OPTIMIZATION OF ELECTRIC MACHINES USING RESPONSE SURFACES

A. S. Gitman, E. K. Strelbitskii

The problem of finding optimal ratios of the dimensions of electrical machines goes back to M. Vidmar, who optimized quadratic forms of two variables [1]. In such a formulation, the problem lost its applied meaning and was no more than an elegant illustration of the existence of an extremum in cost. Vidmar himself understood the imperfection of his theory of economic proportionality; later researchers supplemented this theory with the concept of cost validity, which made it possible to take operating conditions into account [2]. Nevertheless, this accounting did not take the problem beyond the framework of a two-dimensional space and did not eliminate the need for successive approximations. The reason for the non-instrumentality of this theory and for its considerable complication in passing to a space with a larger number of variables lies in the presence of substantial interaction effects among the independent variables. The presence of these effects made it possible to find analytically only local optima with two independent variables. The search for the global optimum inevitably led to a variant method of calculation. The appearance of high-speed electronic digital computers considerably increased the speed and accuracy of determining the optimum [3]. Nevertheless, electronic digital computers did not remove the essential shortcoming of the variant method—the obtaining of only a numerical solution, which subsequently cannot be subjected to analytical transformations when coordinating individual design issues (thermal and ventilation calculations, accounting for changes in specifications). This shortcoming becomes still more significant if one bears in mind the obvious necessity of subsequently carrying out broader optimization with statistical accounting for the diversity of actual operating modes of machines in service.

Proceeding from the foregoing, it seems expedient to find a sufficiently simple and sufficiently adequate analytical form that would reflect the presence of all essential relationships among the variables. This proves possible with the aid of the proposed method of mathematical planning of the calculation, developed on the basis of the recently emerged theory of factor analysis [4]. The effectiveness of the new method increases with the growth in the number of independent variables included in the consideration. Essential here is the possibility of obtaining a mathematical-economic model. This model is final under the adopted method of evaluating economic efficiency, and various particular problems (changes in technical

conditions, and the range of materials used) are solved analytically, with the solutions always obtained as optimal ones.

Mathematically, the problem consists in determining the coefficients in the regression equation

\[ \hat y=\psi(x_1,x_2,\ldots,x_h)=b_0+\sum_i^h b_i x_i+\sum_i^h \sum_j^h b_{ij}x_i x_j . \tag{1} \]

\(y\) is the actual cost of the machine,

\(\hat y\) is the estimate of \(y\) according to equation (1),

\(x_i\) are the design dimensions of the machine,

\(h\) is the number of dimensions taken into account.

The terms \(b_i x_i\) characterize linear effects, \(b_{ii}x_i^2\) quadratic effects, and \(b_{ij}x_i x_j\) effects of pairwise interaction.

Equation (1) is called the response function, and the corresponding geometric image in the space of the variables \(x_i\) is called the response surface.

For most problems it is sufficient to have a linear or quadratic form (1).

The linear form is usually used for gradient ascent into the region of the extremum.

Motion by the method of steepest ascent ends when we enter an almost stationary region that cannot be described by means of a linear approximation. In this part of the response surface, the dominant ones are the regression coefficients characterized by interaction effects. An almost stationary region can usually be described by second-order polynomials. A priori, the quadratic form may be considered sufficiently adequate for our problem. It is not difficult to note that the quadratic dependences of M. Widmar are the intersection of the response surface [1] with hyperplanes parallel to the axes of the variables varied in the optimization.

The coefficients of equation (1) are found by the least-squares method when the independent variables are varied \(N\) times. Equating to zero the partial derivatives of the quadratic form

\[ \sum_{u=1}^{N} (y_u-\hat y_u)^2=\min , \tag{2} \]

taken with respect to the variables \(b_i\), we obtain a system of so-called normal equations.

The coefficients \(b_i\) are the solution of the system obtained.

To simplify the notation, let us turn to matrix algebra. We shall call the rectangular \((h,N)\)-matrix \(X\) the matrix of independent variables, the column matrix \(Y\) the vector of observations, and the column matrix \(B\) the vector of regression coefficients.

In matrix form the system of normal equations is written as follows:

\[ X^{*}XB=X^{*}Y \tag{3} \]

The solution of system (3) is the expression

\[ B=(X^{*}X)^{-1}(X^{*}Y) \tag{4} \]

In the case of linearity of the true relation between \(Y\) and \(X\), solution (4) does not depend on the arrangement of the experimental points in the factor space. However, if in the matrix \(X\) the scalar products for all column vectors are equal to zero \(\left(\sum_u^N x_{iu}x_{ju}=0\right)\), i.e., the matrix \(X\) is planned orthogonally, then the matrix \((X^{*}X)\) is diagonal, and we are freed from the necessity of solving system (3).

The orthogonality criterion is not a sufficiently strong optimality criterion for a second-order design.

The information about the response surface contained in the regression equation can be estimated by means of the inverse standardized variance \(1/N\sigma^2(\hat y)\). For different points of the factor space this quantity assumes different values. It therefore makes sense to consider the information contours of the surface of equal information. A design possessing the property of a uniform distribution of information is called a rotatable design. When the factor axes are rotated in space, the accuracy of determining the regression coefficients does not change.

In the mathematical theory of experiment, methods have now been developed for constructing second-order rotatable designs.

Thus, finding the analytical representation (1) is reduced to a verification calculation of \(y\) for various combinations \(x_i\), specified according to a definite design, and to the subsequent determination of the regression coefficients.

To take into account the constraints imposed by technical conditions, let us consider other response surfaces:

\[ \begin{aligned} y_1 &= \varphi_1(x_1,x_2,\ldots,x_h)\\ y_i &= \varphi_i(x_1,x_2,\ldots,x_h)\\ &\ldots \ldots \ldots \ldots\\ y_m &= \varphi_m(x_1,x_2,\ldots,x_h), \end{aligned} \tag{5} \]

where \(y_i\) is the \(i\)-th technical condition (power, starting torque, etc.). We obtain these response surfaces by a verification calculation with the same design as for \(\psi(x_1,x_2,\ldots,x_h)\).

Optimization of (1) under constraints (5) can be carried out, for example, by using the method of undetermined Lagrange multipliers. This method reduces to solving the system:

\[ \begin{aligned} \frac{\partial \psi}{\partial x_1} + \lambda_1 \frac{\partial \varphi_1}{\partial x_1} + \lambda_2 \frac{\partial \varphi_2}{\partial x_1} + \ldots + \lambda_m \frac{\partial \varphi_m}{\partial x_1} &=0,\\ \frac{\partial \psi}{\partial x_2} + \lambda_1 \frac{\partial \varphi}{\partial x_2} + \lambda_2 \frac{\partial \varphi_2}{\partial x_2} + \ldots + \lambda_m \frac{\partial \varphi_m}{\partial x_2} &=0,\\ &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots\\ \frac{\partial \psi}{\partial x_h} + \lambda_1 \frac{\partial \varphi_1}{\partial x_h} + \lambda_2 \frac{\partial \varphi_2}{\partial x_h} + \ldots + \lambda_m \frac{\partial \varphi_m}{\partial x_h} &=0,\\ \varphi_1(x_1,x_2,\ldots,x_h)&=y_1\\ &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots\\ \varphi_m(x_1,x_2,\ldots,x_h)&=y_m \end{aligned} \tag{6} \]

The solution of system (6) is the vector of optimal dimensions \(X_{\max}\).

System (6) can be solved by computational mathematics, for example, by the method of steepest descent.

Next let us represent \(X_{i\max}\) in the form of a quadratic polynomial in \(Y\):

\[ x_{1\max}=a_{10}+\sum_i^m a_{1i}y_i+\sum_i^m\sum_j^m a_{1ij}y_i y_j, \]

\[ x_{2\max}=a_{20}+\sum_i^m a_{2i}y_i+\sum_i^m\sum_j^m a_{2ij}y_i y_j, \tag{7} \]

\[ \ldots \ldots \ldots \ldots \ldots \]

\[ x_{h\max}=a_{h0}+\sum_i^m a_{hi}y_i+\sum_i^m\sum_j^m a_{hij}y_i y_i \]

To find the coefficients of form (7), we shall again apply rotatable planning, this time varying the components \(y_i\).

Let us rewrite system (7) in matrix form

\[ x_m=AY \tag{8} \]

The matrix \(A\) may be regarded as a mathematical-economic model of a segment of the series, since it transforms the vector of technical conditions \(Y\) into the vector of dimensions \(X_{\max}\) corresponding to the optimal machine.

Planning the calculation sharply reduces the number of variants; for example, for 7 independent variables it is sufficient to carry out a verification calculation for 163 combinations. In addition, besides the numerical solution (6), we also obtain a number of important analytic dependences (1) and (5), which we can use for other purposes, for example in solving accuracy problems.

REFERENCES

1. M. Vidmar. Economic Laws of the Design of Electrical Machines, GONTI, 1930.

2. I. M. Postnikov. Design of Electrical Machines. Gostekhizdat of the Ukrainian SSR, 1960.

3. B. M. Kagan, T. M. Ter-Mikaelyan. Solution of Engineering Problems on Digital Computing Machines, Energiya, 1964.

4. V. V. Nalimov, N. A. Chernova. Statistical Methods for Planning Extreme Experiments, Nauka, 1965.