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79
Electrical Machines and Apparatus
UDC 621.313:536.2.24:539.2 doi: 10.20998/2074-272X.2018.6.02
V.F. Bolyukh, I.S. Schukin
AN OPTIMIZATION APPROACH TO THE CHOICE OF PARAMETERS OF LINEAR PULSE INDUCTION ELECTROMECHANICAL CONVERTER
Purpose. The purpose of the paper is to select the main parameters of the linear pulse induction electromechanical converters (LPIEC) for high-speed and power use with the use of the optimization approach, which provides an increase in speed and power indicators with limited electric, thermal and mass-dimensions. Methodology. A technique for finding the maximum of the integral efficiency criterion of LPIEC in the search space using a global optimization method that randomly searches for parameters, preventing entry into a local maximum, and a local method ensuring the contraction of the range of parameters with a global maximum to minimum dimensions is developed. As a global optimization method, genetic algorithms are used, and the Nelder-Mead method is used as the local method. Results. The LPIEC inductor should have a maximum external and minimum internal diameter, and its height should be less than that of the LPIEC of the basic design. The armature should have a maximum outer diameter, and the thickness of its wire should be minimal. The armature should be made with a significantly higher height, a greater number of turns and a wider wire. The height of the LPIEC inductor for power purposes should be almost the same as that of the LPIEC of the basic design. In this case, the number of turns of the inductor and the cross section of its wire should be approximately the same. The armature should be made with a slightly larger inner diameter and a significantly higher height. This armature should have a larger number of turns of wire, which must be stacked in 4 layers, and a large width of the wire. The average energy value and voltage of the capacitive energy storage for the LPIEC for high-speed and power applications should be higher than for the LPIEC of the basic design. Originality. An optimization approach to the choice of LPIEC parameters with a multi-turn squirrel arm is developed, which consists in finding the maximum of an integral efficiency criterion that takes into account the maximum speed and efficiency in a high-speed converter, the amplitude and magnitude of the electrodynamic force pulse in a power converter, with minimum temperature excesses, the mass of active elements and current of the inductor. The optimization uses a chain mathematical model that takes into account the interconnected electrical, magnetic, thermal and mechanical processes of the LPIEC. Practical value. The electric parameters of the capacitive energy storage device and the geometric parameters of the LPIEC are determined, which ensure the largest values of the integral efficiency criterion depending on the adopted version of the efficiency evaluation strategy. In optimized speed and power transfer converters, the integral efficiency criteria are 2.2 times higher on average than in the basic performance of the LPIEC. References 14, tables 6, figures 2.
Key words: linear pulse induction electromechanical converter, chain mathematical model, integral efficiency criterion, optimization approach, genetic algorithms, Nelder-Mead method.
Разработан оптимизационный подход к выбору параметров линейного импульсного индукционного электромеханического преобразователя (ЛИИЭП) с многовитковым короткозамкнутым якорем. Он состоит в нахождении максимума интегрального критерия эффективности, учитывающего максимальную скорость и КПД преобразователя скоростного назначения, амплитуду и величину импульса электродинамических усилий в преобразователе силового назначения при минимальных превышениях температур, массе активных элементов и токе индуктора. При этом используется цепная математическая модель, которая учитывает взаимосвязанные электрические, магнитные, тепловые и механические процессы ЛИИЭП. Разработана методика поиска максимума интегрального критерия эффективности ЛИИЭП в поисковом пространстве с использованием глобального и локального методов оптимизации. В качестве глобального метода используются генетические алгоритмы, а в качестве локального - метод Нелдера-Мида. Установлены электрические параметры емкостного накопителя энергии и геометрические параметры ЛИИЭП, обеспечивающие наибольшие значения интегрального критерия эффективности в зависимости от принятого варианта стратегии оценки эффективности. В оптимизированных преобразователях скоростного и силового назначения интегральные критерии эффективности в среднем в 2,2 раза выше, чем в ЛИИЭП основного исполнения. Библ. 14, табл. 6, рис. 2.
Ключевые слова: линейный импульсный индукционный электромеханический преобразователь, цепная математическая модель, интегральный критерий эффективности, оптимизационный подход, генетические алгоритмы, метод Нелдера-Мида.
Introduction. Linear pulse electromechanical converters are widely used to accelerate the actuator (A) to high speed in a short active area and/or to create powerful power pulses on the object of action with a slight movement of A, made, for example, as a striker [1-4]. Such converters of high-speed and power purposes are used in many branches of science and technology as electromechanical accelerators and shock-power devices [5].
The most widely used converters are of induction type, which have a coaxial disk configuration. Such
linear pulsed induction electromechanical converters (LPIEC) contain an accelerated electrically conductive armature that magnetically interacts with a fixed inductor [6-8]. When the inductor is excited, a current is induced from a capacitive energy storage (CES) in an electrically conductive armature. The interaction of the inductor's magnetic field with induced current leads to the occurrence of electrodynamic forces (EDF), causing axial displacement of the armature with A. In this case, it is considered expedient to excite the inductor by a
© V.F. Bolyukh, I.S. Schukin
polar aperiodic pulse, which allows the use of electrolytic capacitors with an increased specific energy index for the CES [5].
However, at operation with a rapid change in the electromagnetic, mechanical and thermal parameters, the efficiency of the power and speed indicators of LPIEC is not high enough. One of the ways to improve these indicators is to use an optimization approach to the choice of the main parameters of LPIEC. With this choice, it is advisable to use the integral efficiency criterion, which should include the main speed or power indicators of LPIEC, taking into account its electrical, thermal and mass-dimensional indicators.
The goal of the paper is the choice of the basic parameters of LPIEC for speed and power purposes using an optimization approach that provides an increase in speed and power indicators with limited electrical, thermal and mass-dimensional indicators.
To increase the speed of the computational algorithm we use the chain mathematical model of LIIEP, which uses lumped parameters of the inductor and armature [9, 10]. This model takes into account interconnected electrical, magnetic, thermal and mechanical processes. To eliminate the influence of the skin effect, we consider an armature made as a short-circuited multi-turn winding tightly wound with a relatively thin copper wire.
Parameters and indicators of LPIEC. The optimization process consists in finding a set of parameters that provide the maximum values of velocity and kinetic energy in LPIEC for high-speed assignment and maximum values of amplitude and value of the impulse of EDF in LPIEC for power purposes. These indicators should be provided with the minimum temperature increases and mass of the n-th active elements (n = 1, 2 are the indices of the inductor and armature, respectively) and the minimum current of the inductor, which is important for the control system.
Main LPIEC parameters
CES electrical parameters:
• U0, W0 - CES voltage and energy, respectively.
LPIEC geometrical parameters for the n-th active
element:
• wn - number of turns of the wire;
• d0n - diameter of round wire;
• hzn, hrn - height and width of rectangular wire;
• Dex n, Din n - outer and inner diameters;
• Hn - axial height.
LPIEC additional parameters:
• m2, me - mass of the armature and A, respectively;
• Az0 = 0.5 -(( + H2 )+A0 - initial axial displacement between the centers of the n-th active elements, where A0 - initial gap between active elements;
• converter shape: disk or cylindrical feedthrough (armature inside the inductor or vice versa);
• type of armature (multiturn, massive, combined);
• circuit of the formation of a current pulse in the inductor when excited from the CES;
• initial temperature T0n of the n-th active element;
• mechanical factors: the forces of resistance to the movement of the armature, friction, etc.;
• parameters of lead wires and connecting elements. These parameters are subject to parametric and
functional limitations:
for energy source U0 < U0max - on CES voltage;
W)min < 0.5 • C• U0 < Womax - on CES energy,
where C = 2W0U 0 - CES capacitance;
for load
0 < me < memax - on mass of the accelerated A; fcmin < fc < fcmax - on braking and opposing forces; for electronic control system
h <
- limitation on the amplitude of the inductor
excitation current;
for geometrical parameters
1 < wn < Ent
f D - D
q 5 ^ ex n ^m n hrn + 2hs J
f
• Ent
Hn
V hzn + 2hs J
- for the
number of turns of rectangular wire, where Ent(f) - the largest integer not exceeding f hs - winding conductor insulation thickness;
(
1 < Ent
Q.5
D - D
ex n m n hrn + 2hs J
< Kw n max - for the number of
layers of rectangular wire; 0 < AZ0 < 0.5 - (Hi + H2); 0 < Dm 2 < Dm 2^ ;
Din2 + 2 - (hr2 + 2 - hs ) < Dex2 < Dni " A0 - for cylindrical feedthrough converter, there Din 2max - maximum value of the armature inner diameter;
When using a round wire in parametric constraints, instead of hn and hn it is necessary to use its diameter d0n;
-l(t)i2(t)^Z) < fz
dz
z max
- on the amplitude of EDF
active along the z-axis; where in(t) - current of the n-th active element, M12 - mutual inductance between active
elements; Fzmin >
F
j fzdt
> F.,
z max
- on the EDF impulse
Q
value, where tp is the duration of the EDF action;
Wkin min
> 0.5 -(m2 + me)Vp >
Wkin max - on the kinetic
energy, where Vp is the armature velocity at the end of the operation process; Vmin > v > Vmax - on the velocity of displacement of the armature with A; 0n < 0n max - on
the maximum permissible temperature rise of the n-th active element.
LPIEC of basic design. As the basic design, we consider LPIEC with following parameters: [11]:
Inductor: outer diameter Dex1=100 mm, inner diameter Din1=10 mm, height H1=10 mm. The inductor is made as a two-layer winding with external electrical leads; rectangular wire cross section hz1 xhri=1.8x4.8 mm2, number of turns of the wire w1= 46.
max
t
Armature: outer diameter D,^ 100 mm, inner diameter Din2=6 mm, height H2=2.5 mm. The armature is made as a multilayer short-circuited winding, the cross section of a rectangular copper wire hz2xhr2=1.0x1.2 mm2, number of turns of the wire w2= 80.
CES: capacitance C0=3 mF, voltage U0=0.4 kV.
Initial distance between inductor and armature A0=1 mm.
In LPIEC for speed purposes, the return spring's coefficient of elasticity is KP=50 kN/m. Mass of A me=0.5 kg. We believe that in LPIEC of power designation the counteracting force is significant and there is no movement of A.
In LPIEC of the basic design of speed purpose, the following indicators are implemented: the amplitude of the inductor current iim=2.57 kA, the maximum current density in the inductor's conductor j1m=297.5 A mm2, the maximum current density in the armature j2m=764.56 A/mm2, the amplitude of EDF f„=13.983 kN, the value of the impulse of EDF Fz=5.674 Ns, the maximum speed of the armature with the inductor Vm=8.43 m/s, the efficiency n=10.32 %, the temperature rise of the inductor 6^=0.37 °C, the temperature rise of the armature 62=0.97 °C. The mass of copper in the inductor is m1=0.69 kg, the mass of copper in the armature is m2=0.17 kg.
The following indicators are implemented in the LPIEC of the basic design of power purpose: the amplitude of the inductor current i1m=2.953 kA, the maximum current density in the inductor's conductor j1m=341.78 A/mm2, the maximum current density in the armature j2m=893.51 A/mm2, the amplitude of EDF fzm=20.171 kN, the magnitude of the impulse of EDF Fz=9.076 Ns, the temperature rise of the inductor 61=0.4 °C, the temperature rise of the armature 62=1.45 °C.
Integral efficiency criterion. Since the efficiency of LPIEC of speed or power purpose is characterized by a number of versatile indicators, we introduce an integral efficiency criterion, which takes into account the maximum speed or power indicators with minimum values of the inductor current amplitude, temperature rises and the total mass of copper wire of active elements. In a dimensionless form, it can be written as follows.
K* =4l+PA + P3B* +4+4+4; JP} = 1,
'1m mT. j=1
where fa are the weights of the corresponding indicator; J = 6 is the number of functional indicators, normalized relative to LPIEC of the basic design (marked with asterisks);
2
ms = 05nycuY,(Dexn + D'nn)hmhznWn is the total mass
n=1
of the copper wire in the n-th active elements, where yCu is the copper wire's density; A=fzm, B=Fz - for LPIEC of power purpose; A=Vzm, B=n - for LPIEC of speed purpose; fm is the EDF amplitude; Vzm is the maximum speed of the armature with A;
is
the LPIEC
jj = 100 (m2 + me № + KP^2 %
C0U02 ' °
efficiency.
Note that LPIEC of a basic design of speed or power designation has K*=1. The best will be the converter with the maximum value of K* showing how many times it is more efficient than LPIEC of the basic design.
A technique of finding the maximum of the target function. The integral efficiency criterion of LPIEC K* is a target function of the optimization process. The strategy for finding the maximum of the target function of m variables in the search space is to share the global optimization method that performs a random search for LPIEC parameters in a given space, preventing it from falling into a local extremum, and a local method that provides tightening the parameter area with a global extremum to the minimum sizes.
As a global optimization method, we use genetic algorithms based on the mechanisms of population genetics [12, 13]. According to this method, each attribute of an object in the phenotype corresponds to one gene in the genotype, which is a bit string of fixed length. The sign is divided into tetrads, converted by the Gray code. When encoding a binary string of i bits of the variable xk, which belongs to the segment [xmm, xmax], each string sk expresses the value of the variable xk:
xk xmin + sk(xmax xmin) / 2 ,
where sk is the value of the binary number encoded by this string.
Operating on a set (population) of possible solutions P = (x1,...,xm), the set of parameters xh of structured in a certain way in the form of a chain of finite length is processed, and subsequent generations of the solution population are generated using genetic operators. Thus, a randomized search with centralized control is implemented, using selection and genetic mechanisms of reproduction, with an arbitrary choice of points of application of operators.
Genetic algorithms can be represented as follows:
GA = (r0, m, l, S, Q, i, #),
where P0 = (a{\... ,a°) is the initial population; a0 is the problem solution in the form of a chromosome, i = 1, m ; m is the population dimension: l is the length of each chromosome of the population; S is the selection operator; Q is the recombination mapping recombination (crossover, mutation); i is the optimality function; £ is the break criterion.
The work of genetic algorithms is an iterative process that continues until a given condition is met, for example, slowing down the growth of the efficiency
criterion K*=1 to a given value. P0 is a randomly generated initial population. At each iteration cycle, selection, crossover, and mutation operators are implemented. The selection operator S generates an
intermediate population R from the population P by
selecting and generating new copies of the elements P: R = S(P). The optimality function i, which provides feedback on the results of optimization during generation t, is used to select individuals in the population. The selection is made based on the probabilities
Ps (af) calculated for each individual:
ps (at)=matL.
• random crossover point selection x e {1,
• formation of two new individuals
, i -1};
a=(«y
1,x a2,x+1 •
a2,l
Xi+1 = Xk + r\xr1 -Xk) i e [1, n +1], i * k, Xk+1 = X where ye (0.1), y~ 0.5 is the reduction factor.
X>(aj )
j=1
After the selection is completed, for the element a■ e Rl a partner is selected from Rl for recombination and a new chromosome is built.
The crossover with probability pc is performed as follows:
• random selection of crossbreeding partners
«1 = (an •..«!/) e R , «2 = («2,1— «2,i) e R ;
a2 = (a2,1 • • • a2,x a1,x+1 • - a1,/) .
A mutation is a random change in the chromosome
bit:
• random selection with probability pM of positions {xb — ., xk} c {1,—,/} inside the bit string
a = (a1 •.. ai) e Rf, prone to mutation;
• formation of a new individual
a = (a1 —axx _1 axj axj+1 •■axi-1 axt axt +1 •■■al) , (i = l, k) .
As a local optimization method for finding the maximum of the optimality criterion K (X) in the n-dimensional Euclidean space Rn
max K*(X) = K* (X*), X e Rn the Nelder-Mead method is used changing the current simplex [14].
As a result of reflection of the k-th vertex of the simplex with the coordinates of the vertices Xrj, i e [1, n +1] ,a simplex is formed with the coordinates of the vertices
X[+1 = X[, i e [1, n +1], i * k Xrk+1 = 2Xc _Xrk, 1 n+1
where XC =- YX[ is the vector of coordinates of the
n
1=1,1
center of gravity of the remaining vertices of the simplex.
As a result of the reduction of the vertices of the simplex Xr to the vertex Xk, we obtain the simplex with the coordinates of the vertices
After the operation of compressing the simplex Xr in the direction (Xr - XC), we obtain a simplex with the coordinates of the vertices
Xf+1 = Xk, i e [1, n +1], i * k, Xk+1 = XC +p{Xk _ XC ), where fie (0.1), ¡3« 0.4 - 0.6 is the compression factor.
As a result of extension of the simplex Xir in the
direction (Xkr - XCr ) , we obtain a simplex with the coordinates of the vertices
Xf+1 = Xr, i e [1, n +1], i * k, Xkr+1 = XrC +a(Xrk -XrC), where a« 2.8 - 3.0 is the extension factor.
Since the deformation procedure is repeated many times, the polyhedron adapts to the local relief of the target function and shrinks, ensuring the convergence of the algorithm in the local maximum, allowing by the size of the polyhedron ct1 to judge the stage of the search for the converter parameters.
Realization of the task of choosing the parameters of LPIEC. Consider a LPIEC of a disk configuration with a multiturn armature that is excited by a polar aperiodic pulse (CES is shunted by a reverse diode). The following are used as independent variables that are included in the LPIEC design variables vector: outer Dexn and inner Dinn diameters, height Hn, number of turns wn, height hzn, and width hrn of a rectangular wire of the n-th active element; voltage U0 and energy W0 of CES. Restrictions on these parameters impose the boundaries of the search space (Table 1).
Table 1
Functional and parametric limitations of LPIEC parameters
Parameter Value
CES energy W0, J 150...500
CES voltage U0, V 150...500
Outer diameter of the n-th active elements Dexn, mm 50.100
Inner diameter of the inductor Din1, mm 10.20
Inner diameter of the armature Din2, mm 2.20
Inductor height Hi, mm 5.22
Armature height H2, mm 1.10
Number of inductor layers Kw1 2
Number of turns of inductor w1 30.75
Inductor wire height hz1, mm 1.2
Inductor wire width hr1, mm 2.10
Number of armature layers Kw2 1.8
Number of turns of armature w2 20.200
Armature wire height hz2, mm 0.5.1.5
Armature wire width hr2, mm 1.0.3.0
Wire insulation thickness hs, mm 0.1
Initial gap between the n-th active elements A0, mm 1.0
For optimization calculations, a computational algorithm was applied, which includes the following steps [7].
1. A genetic representation of the polyhedron is specified by a set of N+1 parameters - vectors of design
variables P0 =(x°,...,), *(x1v..,xN) e^N.
2. From K source polyhedra pf =(x01,...,x®N+1j,
i = 1,...,K the population Di (p0jis randomly formed.
3. The operators of reflection, extension, compression, and reduction are applied to each polyhedron p0 to perform a specified number of steps 5 in the search space.
4. The value of the target function
F' X, j )i = 1,..K, j = 1,..N +1 is determined in each vertex of the polyhedron as well as its the «best» vertex x. b,i = 1,..K .
5. The polyhedra are ranked relative to the value of the target function of their best vertexes. Fl (xi,b)i = 1,..K .
6. The polyhedron with the worst parameters is eliminated.
7. A new polyhedron P{K is formed by applying genetic operators of the crossover and mutation, acting with probability pmut, to two randomly selected polyhedra from the remaining (K-1).
8. The value of the target function
F[xK, j j, j = 1,..N +1 and the «best» vertex of the polyhedron PfK are determined.
9. Polyhedrons Pjare ranked by size a(pitj[i = 1,..K .
10. Threshold a1 is determined for getting into the search group by size of the h-th population ap j.
11. To (K-h) populations operators of reflection, extension, compression and reduction are applied.
12. Return to step 4.
The results of the choice of parameters of LPIEC of speed purpose. The choice of parameters is largely determined by the adopted version of the strategy for assessing the effectiveness of LPIEC. Consider four options for the strategy (Table 2).
Table 2
The weights of the options for the evaluation strategy for the LPIEC efficiency, p.u.
Strategy options ßi ß 2 ß 3 ß 4 ß 5 ß 6
I 0.2 0.2 0.2 0.1 0.1 0.2
II 0.1 0.3 0.3 0.1 0.1 0.1
III 0.05 0.5 0.2 0.1 0.1 0.05
IV 0.05 0.2 0.5 0.1 0.1 0.05
inductor; outer Dex2 and inner Din2 diameters, height H2, number of turns w2* and number of layers Kw2*, thickness hz2* and width hr2* of the wire of the armature (Table 3).
Table 3
Relative parameters of LPIEC of speed purpose
Parameter Strategy options for estimation of LPIEC efficiency Average value
I II III IV
1.0 1.0 1.0 1.0 1.0
Din1 1.1 1.0 1.0 1.0 1.0
h: 0.62 0.62 0.82 0.66 0.68
W1* 1.63 1.39 0.87 0.95 1.21
hi 0.55 0.67 1.11 1.11 0.86
hï" 0.625 0.625 0.830 0.662 0.686
Dex2 1.0 1.0 1.0 1.0 1.0
Din2 1.2 1.3 1.1 1.0 1.15
H2* 4.8 6.8 1.6 2.52 3.93
w2* 2.2 2.1 1.2 1.5 1.75
Kw2 2.0 2.0 1.0 1.5 1.63
hi* 1.0 1.0 1.0 1.0 1.0
hr2* 2.08 3.33 1.5 1.66 2.14
The operation of LPIEC is estimated by the following relative indicators: inductor current amplitude i1m*, maximum current density in conductors of inductor jXm* and armature j2m*, maximum speed Vm* and efficiency n*, temperature rise of the inductor 61* and armature 62* at the end of the working process, total mass of the wire m2* and the criterion of efficiency K* (Table 4).
Table 3, 4 also show the average values of the parameters and indicators of LPIEC for speed purpose.
Table 4
Relative indicators of LPIEC of speed purpose
In option I, all LPIEC indicators are estimated equally (the total value of the temperature rise indicator is 0.2). In option II, the maximum speed Vm and the efficiency n of the converter are estimated to be the highest and equivalent. In option III, the maximum speed Vm is most highly estimated, and in option IV -efficiency n.
As a result of calculations for each of the options of the strategy for evaluating the effectiveness, the relative parameters of LPIEC for speed purpose were obtained: energy W0*, voltage U0*, and capacitance C* of CES; outer Dex1 and inner Dini* diameters, height H1*, number of turns w1*, thickness hz1* and width hr1* of the wire of the
Indicator Strategy options for estimation of LPIEC efficiency Average value
I II III IV
W0* 0.729 0.937 2.08 1.458 1.301
U0* 0.437 0.562 1.25 1.25 0.875
C* 3.8 2.96 1.33 0.933 2.256
* 0.345 0.565 1.553 1.3 0.941
* J\m 0.992 1.268 1.677 1.754 1.423
* Jim 0.141 0.12 0.958 0.568 0.447
V * m 0.449 0.516 1.592 1.257 0.965
* n 0.537 0.727 1.403 1.573 1.061
61* 1.953 2.834 4.1 3.712 3.15
Ö2* 0.053 0.037 0.86 0.34 0.323
* mE 1.456 1.856 0.976 1.03 1.33
K* 2.858 3.331 1.3 1.376 2.216
Based on the results obtained, the following conclusions can be drawn. Inductor of LPIEC must have the maximum outer DexI=0.I m and minimum inner DinI=0.0I m diameters. The armature should have the maximum outer diameter Dex2=0.I m, and the
thickness of its wire should be minimum hz2=1 mm. These parameters accept the limiting functional limitations (Table 1) and correspond to the parameters of LPIEC of basic design in all options of the effectiveness evaluation strategy.
The height of the inductor should be less than that of LPIEC of the basic design and, depending on the strategy options, averages Hi=6.8 mm. Here, the number of turns of the inductor in options I and II should be greater, and in options III and IV less than of the basic design of LPIEC, averaging wj=56 turns of the wire with cross section hzi*hr1=1.5x3.3 mm2. The armature of the optimized converter should be made with large inner diameter Din2=7 mm and significantly larger height H2=9.8 mm, with large number of turns w2=140 and wire width hr2=2.6 mm.
CES of optimal LPIEC should have less energy in options I and II, while its average value should be higher than that of LPIEC of basic design and be W0=312 J. Voltage of CES in options I and II is low and averages U0=200 V, and in options III and IV - the maximum U0=500 V. The average values of maximum current densities as compared to LPIEC of the basic design in the inductor conductors are increased to j1m=423 A/mm2, and in the armature conductors are reduced to j12m=342 A/mm2.
Maximum speed and efficiency are reduced in strategy options I and II and increased in options III and IV. For example, in option III of the strategy, maximum speed is Vm=13.4 m/s, and the efficiency in option IV is n = 16.2 %.
Compared to LPIEC of the basic design, in optimized converters, the temperature rise of the inductor increases, and the armature temperature rise decreases, averaging 6»1=1.17 °C and 02=0.31 °C. The weight of copper wire increases on average to ms=1.14 kg. Compared to the basic design of LPIEC, the integral efficiency criteria of optimized converters increase on average to a value K*=2.2.
Fig. 1 shows the electromechanical characteristics of LPIEC of speed purpose, optimal in strategy option IV. A feature of these characteristics is that the maximum values of current densities in the windings of the inductor and the armature occur almost simultaneously, which causes the nature of the change in EDF f.. Movement of the armature with A begins in 0.2 ms after the start of the working process.
The results of the choice of parameters of LPIEC of power purpose. Let us consider four options of the strategy for evaluating the effectiveness of LPIEC for power purpose (Table 2). In option II, the amplitude fzm and the value of the impulse Fz of EDF are estimated most highly and equally. In option III, the amplitude of EDU fzm is most evaluated. In option IV, the value of the impulse of EDU Fz is most evaluated.
Table 5, 6 show the average values of parameters and indicators of LPIEC for power purpose.
j, A/mm2; uc, V; Az, mm; v, m/s SCO—-1 I -—1-
-6QQ J------
0.25 0.5 0.75 1.0 ms 1.5
Fig. 1. Electromechanical characteristics of the optimal LPIEC of speed purpose (option of strategy IV)
Table 5
Relative parameters of LPIEC of power purpose
Parameter Strategy options for estimation of LPIEC efficiency Average value
I II III IV
Dex1 i.0 1.0 1.0 1.0 1.0
Dinl i.i 1.0 1.0 1.0 1.0
h; i.04 1.04 1.06 0.86 1.0
W1* 0.87 1.39 1.09 0.91 1.07
hzi* i.ii 0.66 0.89 1.11 0.94
hri i.04 1.04 1.06 0.88 1.01
Dex2 i.0 1.0 1.0 1.0 1.0
Din2 i.2 1.2 1.3 1.4 1.275
H2* 5.0 5.0 6.0 3.84 4.96
W2* 2.1 2.3 2.2 2.2 2.2
Kw2 2 2 2 2 2
hz2* 1.0 1.0 1.0 1.0 1.0
hr2* 2.5 2.5 2.9 1.92 2.46
Table 6
Relative indicators of LPIEC of power purpose
Indicator Strategy options for estimation of LPIEC efficiency Average value
I II III IV
W„* 0.625 2.08 1.875 2.08 1.665
U* 0.375 1.25 1.125 1.2 0.988
c* 4.444 1.333 1.481 1.45 2.177
* 0.818 0.993 1.234 1.616 1.165
* j1m 0.703 1.43 1.333 1.662 1.282
* Jim 0.146 0.288 0.243 0.428 0.276
f * Jzm 0.382 1.47 2.018 1.361 1.308
F* 0.891 3.147 2.918 3.318 2.568
61* 1.062 3.382 3.181 4.198 2.956
Ö2* 0.051 0.19 0.125 0.291 0.164
* mE 1.832 1.832 2.032 1.456 1.788
K* 2.664 2.147 2.159 2.047 2.254
The height of the inductor of this converter should be almost the same as that of LPIEC of the basic design, and should be on average H1=10 mm. At the same time, the number of inductor turns in strategy options I and IV should be less, and in strategy options II and III more than in the basic design of LPIEC, averaging w1=50 turns. The cross section of the inductor wire hz1xhr1=1.7x4.8 mm2 should be almost the same. The armature should be made with slightly larger inner diameter Din2=7.6 mm and significantly larger height H2=12.4 mm. This armature should have greater number of turns of wire w2=176, which should be laid in 4 layers, and large wire width hr2=3.0 mm.
CES of LPIEC for power purpose should have less energy only in option I of strategy, in which all indicators are evaluated equally. The average value of the energy of CES should be higher than that of LPIEC of the basic design and be W0=400 J. The voltage of CES in the option of strategy I is low and is U0=150 V, and in other variants it is increased and is U0=450-500 V. The capacitance of CES increases in all variants of the strategy, averaging C=6530 pF.
The average values of the maximum current densities compared with LPIEC of the main design in inductor conductors are on average increased to j1m=438 A/mm2, and in armature conductors reduced to j2m=246 A/mm2. The amplitude and value of the impulse of EDF are increased except for option of strategy I. Thus, in the option of strategy III, the amplitude of EDF is fzm=40.7 kN, and in the option of strategy IV, the value of the impulse of EDF is Fz=30.11 N-s. In all options of the strategy, compared to LPIEC of the basic design, the temperature rise of the inductor increases, and the armature temperature rise decreases, averaging 61=1.18 °C and 62=0.24 °C, respectively. The mass of copper wire increases on average to ms=1.54 kg. Compared to the basic design of LPIEC, the integral efficiency criteria of optimized converters of power purpose increase on average to the value K*=2.25.
Fig. 2 shows the electrodynamic characteristics of LPIEC for power purpose, optimal in strategy option IV. Compared with LPIEC speed purpose, in this converter the electrodynamic processes proceed with large delay in time, with large amplitudes of the current densities in the inductor and armature, and also EDF.
Conclusions
1. An optimization approach has been developed for choice parameters of LPIEC with a multi-turn short-circuited armature, which consists in finding the maximum of the integral efficiency criterion that takes into account the maximum speed and efficiency of the converter of speed purpose, amplitude and value of the impulse of EDF of the converter of power purpose with minimum temperature rises, mass of active elements and current in inductor. The optimization uses a chain mathematical model that takes into account the interconnected electrical, magnetic, thermal, and mechanical processes of LPIEC.
j, A/mm2; uc, V; fz, kN; Fz, Ns
Fig. 2. Electrodynamic characteristics of the optimal LPIEC of power purpose (option of strategy IV)
2. A technique has been developed for finding the maximum of the integral criterion of LPIEC efficiency in the search space using a global optimization method that performs a random search for parameters, preventing it from falling into a local maximum, and a local method that provides a parameter region tightening with a global maximum to the minimum size. Genetic algorithms are used as a global optimization method, and the Nelder-Mead method is used as a local optimization method.
3. The values of the electrical parameters of the capacitive energy storage and the geometrical parameters of LPIEC are determined, which provide the highest values of the integral efficiency criterion depending on the adopted option of the effectiveness evaluation strategy. In optimized converters for speed and power purposes, the integral efficiency criteria are on average 2.2 times higher than in LPIEC of basic design.
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Received 04.09.2018
V.F. Bolyukh1, Doctor of Technical Science, Professor, I.S. Schukin2, Candidate of Technical Science, Associate Professor,
1 National Technical University «Kharkiv Polytechnic Institute», 2, Kyrpychova Str., Kharkiv, 61002, Ukraine,
phone +380 57 7076427, e-mail: [email protected]
2 Firm Tetra, LTD,
2, Kyrpychova Str., Kharkiv, 61002, Ukraine,
phone +380 57 7076427,
e-mail: [email protected]
How to cite this article:
Bolyukh V.F., Schukin I.S. An optimization approach to the choice of parameters of linear pulse induction electromechanical converter. Electrical engineering & electromechanics, 2018, no.6, pp. 18-25. doi: 10.20998/2074-272X.2018.6.02.
