ON SOME PROBLEMS IN THE DESIGN OF ELECTRICAL MACHINES ON DIGITAL COMPUTERS

E. K. STRELBITSKII, A. S. GITMAN

(The work was presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

After the appearance of electronic digital computing machines (digital computers) in the field of electrical-machine design, a new stage began. If previously a design engineer, having obtained a more or less satisfactory variant of the calculation of an electrical machine, would stop further search, then the use of digital computers significantly expanded the possibilities in design. The application of digital computers to the design of series of electrical machines with the search for optimal variants became especially attractive. The first such work was begun at VNIIEM in 1956 in developing the AO 2 series of asynchronous motors. About 10 years have passed since then, and it is now possible to draw some conclusions and consider problems that still remain unresolved.

In a short period of time, a large number of mathematical methods were proposed for the design of electrical machines, and the number of dimensions of the space in which the search for an optimum is carried out increased considerably. It is quite obvious that the expansion of the use of mathematical methods was closely connected with the appearance of more powerful digital computers. If, in designing the AO 2 series, the search for the optimum was carried out in a space of three or four dimensions, then problems are now being solved in a seven-dimensional space. If, for designing the AO 2 series, the method of traversing the nodes of a spatial grid was used (the blind-search method), then by the present time a method has been proposed that uses elements of dynamic programming [2], as well as a random-search method with self-learning [3]. Never before had such rapid renewal of methods taken place in such a short period of time. Nevertheless, it should be noted that the improvement of methods is one-sided in character. If one abstracts from specific features, all the proposed methods can be reduced to the trial-and-error method, and design is still carried out only by the method of analysis. It is known that an indicator of the degree of development of any theory is its ability to solve the posed problems by the method of synthesis. In this sense, the theory of designing electrical machines with the search for an optimum has hardly been developed. It should be expected that in the near future, for the design of electrical machines, it will be necessary to sharply increase the number of dimensions of the space in connection with the new problems facing electrical-machine building. In particular, in design it will be necessary to consider factors affecting reliability, to take into account the actual loads in operation, etc. It may turn out that the most acceptable methods

the search will be methods based on synthesis, since in solving multidimensional nonlinear programming problems by methods of analysis, greater difficulties are encountered.

Another unsolved problem is the question of the uniqueness of the solution. Since the overwhelming majority of methods do not make it possible to assess whether the optimum value found is global, there is always room for doubt that the best solution has been obtained. The question of the uniqueness of the solution becomes of primary importance if one takes into account that the manufacture of many types of electrical machines is carried out in series, and kopecks saved on each machine quickly turn into millions of rubles. Of the widely used methods, only two—the method of traversing the nodes of a spatial grid and the Monte Carlo method—make it possible to give an estimate of the accuracy of determining the global extremum: in the first case with accuracy up to the step, and in the second with a certain probability. But with a relatively large number of variables, both methods prove unsuitable because of the excessively large expenditure of machine time. When using the method of traversing the nodes of a spatial grid with steps amounting to 10% of the interval of variation of the variables, the required number of variants for finding the global optimum with \(k\) variables is

\[ N_j = 10^k . \]

For the Monte Carlo method, the required number of variants is determined by the formula [1]

\[ N = \frac{\ln(1-p)}{\ln(1-\Delta)}, \]

where

\[ \begin{aligned} p &\text{ is the probability of determining the global optimum;}\\ \Delta &\text{ is the volume of the } k\text{-dimensional cube, expressed as a fraction of the total volume of the search region.} \end{aligned} \]

If each edge of the \(k\)-dimensional cube is taken equal to 0.1 of the edge of the overall \(k\)-dimensional cube, then the volume \(\Delta\) for \(k\) variables will be equal to \(10^{-k}\). For a satisfactory search for the optimum of electrical machines, at least seven independent variables are necessary. Then the required number of variants in the first case will be

\[ N = 10^7, \]

and in the second—at probability \(p = 0.95\),

\[ N = 3 \cdot 10^7 . \]

Taking into account that even on the M-20 digital computer the computation time for one variant of an induction motor is 0.25 sec, we obtain a total time for \(N = 10^7\) variants equal to approximately 700 hours. Solving such problems on modern digital computers is hardly expedient.

The possibility of the existence of a number of local extrema is evidenced by an example obtained in the design of induction motors. For a two-pole motor with a rated power of 7.5 kW, the search was carried out with six variables:

\[ \begin{aligned} \text{bore diameter} &\ — && D_i,\\ \text{number of conductors in the stator slot} &\ — && S_p,\\ \text{stator tooth width} &\ — && b_{z1},\\ \text{rotor tooth width} &\ — && b_{z2},\\ \text{stator tooth height} &\ — && h_{z1},\\ \text{rotor tooth height} &\ — && h_{z2}. \end{aligned} \]

As the optimum criterion, the sum of the costs of manufacturing the motor and of the energy losses during operation of the mot-

motor [1]. The range of admissible values was determined by constraints: the ratios of the maximum and initial starting torques, the starting-current ratio, the rate of temperature rise in the short-circuit mode, and the temperature rise in the stator winding in the rated mode.

The search was carried out by two methods: the method of multistage processes [2] and the Gauss–Seidel method [4]. The flowcharts of the solution are shown in Fig. 1 and Fig. 2. In both cases the steps of the variables were taken—

Text in Fig. 1:

• Input of initial data
• Selection of \(D_i\)
  \(D_i(k+1)=D_i(k)+\Delta D_i\)
• Check
  \(c(D_i(k+1))<c(D_i(k))\)
• Yes
• No
• Optimization of the choice of \(\Delta D_i\), taking constraints into account
• Selection of \(S_n\)
  \(S_n(k+1)=S_n(k)+\Delta S_n\)
• Optimization of the choice of \(\Delta S_n\), taking constraints into account
• Check
  \(c(S_n(k+1))<c(S_n(k))\)
• Yes
• No
• Selection of \(b_{z1}\)
  \(b_{z1}(k+1)=b_{z1}(k)+\Delta b_{z1}\)
• Check
  \(c(b_{z1}(k+1))<c(b_{z1}(k))\)
• Yes
• No
• Optimization of the choice of \(\Delta b_{z1}\), taking constraints into account
• Selection of \(b_{z2}^{0}\)
  \(b_{z2}^{0}(k+1)=b_{z2}^{0}(k)+\Delta b_{z2}^{0}\)
• Check
  \(c(b_{z2}^{0}(k+1))<c(b_{z2}^{0}(k))\)
• Yes
• No
• Optimization of the choice of \(\Delta b_{z2}^{0}\), taking constraints into account
• Selection of \(h_{z1}\)
  \(h_{z1}(k+1)=h_{z1}(k)+\Delta h_{z1}\)
• Check
  \(c(h_{z1}(k+1))<c(h_{z1}(k))\)
• Yes
• No
• Optimization of the choice of \(\Delta h_{z1}\), taking constraints into account
• Selection of \(h_{z2}\)
  \(h_{z2}(k+1)=h_{z2}(k)+\Delta h_{z2}\)
• Check
  \(c(h_{z2}(k+1))<c(h_{z2}(k))\)
• Yes
• No
• Optimization of the choice of \(\Delta h_{z2}\), taking constraints into account
• Printing of the optimal variant

Fig. 1. Flowchart of the solution by the Gauss–Seidel method.

were taken to be identical, i.e., the region of admissible values was divided by one and the same grid (in the block diagrams, the increment \(\Delta\) is understood as a generalized step, a multiple of the unit step). The results of the solution are presented in Table 1.

Flowchart text

• Input of initial data.
• Selection of \(D_i\):

\[ D_i(k+1)=D_i(k)+\Delta D_i \]

• Optimization of the choice of \(\Delta D_i\), taking the constraints into account.
• Selection of \(S_n\):

\[ S_n(k+1)=S_n(k)+\Delta S_n \]

• Optimization of the choice of \(\Delta S_n\), taking the constraints into account.
• Selection of \(b_{z1}\):

\[ b_{z1}(k+1)=b_{z1}(k)+\Delta b_{z1} \]

• Optimization of the choice of \(\Delta b_{z1}\), taking the constraints into account.
• Selection of \(b_{z2}\):

\[ b_{z2}(k+1)=b_{z2}(k)+\Delta b_{z2} \]

• Optimization of the choice of \(\Delta b_{z2}\), taking the constraints into account.
• Selection of \(h_{z1}\):

\[ h_{z1}(k+1)=h_{z1}(k)+\Delta h_{z1} \]

• Optimization of the choice of \(\Delta h_{z1}\), taking the constraints into account.
• Selection of \(h_{z2}\):

\[ h_{z2}(k+1)=h_{z2}(k)+\Delta h_{z2} \]

• Optimization of the choice of \(\Delta h_{z2}\), taking the constraints into account.
• Printing of the optimal variant.
• Check:

\[ C(D_i(k+1))<C(D_i(k)) \]

yes / no

• Check:

\[ C(S_n(k+1))<C(S_n(k)) \]

yes / no

• Check:

\[ C(b_{z1}(k+1))<C(b_{z1}(k)) \]

yes / no

• Check:

\[ C(b_{z2}(k+1))<C(b_{z2}(k)) \]

yes / no

• Check:

\[ C(h_{z2}(k+1))<C(h_{z2}(k)) \]

yes / no

• Check:

\[ C(h_{z1}(k+1))<C(h_{z1}(k)) \]

yes / no

Fig. 2. Flowchart of the solution by the method of multistage processes.

Table 1

It is known that a sufficient condition for the coincidence of global and local extrema is the requirement of convexity of the objective function and of the region of admissible values [5]. If, as follows from Table 1, different methods on one and the same grid yield different optimum values, then either one of the requirements, or both requirements at once, concerning convexity have been violated. Of course, generalizations cannot be made on the basis of isolated examples. For the present, one conclusion may be drawn: very serious attention should be paid to the problem of uniqueness.

References

1. B. M. Kogan, T. M. Ter-Mikaelyan. Solving Engineering Problems on Digital Computers, Energiya, 1964.

2. J. Avetisyan, A. I. Bertinov. Multistage processes for selecting optimal dimensions of electrical machines, Elektrichestvo, No. 6, 1966.

3. T. G. Soroker. Application of automatic digital computers in the design of new series of asynchronous motors, VNIEM, 1966.

4. J. A. Rastrigin. Random Search in Problems of Optimization of Multiparameter Systems, Znanie, Riga, 1965.

5. S. I. Zukhovitskii, L. I. Avdeev. Linear and Convex Programming, Nauka, 1964.