Application of the Statistical Design Method for Modeling Predicted Parameters of Asynchronous Motors
Abstract Generated abstract
This paper addresses the forecasting of asynchronous motor parameters whose values depend on power, speed, and time, making separate one-factor regression curves impractical for large information arrays. It proposes using statistical experimental design, specifically a second-order composite rotatable design with coded variables, to construct a quadratic regression model including interactions among power, speed, and time. The method is illustrated for the parameter weight per unit power, using geometric constructions to obtain design-point values from existing regression curves and then estimate model coefficients. Model checking in the example gives an acceptable adequacy test at the 5 percent significance level, and the reported average accuracy of determining parameter values is about 92 to 93 percent.
Full Text
APPLICATION OF THE STATISTICAL PLANNING METHOD FOR MODELING PREDICTED PARAMETERS OF ASYNCHRONOUS MOTORS
O. P. MURAVLEV, O. L. RAPOPORT
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
Among the numerous problems that have arisen in connection with the creation of a forecasting system for the electrical-machine-building industry, an important place is occupied by the development of forecasts for the parameters of asynchronous motors. The technical characteristics of these machines for various series produced by different countries and firms constitute extensive bodies of information. Therefore, it is most expedient to adopt, as the basic method for processing information arrays, the statistical method, as a result of which regression equations are obtained that concentrate within themselves the entire volume of the initial information.
The parameters of electrical machines, in particular asynchronous motors, depend on power and speed and change with time. Moreover, several speeds correspond to each power. All this leads to the fact that all the initial information on the parameters must be divided into groups, each corresponding to definite values of power and speed. Processing the statistical sets of parameters in each such group will make it possible, for each parameter that must be forecast, to construct a model in the form of a regression equation for the investigated parameter over time at a given power and speed. If \(P\) is the predicted parameter and \(t\) is time, then at constant power
\[ (P_{\text{kW}}=\text{const}) \]
and constant speed
\[ \left(n\ \frac{\text{rev}}{\text{min}}=\text{const}\right) \]
the dependence \(P(t)\) is called one-factor. The number of regression equations \(P(t)\) for one parameter \(P\) (for example, weight) is determined by the number of motor frame sizes considered and by the number of pole pairs. If \(n_1\) is the number of powers taken into account in the range considered, and \(n_2\) is the number of speeds, then the number of regression equations for one predicted parameter is determined by the formula
\[ c = n_1 \cdot n_2 . \tag{1} \]
Forecasting a large number of parameters and using the curves \(P(t)\) is not practically convenient because of the large number of these curves. Therefore, in this article it is proposed to use the method of experiment planning to obtain a model that relates the predicted parameter to power, speed, and time in the form of a second-degree polynomial of the type:
\[
P = b_0 + b_1x_1 + b_2x_2 + b_3x_3 + b_4x_1^2 + b_5x_2^2 + b_6x_3^2 + b_7x_1x_2 +
\]
\[
+ b_8x_1x_3 + b_9x_2x_3 ,
\tag{2}
\]
where
\[ x_1=P_{\mathrm{kW}} \quad \text{— power of asynchronous motors;} \]
\[ x_2=n\ \frac{\mathrm{rev}}{\mathrm{min}} \quad \text{— speed;} \]
\[ x_3=t_{\text{years}} \quad \text{— time;} \]
\[ P \quad \text{— predicted parameter.} \]
Such a polynomial model is of no interest from the standpoint of cognition [1], but in practical terms such a model is very useful in planning an experiment. It is convenient in that it makes it possible to improve the approximation by increasing the order of the polynomial without thereby disturbing the relationships among the parameters of the approximating function; this makes it possible to use such statistical procedures as the least-squares method for estimating parameters, the selection of an optimal arrangement of points in the space of independent variables, etc.
Before considering the method for constructing such a model, it is necessary to make several assumptions.
-
The initial sets of parameters at each point on the time axis \(t\) are distributed according to the normal law. For them, standard deviations \(\sigma\{P\}\) and mean values \(P\) are found, from which adequate regression lines are drawn.
-
These regression lines are smooth functions with no more than one extremum. Otherwise, it becomes necessary to divide \(P(t)\) into intervals with respect to the argument \(t\), each containing only one extremum.
Studies carried out on asynchronous motors have proven the validity of these assumptions, and they may be extended to machines of other classes when predicting their parameters.
Proceeding from the conditions of obtaining a quadratic model with all interactions, of extracting the maximum information from the regression curves, and of the orthogonality of the design in order to eliminate ambiguity in determining the coefficients of the model, we used a second-order compositional rotatable design of type \(2^3\). The complications that arise in this case are not connected with the difficulty of carrying out a “live” experiment and are resolved by computational and graphical means. A large saving of time is provided by the use of computers. The design matrix is presented in Table 1 for three variables.
The design is carried out at five levels of each independent variable. To simplify the calculations, the values of the levels of the factors under study are transformed into coded units. The values of the levels are established so that, when converted to the coded scale, they correspond to \(-\alpha\), \(-1\), \(0\), \(+1\), \(+\alpha\). The levels are calculated as follows:
1) the limits of variation of power, speed, and time are established, determining the number of predicted standard sizes of motors;
2) the values of the zero level are found in accordance with the ranges of variation of power, speed, and time;
3) the limits of variation of these parameters are denoted by the levels \(\pm \alpha\);
4) the values of the levels \(+1\) and \(-1\) are determined from the equations:
\[ \begin{aligned} x_i^{+1} &= x_i^0 + h,\\ x_i^{-1} &= x_i^0 - h, \end{aligned} \tag{3} \]
Table 1
Matrix of composite rotatable planning for three variables
| Exp. no. | $x_0$ | $x_1$ | $x_2$ | $x_3$ | $x_1^2$ | $x_2^2$ | $x_3^2$ | $x_1x_2$ | $x_1x_3$ | $x_2x_3$ |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | $-1$ | $-1$ | $-1$ | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 1 | $-1$ | $-1$ | 1 | 1 | 1 | $-1$ | $-1$ | 1 |
| 3 | 1 | $-1$ | 1 | $-1$ | 1 | 1 | 1 | $-1$ | 1 | $-1$ |
| 4 | 1 | 1 | 1 | $-1$ | 1 | 1 | 1 | 1 | $-1$ | $-1$ |
| 5 | 1 | $-1$ | $-1$ | 1 | 1 | 1 | 1 | 1 | $-1$ | $-1$ |
| 6 | 1 | 1 | $-1$ | 1 | 1 | 1 | 1 | $-1$ | 1 | $-1$ |
| 7 | 1 | $-1$ | 1 | 1 | 1 | 1 | 1 | $-1$ | $-1$ | 1 |
| 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 9 | 1 | $-\alpha$ | 0 | 0 | $\alpha^2$ | 0 | 0 | 0 | 0 | 0 |
| 10 | 1 | $+\alpha$ | 0 | 0 | $\alpha^2$ | 0 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | $-\alpha$ | 0 | 0 | $\alpha^2$ | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | $+\alpha$ | 0 | 0 | $\alpha^2$ | 0 | 0 | 0 | 0 |
| 13 | 1 | 0 | 0 | $-\alpha$ | 0 | 0 | $\alpha^2$ | 0 | 0 | 0 |
| 14 | 1 | 0 | 0 | $+\alpha$ | 0 | 0 | $\alpha^2$ | 0 | 0 | 0 |
| 15 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 19 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 20 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
where
\[ h=\frac{x_{i\max}-x_i^0}{+\alpha} =\frac{x_{i\min}-x_i^0}{-\alpha}; \tag{4} \]
$h$ is the variation step,
$x_i^0$ is the value of the factor at the zero level,
$x_{i\max}$ and $x_{i\min}$ are the limits of factor variation.
Table 2 gives the calculation data for the range of variation of power from 2 to 5.5 kW, speed from 1000 to 3000 rpm, and time from 1950 to 1970.
Table 2
| Variables | Variation step | Values of variable levels | Values of variable levels | Values of variable levels | Values of variable levels | Values of variable levels |
|---|---|---|---|---|---|---|
| Variables | Variation step | $-1.682$ | $-1$ | 0 | $+1$ | $+1.682$ |
| Power $(x_1)$ | 1.04 | 2.0 | 2.66 | 3.75 | 4.79 | 5.5 |
| Speed $(x_2)$ | 594 | 1000 | 1406 | 2000 | 2594 | 3000 |
| Time $(x_3)$ | 5.94 | 1950 | 1954.1 | 1960 | 1965.9 | 1970 |
The coefficients of equation (2) for three variables are calculated from the results of 20 experiments. In doing so, a difficulty arises in determining the value of the predicted parameter $P$, associated with the discrete character of the variables $x_1$, $x_2$, $x_3$. We propose a solution that makes it possible, by means of geometric constructions, to determine the values of $P$ for any experiment according to the plan in Table 1. For this purpose, regression curves $P(t)$, obtained from statistical analysis of the initial information for different values of power and speed, are taken as the initial data.
The idea is to find the form of the dependences $P(x_1)$ and $P(x_2)$. Their construction can be easily seen from the figures provided. In the right-hand parts of Figs. 1–3, the initial curves are shown for the predicted para-
Fig. 1
Fig. 2
of the parameter “weight per unit power \(G/P\).” On the time axis \(t\) there are points corresponding to the levels \(x_3\) of Table 2, and perpendiculars are restored. The sections of the curves \(P(t)\) are unfolded in the left-hand parts of Figs. 1–3. In this case \(x_1\) is taken as the abscissa axis, and the curves are constructed for different levels of \(x_3\). The curves \(P(x_1)\) obtained in this way, like \(P(t)\), form three groups, in each of which \(x_2=\mathrm{const}\), but between the groups \(x_2\) is different \((1000\ \text{rpm};\ 1500\ \text{rpm};\ \text{and }3000\ \text{rpm for our example}\)
Fig. 3
example). It is now necessary to obtain the curves \(P(x_2)\), but for this it is necessary that, for each newly obtained group of curves, the time \(x_3\) be fixed. For this purpose, in the left-hand parts of Figs. 4–8, the curves \(P(x_1)\) with \(t=\mathrm{const}\) are transferred onto the same axes \(P\) and \(x_1\). Groups of curves \(P(x_1)\) with different speeds are obtained. Then the power axis \(x_1\) is divided into the levels of Table 2, and vertical sections of the curves \(P(x_1)\) are drawn. It is necessary to stipu-
Fig. 4
Fig. 5.
\(t_{-1}=1954.1;\quad G/p\,[\mathrm{kg}^2/(\mathrm{kW}\cdot \mathrm{m})]\)
Fig. 6.
\(t_{0}=1960;\quad G/p\,[\mathrm{kg}^2/(\mathrm{kW}\cdot \mathrm{m})]\)
…it can be seen that, in the left-hand parts of Figs. 1–3, many curves \(P(x_1)\) in this case have two extrema each; therefore the range of variation of \(x_1\) under consideration is divided into two intervals: from 2 to 5.5 kW and from 5.5 to 10 kW (in the article, the levels of \(x_1\) are given only for the first interval, 2–5.5 kW, in Table 2).
The sections of the curves \(P(x_1)\) must be expanded in the right-hand parts of Figs. 4–8 along the \(x_2\) axis, obtaining the desired curves \(P(x_2)\) for different \(x_1\) at fixed \(x_3=t\). The values of the levels from Table 2 must be plotted on the \(x_2\) axis.
Fig. 7
Fig. 8
To determine the predicted parameter in accordance with the experimental plan, it is now necessary to take the graph corresponding to the fixed level \(x_3\), find on the axis the point of the level \(x_2\), and draw a vertical line to intersect the curve corresponding to the level \(x_1\). The ordinate of the point of intersection gives the desired value of the predicted parameter \(P\) according to the plan of Table 1.
Similar constructions can be carried out for a larger number of variables \(x\), if this is not restricted by the planning method. It should be noted that the error in finding the values of the parameter \(P\) is determined by the number of values of power and speed adopted for prediction, as well as by the accuracy with which the geometric constructions are carried out. Work on predicting the parameters of general-purpose asynchronous motors has shown that, for the adopted values \(x_1\), \(x_2\), and \(x_3\), the accuracy of determining the value of the predicted parameter \(P\) is on average 92–93 percent.
The coefficients of equation (2), adopted as the model, are determined by the formulas [2]:
\[ b_0=\frac{A}{N}\left[2\lambda_4^2(k+2)(Oy)-2\lambda_4 c\sum_{i=1}^{k}(iy)\right]; \tag{5} \]
\[ b_i=\frac{c}{N}(iy); \tag{6} \]
\[ b_{ii}=\frac{A}{N}\left[c^2\left[(k+2)\lambda_4-k\right](iiy)+c^2(1-\lambda_4)\sum_{i=1}^{k}(iiy)-2\lambda_4 c(Oy)\right]; \tag{7} \]
\[ b_{ij}=\frac{c^2}{N\lambda_4}(ijy), \tag{8} \]
where
\[ c=\frac{N}{\sum_{u=1}^{N}x_{iu}^{2}};\qquad A=\frac{1}{2\lambda_4\left[(k+2)\lambda_4-k\right]}; \tag{9} \]
\[ \lambda_4=\frac{k(n_0+n_{\mathrm{p}})}{(k+2)n_{\mathrm{p}}}; \tag{10} \]
\(N\) is the number of experiments;
\(n_0\) is the number of zero points;
\(n_{\mathrm{p}}\) is the number of peripheral points;
\(k\) is the number of parameters (factors).
Turning to the numerical example of finding a model of the parameter
\[ \frac{G}{P}\left[\frac{kg}{kW}\right] \]
as a function of power, speed, and time, one can write the values of the coefficients of equation (2), which were obtained after carrying out the corresponding geometric constructions and calculations.
\[ P=\frac{G}{P}=12.2735-1.0822x_1-1.4962x_2-1.5904x_3+0.5299x_1^2+ \]
\[ +1.3145x_2^2+0.1126x_3^2+0.155x_1x_2+0.4299x_1x_3+0.020x_2x_3. \tag{11} \]
In order to convert equation (11) into the actual values of the variables, it is necessary to make a substitution, replacing \(x_i\) by the expression
\[ x_i=\frac{X_i-x_{0i}}{h_i}, \tag{12} \]
where
\(x_i\) are the variable factors in coded units (in our case these are \(x_1\), \(x_2\), and \(x_3\));
\(X_i\) are the same variables in actual units;
\(h_i\) is the variation step of the \(i\)-th factor;
\(x_{0i}\) is the value of the zero level.
To assess the correctness of the model, we obtained values of $\hat{P}$ for each trial of the experimental design, which are compared with the values of the predicted parameter $P$ from the geometric constructions. They are given in Table 3.
Table 3
| No. | $G/P$ experimental value | $G/P$ calculated value | No. | $G/P$ experimental value | $G/P$ calculated value |
|---|---|---|---|---|---|
| 1 | 18.10 | 19.00 | 11 | 21.00 | 19.51 |
| 2 | 14.86 | 15.67 | 12 | 11.80 | 13.47 |
| 3 | 16.60 | 15.67 | 13 | 14.95 | 15.27 |
| 4 | 13.80 | 12.95 | 14 | 11.05 | 9.92 |
| 5 | 13.50 | 14.92 | 15 | 12.25 | 12.27 |
| 6 | 11.80 | 13.31 | 16 | 12.25 | 12.27 |
| 7 | 11.90 | 11.66 | 17 | 12.25 | 12.27 |
| 8 | 16.00 | 15.59 | 18 | 12.25 | 12.27 |
| 9 | 11.00 | 10.66 | 19 | 12.25 | 12.27 |
| 10 | 12.35 | 11.95 | 20 | 12.25 | 12.27 |
Verification of the adequacy of the obtained model showed that the calculated $F$-ratio, 4.37, is less than the tabulated value $F = 4.74$ for the 5% significance level.
Conclusions
1) This article proposes a method for obtaining models of predicted parameters of asynchronous motors as a function of several variables taking discrete values, using the method of experimental design. This method may also be applied to modeling information arrays of parameter values for other machines.
2) To implement the design, geometric constructions are proposed that make it possible to determine the values of the predicted parameter in accordance with the experimental design. In this case, the error in their determination is no more than 7–8%.
References
- V. V. Nalimov. Theory of Experiment. Moscow, 1971.
- V. V. Nalimov, N. A. Chernova. Statistical Methods for Planning Extremal Experiments. Moscow, 1965.