Some Issues in the Application of Statistical Optimization Methods to the Design of Electromagnetic Automation Devices
Abstract Generated abstract
The paper discusses the use of experimental statistical optimization methods in the design of electromagnetic automation devices, where purely analytical mathematical programming is limited by approximate dependencies and experimentally determined coefficients. It proposes a two stage procedure in which preliminary optimal geometry is calculated analytically, then refined through physical modeling and planned extremal experiments using quadratic response approximations and constrained optimization. The approach is illustrated for transformers controlled by shunt biasing, with active output power optimized under an overheating constraint while varying primary voltage, load current, and magnetizing current. The study shows that such methods can refine device parameters while accounting for practical factors such as leakage fluxes, air gaps, heat transfer, and losses that are difficult to include in analytical design formulas.
Full Text
PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
OF THE ORDER OF THE OCTOBER REVOLUTION AND THE ORDER
OF THE RED BANNER OF LABOR
NAMED AFTER S. M. KIROV
Volume 267 1975
SOME ISSUES IN THE APPLICATION OF STATISTICAL METHODS OF OPTIMIZATION TO THE DESIGN OF ELECTROMAGNETIC AUTOMATION DEVICES
M. S. ALEINIKOV, T. Yu. ALEINIKOVA, V. G. KISELEV
V. P. OBRUSNIK, N. S. BOLNYKH
(Presented by the scientific and technical seminar of the Department of Instruments and Devices of Automation Systems)
In the optimal design of electromagnetic devices, two approaches are possible: in the first, methods of mathematical programming are used; in the second, methods of planning extremal experiments are used. Methods of mathematical programming require knowledge of the analytical dependences that determine the quality function and the constraints, which are obtained by solving differential equations for the physical quantities of the geometry parameters of electromagnetic devices. Often such analytical dependences have a complex form and are approximate in character. In addition, as a rule, they contain coefficients determined experimentally for a particular type of device.
All this suggests the use of experimental-statistical methods for the optimal design of electromagnetic automation devices. Depending on the goal set, the following tasks may be solved:
-
Obtaining the maximum output parameter (power, speed of response) of a standard device [1].
-
Optimal design for specified electrical parameters [2, 3, 4].
-
Refinement of the coefficients entering into the analytical expression describing the dependence of the optimized parameter of the device on certain electrical, magnetic, and design quantities.
-
Synthesis of automatic-control systems, optimal in the sense of speed of response, provision of a specified control quality, and minimum weight, which use electromagnetic and electromechanical devices as actuating, amplifying-converting, and correcting devices.
In the present work, the application of experimental-statistical methods to solving the problem is considered using as an example transformers controlled by shunt biasing (TRPSh). In [5], analytical expressions are proposed that make it relatively simple, for given electrical quantities, to choose the optimal geometry of ferromagnetic devices used in control systems. In particular, it is also acceptable for TRPSh. The latter are distinctive in that they have a large number of design varieties (more than fifty) and are manufactured in a power range from several tens-
from watts to hundreds of kilowatts. This leads to the need to refine the coefficients entering the analytical expressions by means of which optimal design of the TRPSh is carried out by the method of mathematical programming.
We propose the following procedure for the optimal design of ferromagnetic devices:
-
The optimal geometry is calculated on the basis of analytical expressions by the method of mathematical programming.
-
Physical modeling is carried out, and the device parameters are selected by methods of planning extremal experiments.
The design procedure indicated above makes it possible, first, to verify the validity of the assumptions adopted in deriving the analytical dependences and to determine the limits of applicability of these dependences. Second, it takes into account a number of factors that affect the choice of the optimal geometry of ferromagnetic devices and are difficult to allow for when deriving analytical dependences (leakage fluxes, air gaps in the magnetic circuit, heat transfer, losses in the steel, etc.).
When experimental-statistical methods are used for optimization, it is necessary to maximize the functional
\[ Y_1 = F_1(x_1, x_2, \ldots x_m, k_1, k_2, \ldots k_n) \quad m \ne n. \tag{1} \]
Subject to a constraint of the form
\[ Y_2 = F_2(x_1, x_2, \ldots x_m, k_1, k_2, \ldots k_n) \quad m \ne n, \tag{2} \]
where \(Y_1\) is the parameter being optimized (power, efficiency, cost, etc.);
\(Y_2\) is the constraint (allowable overheating, specified speed of response, shape of the current curve, etc.);
\(x_1, x_2, \ldots x_m\) are easily varied factors (electrical quantities);
\(k_1, k_2, \ldots, k_m\) are factors that are difficult to vary (design parameters).
It is assumed that the functionals (1) and (2) are approximated with sufficient accuracy by a second-degree polynomial:
\[ \hat{y}_1 = b_0 + \sum_{i=1}^{m} b_i x_i + \sum_{ij}^{m} b_{ij} x_i x_j + \sum_{i=1}^{m} b_{ii} x_{ii}^{2}; \tag{3} \]
\[ \hat{y}_2 = a_0 + \sum_{i=1}^{n} a_i x_i + \sum_{ij,\ i<j}^{m} a_{ij} x_i x_j + \sum_{i=1}^{n} a_{ii} x_i^{2}, \tag{4} \]
where \(x_i,\ x_j\) are varied factors,
\(b_0, b_i, b_{ij}, b_{ii}, a_0, a_i, a_{ij}, a_{ii}\) are coefficients to be determined.
The coefficients in (3) and (4) are obtained on the basis of processing and analyzing the results of an experiment set up according to a specially composed design [6]. Since only easily varied factors enter the approximating expressions (3) and (4), the experiment is carried out on a physical model for which the design parameters must be chosen on the basis of a preliminary analytical calculation of the device being optimized. Having obtained an analytical description of the experimental results in the form of expressions (3) and (4), one determines the values of the varied parameters at which the maximum value of the function \(Y_1\) is attained, taking into account the constraint \(Y_2\).
In the presence of one or more constraints, one has to find a conditional extremum. The problem of finding a conditional extremum can be solved by means of computational mathematics, using the method of Lagrange multipliers.
The method of Lagrange multipliers reduces to solving the system of equations
\[ \begin{cases} \dfrac{\partial y_1}{\partial x_1}+\lambda\dfrac{\partial y_2}{\partial x_1}=0,\\[6pt] \dfrac{\partial y_1}{\partial x_2}+\lambda\dfrac{\partial y_2}{\partial x_2}=0,\\ \cdots\\ \dfrac{\partial y_1}{\partial x_m}+\lambda\dfrac{\partial y_2}{\partial x_m}=0,\\[6pt] F(x_1,x_2,\ldots,x_n)=y_2, \end{cases} \tag{5} \]
with respect to the variables \(x_1, x_2, \ldots, x_m, \lambda\) for some fixed value \(Y_2\).
The proposed procedure was tested on the example of designing several TRPSh constructions. Using the simplified method for calculating the optimal geometry of the TRPSh [5], we first established the design parameters and electrical quantities of the devices. The output active power was selected as the parameter to be optimized, and the voltage on the primary winding \(U_1\), the load current \(I_{\mathrm{n}}\), and the magnetizing current \(I_d\) were selected as the variable parameters. The maximum permissible overheating was adopted as the constraint parameter.
Table 1
| Studied factor and its designation | \(U_1 \equiv x_1\) [V] | \(I_{\mathrm{n}} \equiv x_2\) [A] | \(I_d \equiv x_3\) [A] | Code designation |
|---|---|---|---|---|
| Basic level | 180 | 1.12 | 0.72 | 0 |
| Variation interval | 40 | 0.28 | 0.18 | — |
| Upper level | 220 | 1.4 | 0.9 | \(+1\) |
| Lower level | 140 | 0.84 | 0.54 | \(-1\) |
Table 1 gives, as an example, the variation intervals of the studied factors for one of the TRPSh units tested. The variation intervals were chosen on the basis of a preliminary experiment, taking into account the permissible overheating. After the variation interval had been selected, an orthogonal central composite design (OCCD) was implemented; its planning matrix is presented in Table 2 [6].
As a result of processing the experimental data, adequate equations were obtained:
\[
P_{\mathrm{n}}=\hat{Y}_1=145+24.5x_1+6.9x_2+15.4x_3-18.5x_2^2
\]
\[
{}-20.8x_3^2+5.5x_1x_2+18.25x_2x_3;
\tag{6}
\]
\[
\theta=\hat{Y}_2=2.3+0.245x_2+0.183x_3+0.204x_2^2+
\]
\[
{}+0.162x_3^2-0.13x_2x_3.
\tag{7}
\]
Table 2
| Experiment type | Run no. | $x_0$ | $x_1$ | $x_2$ | $x_3$ | $x_1^2$ | $x_2^2$ | $x_3^2$ | $x_1x_2$ | $x_1x_3$ | $x_2x_3$ | $P_{\mathrm{n}}$ | $\theta$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Full factorial experiment | 1 | +1 | −1 | −1 | −1 | 0.27 | 0.27 | 0.27 | +1 | +1 | +1 | 91 | 1.85 |
| Full factorial experiment | 2 | +1 | +1 | −1 | −1 | 0.27 | 0.27 | 0.27 | −1 | −1 | +1 | 130 | 2 |
| Full factorial experiment | 3 | +1 | −1 | +1 | −1 | 0.27 | 0.27 | 0.27 | −1 | +1 | −1 | 69 | 2.66 |
| Full factorial experiment | 4 | +1 | +1 | +1 | −1 | 0.27 | 0.27 | 0.27 | +1 | −1 | −1 | 132 | 2.76 |
| Full factorial experiment | 5 | +1 | −1 | −1 | +1 | 0.27 | 0.27 | 0.27 | +1 | −1 | −1 | 92 | 2.45 |
| Full factorial experiment | 6 | +1 | +1 | −1 | +1 | 0.27 | 0.27 | 0.27 | −1 | +1 | −1 | 135 | 2.61 |
| Full factorial experiment | 7 | +1 | −1 | +1 | +1 | 0.27 | 0.27 | 0.27 | −1 | −1 | +1 | 145 | 2.69 |
| Full factorial experiment | 8 | +1 | +1 | +1 | +1 | 0.27 | 0.27 | 0.27 | +1 | +1 | +1 | 208 | 2.9 |
| Star points | 9 | +1 | −1.215 | 0 | 0 | 0.75 | −0.73 | −0.73 | 0 | 0 | 0 | 151 | 2.56 |
| Star points | 10 | +1 | +1.215 | 0 | 0 | 0.75 | −0.73 | −0.73 | 0 | 0 | 0 | 200 | 2.1 |
| Star points | 11 | +1 | 0 | −1.215 | 0 | −0.73 | 0.75 | −0.73 | 0 | 0 | 0 | 160 | 2.45 |
| Star points | 12 | +1 | 0 | +1.215 | 0 | −0.73 | 0.75 | −0.73 | 0 | 0 | 0 | 135 | 2.95 |
| Star points | 13 | +1 | 0 | 0 | −1.215 | −0.73 | −0.73 | 0.75 | 0 | 0 | 0 | 140 | 2.0 |
| Star points | 14 | +1 | 0 | 0 | +1.215 | −0.73 | −0.73 | 0.75 | 0 | 0 | 0 | 148 | 2.51 |
| Center point | 15 | +1 | 0 | 0 | 0 | −0.73 | −0.73 | −0.73 | 0 | 0 | 0 | 145 | 2.0 |
The overheating was measured by means of thermocouples placed at various points of the windings and cores of the TRPSh. In calculating the coefficients of equation (4), only the maximum overheatings were taken into account (the thermoe.m.f. of the thermocouple at the point of maximum overheating). Substituting equations (6) and (7) into (5) and solving the resulting system of equations on an electronic digital computer by the Kiefer–Wolfowitz method, we obtain the optimal values of the parameters \(P_n, x_1, x_2, x_3\).
As a result of the calculation, the optimal parameters of the output active power, the load and magnetizing currents were obtained for the TRPSh design parameters selected according to the procedure of [5].
Thus, the application of experimental-statistical methods for optimizing automation and electromechanical devices makes it possible to solve the problem of precise analysis-synthesis of the parameters of these devices with allowance for technical requirements.
REFERENCES
-
A. S. Gitman, E. S. Gitman. Search for optimal parameters of a ferroresonant stabilizer by the steepest-ascent method. Izv. TPI, vol. 172, Tomsk, 1967.
-
D. A. Avetisyan, A. I. Bertinov, E. S. Gitman. Application of the mathematical theory of experiment to the selection of optimal parameters of an alternating-current choke. Izv. vuzov, “Elektromekhanika,” 1968, No. 11.
-
M. S. Aleinikov, T. Yu. Aleinikova. Application of a factorial experiment to the selection of optimal parameters of a T-shaped inductive-capacitive filter. Proceedings of the Jubilee Scientific-Technical Conference of the Faculty of Automatic Systems. Tomsk, TSU Publishing House, 1970.
-
M. M. Dmitriev, N. L. Kuznetsov. Application of a factorial experiment to the study of the characteristics of electrical machines. Proceedings of the Jubilee Scientific-Technical Conference of the Faculty of Automatic Systems. Tomsk, TSU Publishing House, 1970.
-
V. P. Obrusnik. A simplified method for calculating the optimal geometry of transformers regulated by magnetization of shunts. Izv. TPI, vol. 132, Tomsk, 1965.
-
V. V. Nalimov, N. A. Chernova. Statistical Methods for Planning Extremal Experiments. Moscow, Nauka Publishing House, 1965.