Application of the Mathematical Method of Experimental Design for the Optimal Design of the Toothed Zone of an Impact Generator
Abstract Generated abstract
This paper applies mathematical experimental design to optimize the rotor tooth zone of a limiting-power impact generator under air-gap inductions from 1 to 2.4 T. Using a second-order rotatable three-factor design, it models field-winding current density as a function of the relative tooth-slot parameter, the fraction of rotor slots occupied by winding, and the total number of rotor slots, including selected quadratic and interaction effects. The resulting regression equations are assessed for adequacy and coefficient significance, then used to identify local optima within the investigated design region. The study finds induction-dependent optimal values for the tooth parameter, an optimal slot-filling ratio at the upper boundary of the tested range, and, except at 1 T, an optimal rotor slot number of 40.
Full Text
APPLICATION OF THE MATHEMATICAL METHOD OF EXPERIMENTAL DESIGN FOR OPTIMAL DESIGN OF THE TOOTH ZONE OF AN IMPACT GENERATOR ROTOR
V. I. Andreev
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
The rotor of an impact generator is one of the main links in obtaining maximum pulse power. As is known, it is manufactured with the same overall dimensions as the rotor of a large turbogenerator, but the method for designing the tooth zone has its own special features. The principal difference lies in the specific operating conditions of the impact generator. Its rotor is overloaded mechanically, electromagnetically, and thermally several times more severely than the corresponding turbogenerator rotor. Therefore, the question of the optimal design of its tooth zone and the choice of the field winding and damper winding in order to increase the induction in the air gap is a pressing one.
In the present work, the tooth geometry of the rotor is optimized for various inductions in the gap. The damper winding is uniformly laid over the entire circumference of the rotor and occupies 40% of the cross section of the field-winding slot [1].
The methods for obtaining the optimal geometrical parameters of the rotor are various. One method is to enumerate all values of the independent variables and directly obtain the optimal value of the output quantity. However, in this case the number of experiments will be a large quantity equal to \(m^n\), where \(m\) is the number of different states of the input quantity, and \(n\) is the number of independent variables. For example, with five different states of three independent variables, the number of experiments will be \(N = 125\). Another optimization method is the stepwise search method (or gradient method) [8]. It somewhat reduces the number of experiments. However, all the methods listed, as well as a number of others, do not make it possible to take into account the influence of paired interactions of the independent variables on the output quantity.
To obtain maximum information and minimum expenditure, we apply the method of mathematical experimental design. The advantages of this method are described rather fully in [5, 6, 7].
The initial data characterizing the tooth geometry are: \(\lambda\)—the ratio of the difference between the maximum and minimum tooth widths to the tooth pitch (explanations concerning the choice of this quantity as an independent variable will be given below), \(\gamma\)—the ratio of the number of rotor slots filled with winding to the total number of slots, and \(N_R\)—the total number of slots. As the output parameter, for different values of induction, the current density \((j)\) in the field winding was taken.
In the event of a sudden short circuit, a powerful damper winding ...
reliably protects the field winding from armature reaction, and therefore the values of the field-winding current remain practically unchanged throughout the entire transient process.
The investigations were carried out for a limiting-power impact generator with the following overall dimensions: \(l_R = 6\ m,\ D_R = 1.25\ m,\ \delta = 0.03\ m\).
If, in a turbogenerator rotor, the minimum tooth cross section is selected on the basis of mechanical strength, technological factors, and also the permissible induction in the narrow section of the tooth (usually \(b_{z\min} = 1.35\ cm\)) [3], then for the rotor of an impact generator it is expedient to take into account only the first two factors [4].
For a rectangular slot:
\[ t_{z\min} - b_{z\min} = t_z - b_z, \]
\[ b_z = \frac{\Pi}{N_R}\left(D_R - D_{R\min}\right) + 1.35 \]
or
\[ b_z = \frac{2\Pi h_{nR}}{N_R} + 1.35, \]
whence
\[ h_{nR} = \frac{N_R}{2\Pi}\left(b_z - 1.35\right) \]
or
\[ h_{nR} = \frac{D_R}{2}\,\frac{\left(b_z - 1.35\right)}{t_z}. \]
Denoting
\[ \lambda = \frac{b_z - 1.35}{t_z}, \tag{1} \]
we obtain
\[ h_{nR} = \lambda\,\frac{D_R}{2}, \]
whence
\[ \lambda = \frac{2h_{nR}}{D_R}, \tag{2} \]
where \(t_z,\ b_z\) are the tooth pitch and the tooth width along the outer circumference of the rotor,
\(t_{z\min},\ b_{z\min}\) are the tooth pitch and the tooth width at the base of the slot,
\(D_{R\min}\) is the rotor diameter at the base of the teeth.
Thus, the coefficient \(\lambda\) is universal for rotors of different diameters. Its use in planning the experiment halves the number of independent variables. The optimum value of \(\lambda\) will correspond, for all diameters, to nonsalient-pole rotors of synchronous machines.
In the present work, on the basis of a priori information, a second-order rotatable design of type \(2^3\) was used.
The adopted limits of variation of the independent variables and the notation are given in Table 1.
Table 1
| Variables | Designations | \(-\alpha\) | \(-1\) | \(0\) | \(+1\) | \(+\alpha\) |
|---|---|---|---|---|---|---|
| \(\lambda\) | \(x_1\) | 0,1 | 0,1405 | 0,2 | 0,2595 | 0,3 |
| \(\gamma\) | \(x_2\) | 0,5 | 0,581 | 0,7 | 0,819 | 0,9 |
| \(N_{\mathrm{B}}\) | \(x_3\) | 40 | 48 | 60 | 72 | 80 |
where \(\alpha = 2^{K/4} = 1.682\), \(K\) is the number of independent variables;
\(y_i\) is the current density, for different values of induction.
The design matrix and the experimental results are presented in Table 2. The regression coefficients of equation (3), found by the formulas [6], are presented in Table 3.
\[
y_i = b_0 + b_1x_1 + b_2x_2 + b_3x_3 + b_{11}x_1^2 + b_{22}x_2^2 + b_{33}x_3^2 +
\]
\[
b_{12}x_1x_2 + b_{13}x_1x_3 + b_{23}x_2x_3 .
\tag{3}
\]
The analysis of the equations for adequacy, carried out at the 5% significance level, showed that the regression equations describe the response surface with an error of no more than 10%, and the error decreases for variants with greater induction.
To simplify the regression equations, we assess the significance of the coefficients, as a result of which all insignificant coefficients are discarded from the equation. Reducing the number of terms in the regression equation simplifies its analytical investigation. The significance of the coefficients is assessed by constructing a confidence interval. A sign that a coefficient is insignificant is that the absolute value of the coefficient is smaller than the value of the confidence interval.
After assessing the significance of the coefficients, the regression equations for the inductions under consideration take the following form:
\[ y_1 = 3,56 - 0,752x_1 - 0,6634x_2 + 0,24x_1^2 - 0,106x_2^2 + 0,171x_1x_2; \]
\[ y_2 = 7,72 - 0,7622x_1 - 1,4044x_2 + 0,391x_3 + 1,585x_1^2 + 0,259x_3^2; \]
\[ y_3 = 24,02 + 1,2118x_1 - 4,5601x_2 + 1,681x_3; \]
\[ y_4 = 40,14 - 2,6374x_1 - 7,9136x_2 + 2,9149x_3 + 2,24x_1^2 + 0,759x_1x_2; \]
\[ y_5 = 55,61 - 5,4593x_1 - 1,10606x_2 + 4,0505x_3 + 3,521x_1^2 + 1,031x_1x_2 + 0,965x_1x_3 . \]
Figure 1 shows the coefficients of the regression equation and the confidence intervals. Using these curves, without reproducing the design matrix, one can write the regression equation in coded values for all intermediate values of induction in the interval from 1 T to 2.4 T and for the same limits of variation of the independent variables. From the graphs in Fig. 1, without carrying out the corresponding calculations of the confidence intervals, one can assess the significance of the coefficients of the regression equations for intermediate values. It is possible to describe mathematically the changes in the regression coefficients as a function of the induction in the air gap; however, this requires considerably more time and involves certain difficulties in the description.
Table 2
| No. | $x_0$ | $x_1$ | $x_2$ | $x_3$ | $x_1^2$ | $x_2^2$ | $x_3^2$ | $x_1x_2$ | $x_1x_3$ | $x_2x_3$ | $B_\delta = 1$ T | $y_1$ | $B_\delta = 1.35$ T | $y_2$ | $B_\delta = 1.7$ T | $y_3$ | $B_\delta = 2.0$ T | $y_4$ | $B_\delta = 2.4$ T | $y_5$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | + | − | − | − | + | + | + | + | + | + | 5.19 | 10.04 | 27.5 | 49.79 | 71.25 | |||||
| 2 | + | + | − | − | + | + | + | − | − | + | 3.58 | 11.67 | 27.63 | 43.45 | 59.09 | |||||
| 3 | + | − | + | − | + | + | + | − | + | − | 3.83 | 7.84 | 20.48 | 34.84 | 51.3 | |||||
| 4 | + | + | + | − | + | + | + | + | − | − | 2.9 | 9.37 | 19.06 | 32.29 | 43.29 | |||||
| 5 | + | − | − | + | + | + | + | + | − | − | 5.27 | 10.75 | 30.69 | 56.28 | 80.61 | |||||
| 6 | + | + | − | + | + | + | + | − | + | − | 3.38 | 12.88 | 31.65 | 50.5 | 68.86 | |||||
| 7 | + | − | + | + | + | + | + | − | − | + | 3.92 | 8.53 | 22.94 | 40.85 | 58.21 | |||||
| 8 | + | + | + | + | + | + | + | + | + | + | 2.72 | 10.3 | 23.82 | 37.35 | 50.56 | |||||
| 9 | + | $-\alpha$ | 0 | 0 | $\alpha^2$ | 0 | 0 | 0 | 0 | 0 | 5.51 | 10.59 | 17.14 | 51.51 | 75.58 | |||||
| 10 | + | $+\alpha$ | 0 | 0 | $\alpha^2$ | 0 | 0 | 0 | 0 | 0 | 2.75 | 12.58 | 26.65 | 40.9 | 54.78 | |||||
| 11 | + | 0 | $-\alpha$ | 0 | 0 | $\alpha^2$ | 0 | 0 | 0 | 0 | 4.63 | 10.2 | 32.19 | 54.79 | 76.41 | |||||
| 12 | + | 0 | $+\alpha$ | 0 | 0 | $\alpha^2$ | 0 | 0 | 0 | 0 | 1.65 | 4.33 | 13.69 | 23.05 | 32.05 | |||||
| 13 | + | 0 | 0 | $-\alpha$ | 0 | 0 | $\alpha^2$ | 0 | 0 | 0 | 3.67 | 7.3 | 21.91 | 36.39 | 50.3 | |||||
| 14 | + | 0 | 0 | $+\alpha$ | 0 | 0 | $\alpha^2$ | 0 | 0 | 0 | 3.34 | 8.37 | 26.98 | 45.43 | 63.38 | |||||
| 15 | + | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3.6 | 7.82 | 24.27 | 40.47 | 56.06 | |||||
| 16 | + | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3.6 | 7.82 | 24.27 | 40.47 | 56.06 | |||||
| 17 | + | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3.6 | 7.82 | 24.27 | 40.47 | 56.06 | |||||
| 18 | + | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3.6 | 7.82 | 24.27 | 40.47 | 56.06 | |||||
| 19 | + | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3.6 | 7.82 | 24.27 | 40.47 | 56.06 | |||||
| 20 | + | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3.6 | 7.82 | 24.27 | 40.47 | 56.06 |
Table 3
| $y_i$ | $B_\delta$, T | $b_0$ | $b_1$ | $b_2$ | $b_3$ | $b_{11}$ | $b_{22}$ | $b_{33}$ | $b_{12}$ | $b_{13}$ | $b_{23}$ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| $y_1$ | 1 | 3.56 | −0.752 | −0.6634 | −0.0556 | 0.24 | −0.106 | 0.024 | 0.171 | −0.069 | 0.00375 |
| $y_2$ | 1.35 | 7.72 | 0.7622 | −1.4044 | 0.391 | 1.585 | 0.058 | 0.259 | −0.0575 | 0.0925 | −0.0375 |
| $y_3$ | 1.7 | 24.02 | 1.2118 | −4.5601 | 1.681 | −0.33 | 0.04 | 0.572 | −0.204 | 0.391 | 0.00125 |
| $y_4$ | 2.05 | 40.14 | −2.6374 | −7.9136 | 2.9149 | 2.24 | −0.335 | 0.368 | 0.759 | −0.0488 | −0.3086 |
| $y_5$ | 2.4 | 55.61 | −5.4593 | −11.0606 | 4.0505 | 3.521 | −0.349 | 0.573 | 1.031 | 0.9625 | −0.6188 |
Fig. 1. Coefficients of the regression equation and confidence intervals
The transition from coded values to physical ones was carried out by the method of [6]. The equations in physical units took the following form:
\[ \begin{gathered} y_1=12.428-56.64x_1+0.04x_2+67.8x_1^2-7.48x_2^2+24.2x_1x_2;\\ y_2=35.865-166.4x_1-11.8x_2-0.1834x_3+448x_1^2+0.0018x_3^2;\\ y_3=38.34+20.4x_1-38.3x_2+0.14x_3;\\ y_4=121.22-372.3x_1-87.9x_2+0.243x_3+63.3x_1^2+107x_1x_2;\\ y_5=195.13-673.2x_1-122x_2+0.068x_3+995x_1^2+146x_1x_2+1.35x_1x_3. \end{gathered} \tag{5} \]
An analysis of the influence of the independent variables on the optimization parameter, of their pairwise interaction and quadratic effect, could in general have been carried out in coded values as well; however, in that case their physical interpretation is required, and therefore it is better to carry out the analysis in physical units. The regression equation itself has no physical meaning, but its analytical or graphical investigation is important, since it makes it possible to draw certain conclusions about the physical picture of the process.
We find the values of \(x_1, x_2, x_3\), at a fixed induction in the gap, for which the current density (the output parameter) is minimal. A preliminary analysis showed that there is no global optimum in the region under study. Therefore the obtained regression equations are investigated for local extrema. Parameters that have no extremum in the region under study take their optimal value at the boundaries of the interval. Such parameters include \(x_2\) in the second equation of system (5); \(x_1, x_2, x_3\) in the third equation; \(x_2, x_3\) in the fourth; and \(x_2, x_3\) in the fifth equation of this same system. The second partial derivatives of the functions \(y_i\) with respect to \(x_1\) in most equations of system (5) have a positive sign. This indicates that the function has a minimum value at the given extremum point. The exception is the third equation of this system, which has no extrema at all and describes a plane. The obtained regression equations can be reduced to canonical form and then investigated graphically. In this case, however, their interpretation is difficult [5]. Moreover, not all of the obtained regression equations can be reduced to canonical form.
With a small number of factors, it is more rational to present the results of the investigation by two-dimensional sections of the response surface. The most characteristic curves for different inductions are shown in Figs. 2 and 3. Figure 2a shows the dependences \(j=f(\lambda)\) for \(B_\delta=1\) T at different \(\gamma\). The curves in Figs. 2b, 2c, 3a, and 3b are analogous, but with allowance for \(N_R\), which was insignificant in the first equation.
The obtained curves are highly illustrative. In addition, of some interest is the representation of the response surface, which is shown in Fig. 3c.
On the basis of the foregoing, the following conclusions may be drawn:
-
The optimal value of \(\lambda\) for an induction \(B_\delta=1\) T is 0.26; for \(B_\delta=1.35\), \(\lambda=0.18\); for \(B_\delta=1.7\) T, \(\lambda=0.1\); for \(B_\delta=2.05\) T, \(\lambda=0.22\); for \(B_\delta=2.4\) T, it lies in the range from 0.22 to 0.27 and depends on \(N_R\) and \(\gamma\).
-
The quantity \(\gamma\) has no extrema in this region and therefore takes the optimal value at the end of the interval, which can be estimated unambiguously for all values of the induction as \(\gamma=0.9\).
-
For an induction of 1 T, \(N_R\) does not affect the current density; therefore it may be taken as any value in the interval under study. For the remaining inductions, the optimal value will be unambiguous and will be \(N_R=40\).
Fig. 2. Dependence of the current density \(j\) on \(\lambda\) for:
a) \(B_\delta = 1\,\mathrm{T}\); b) \(B_\delta = 1.35\,\mathrm{T}\); c) \(B_\delta = 1.7\,\mathrm{T}\).
Visible graph annotations:
- Vertical axis: \(j,\;10^2\,\mathrm{A}/\mathrm{mm}^2\)
- Horizontal axis: \(\lambda\)
- a) \(B_\delta = 1\,\mathrm{T}\): \(\gamma = 0.5\), \(\gamma = 0.9\)
- b) \(B_\delta = 1.35\,\mathrm{T}\): \(N_R = 40\), \(\gamma = 0.5\), \(\gamma = 0.9\)
- c) \(B_\delta = 1.7\,\mathrm{T}\): \(\gamma = 0.5,\;N_R = 80\); \(\gamma = 0.5,\;N_R = 40\); \(\gamma = 0.9,\;N_R = 80\); \(\gamma = 0.9,\;N_R = 40\)
Fig. 3. Dependence of the current density \(j\) on \(\lambda\) for:
a) \(B_\delta = 2.05\ \mathrm{T}\); b) \(B_\delta = 2.4\ \mathrm{T}\); c) \(B_\delta = 1.35\) (response surface)
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