coefficients [7 - 9].
The heat transfer coefficients [in W/(m2-K)] for turbogenerator type TGV-500 have the following values [7 - 9], Fig. 4:
Vt, ^it ^32*
h K
h3 ^24
- for end packet:
hi = 474; h2 = 332; h3 = hs = hv = 354; hA = ¿6 = 248; hi = 225; hg = 221; hio = 209; fei = 1330; hsi = 205;
- for press finger:
hii = hi2 = 280; hi3 = hi4 = 243; h22 = 1280; h32 = 243;
- for copper shield:
his = hi6 = h23 = 280; h33 = 243;
- for press plate:
hiv = 280; hi8 = 200; h24 = 384;
- averaged value for other packets: hin = 24i; h2n = i000; h3n = 200.
With help a mathematical simulation in accordance with the given algorithm (i) - (i5) for turbogenerator type TGV-500 under rated load conditions the following results were obtained.
The instantaneous distribution of magnetic flux density and vector magnetic potential in central zone of a
turbogenerator (for t = 0,02 sec corresponding to one full revolution of rotor, the step size of time is 14 sec, that is, at one step the rotor turns to i,8o) is shown in Fig. 5. The distribution of magnetic permeability is shown in Fig. 6. As can be seen in this figure, the maximum value is 8542 and appears in the stator yoke.
■
Fig. 5
Fig. 8
As can be seen in Fig. 7, the maximum magnetic flux density of stator core under rated load is 3.4 T and appears in the tooth crown, the magnetic field distribution is symmetrical along the pole central line and the maximum values appear in the pole central line [14]. The distribution of magnetic flux density in the press fingers is shown in Fig. 8. The values magnetic flux densities of non-magnetic steel fingers are lower and the maximum value is 0.53 T.
Fig. 7
The press plate and copper shield materials are nonmagnetic too, and the press plate is before to the copper shield. So, their magnetic field distributions are coincident [14]. The maximum magnetic flux densities are 0.36 T and 0.33 T, respectively (Figs. 9 and 10).
Fig. 10
Figs. 11, 12 and 13 shows the distribution of the eddy current density from the main radial field in the press fingers, the plate and the copper shield. For the good conductivity of the copper, the eddy current value of the copper shield is much larger than that of the non-magnetic press fingers and plate [14].
-.4
Fig. 12
Fig. 13
The distribution of magnetic field in turbogenerator end zone under rated load is shown in Fig. 14. The distribution of temperature in stator core end zone is shown in Fig. 15.
Fig. 14
Fig. 15
The maximum temperature in the tooth root of stator end packet is 97 °C, the press plate is 91 °C, the copper shield is 86 °C and the press finger is 85 ° C. The maximum temperature in the tooth zone of end packet is explained by the combined effect of the main radial field, the axial scattering flux of windings frontal part of stator and rotor, as well as by the "buckling" of a portion of the main field from the air gap. In addition, the shield effect of press plate is the cause of field local concentration in the end packet tooth zone.
The temperature distribution in press plate depends on the channels with cooling water. However, they are effective only in beside the plate, and for heating the copper shield are not affected. With the distance from the end packets tooth zone with the maximum
temperature towards the stator yoke the temperature sharply decreases due to the shield effect of press plate and the copper hoop. The temperature also decreases when approaching the gap. The axial component of magnetic flux density in the crown tooth zone is 0.70 -0.75 T (the end of the first packet), however, this zone is intensively cooled with gas circulating in the gap.
In order to verify the reliability of obtained results was performed a comparison of the temperature values for end zone of turbogenerator type TGV-500 under rated load conditions in accordance with experimental data [7 - 9].
Experimental data correspond to the stationary heat of stator core end packet from the turbine side at section of 1/8 width of packet from stator core end
along the axis of a turbogenerator. The differences of calculated and experimental values did not exceed 7%
Conclusions
1. The advantages of constructing models in the Comsol Multiphysics software are shown. The most significant advantages are the ability to solve mul-tiphysical problems, the interface simplicity, the easiness simulation and powerful visualization.
2. A quasi-three-dimensional mathematical model of coupling calculation the electromagnetic field and heat of powerful turbogenerator end zone is developed. The mathematical model differs from are known more complete account of the physical and technical factors and the high accuracy of the calculation results with a relatively simple the software implementation. In particular, developed a method for calculating the end zone turbogenerator electromagnetic field, allows description the currents of stator winding frontal part with consider the geometric shape of frontal connections and, most importantly, the load condition of turbogenerator. It is important to emphasize, that all known today a quasi-three-dimensional models the end zone of powerful electrical machine have a homogeneous character of stator winding frontal part.
3. The results of simulation characterizing the distributions of currents, magnetic flows and temperature in the depth of end zone elements and stator core packets, and interesting to specialists and engineering staff associated with the development of electrical equipment.
4. The presented mathematical model allows at the design stage to evaluate the efficiency of design solutions for the construction of stator end zone of a turbogenerator for different load conditions, including the conditions of reactive power consumption.
Literature
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2. Kensytskiy O.G., Hvalin D.I. Distribution of electromagnetic field in the end zone of turbogenerator under nominal load condition. Tekhnichni visti. 2018. № 1(47), 2(48). P. 11 - 14. (Ukr)
3. Kensytskiy O.G., Hvalin D.I. The end zone turbogenerator electromagnetic field for changes the reactive load. Tekhnichna Elektrodynamika. 2018. № 1. P. 62 - 68. (Ukr)
(Table i), indicating a high reliability of the applied approaches and assumptions.
Table i
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Source of information Temperature, °C
Tooth Yoke
Crown Middle Root Middle
Nominal load condition (P = 495 MW, COS 9 = 0,848 )
Calculation 50,9 51,25 72,37 56,63
Experiment -- 54,90 76,20 53,70
Error, % -- 6,65 5,03 5,46
