Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Vol. 138
1965

Use of Digital Computers in Techno-Economic Studies of Three-Phase Power Transformers¹

G. V. Del

(Presented by the scientific seminar of the Department of Electric Power Stations and Networks)

In [1], three-phase transformers with a continuous-continuous (Cont.-Cont.) combination of windings on the high- and low-voltage sides are studied. It is also indicated there that, for the transformers under consideration, at higher power on the low-voltage side it will be necessary to use a helical winding, which reduces the number of independent variables from three (the Cont.-Cont. combination) to two (Cont.-Hel.). Taking the above additional condition into account, the system of equations under study takes the form:

\[ \left. \begin{aligned} S &= K_B - \frac{x_1 y_1 \Delta_1 h d^2}{(1+\lambda_1 y_1 \Delta_1)(y_1+\delta_1)}, \\[4pt] u_p &= K_p \frac{x_1 y_1 \Delta_1 (x_1+x_2+3\delta_{12})}{B d^2 (1+\lambda_1 y_1 \Delta_1)(y_1+\delta_1)}(d+2x_2+l), \\[4pt] k_{p1}x_1+y_1 &= L_1\Delta_1^2\,\frac{x_1y_1}{1+\lambda_1 y_1\Delta_1}, \\[4pt] k_{p2}x_2+y_2 &= L_2\Delta_2^2\,\frac{x_2y_2}{1+\lambda_2 y_2\Delta_2}, \\[4pt] \frac{x_2y_2\Delta_2}{(1+\lambda_2y_2\Delta_2)(y_2+\delta_2)} &= \frac{x_1y_1\Delta_1}{(1+\lambda_1y_1\Delta_1)(y_1+\delta_1)}, \\[4pt] I_2 w_{k2} &= \frac{x_2y_2\Delta_2}{1+\lambda_2y_2\Delta_2}. \end{aligned} \right\} \tag{1} \]

All notation corresponds to [1].

System (1) was studied on the Minsk-1 electronic digital computer. The calculations were carried out in the order of arrangement of the following formulas:

\[ \Delta_1 = \frac{ \lambda_1+\sqrt{\lambda_1^2+\dfrac{4L_1x_1}{(k_{p1}x_1+y_1)y_1}} }{ \dfrac{2L_1x_1}{k_{p1}x_1+y_1} }, \tag{2} \]

¹ The work was carried out under the supervision of Prof. Dr. I. D. Kutyavin.

\[ y_2 = \frac{I_2 w_{k2}\left(1+\lambda_1 y_1 \Delta_1\right)(y_1+\delta_1)} {x_1 y_1 \Delta_1} -\delta_2, \tag{3} \]

\[ \Delta_2 = \frac{ \left(k_{n2}\lambda_2+\dfrac{y_2}{I_2 w_{k2}}\right) + \sqrt{ \left(k_{n2}\lambda_2+\dfrac{y_2}{I_2 w_{k2}}\right)^2 +4L_2\dfrac{k_{n2}}{y_2} } }{2L_2}, \tag{4} \]

\[ x_2 = \frac{I_2 w_{k2}\left(1+\lambda_2 y_2 \Delta_2\right)} {y_2 \Delta_2}, \tag{5} \]

\[ V = \frac{K_p I_2 w_{k2}\left(x_1+x_2+3\delta_{12}\right)} {2u_p B\left(y_2+\delta_2\right)}, \tag{6} \]

\[ d = V+\sqrt{V^2+2V(2x_2+l)}, \tag{7} \]

\[ h = \frac{S}{K I_2 w_{k2} B}\cdot \frac{y_2+\delta_2}{d^2}, \tag{8} \]

\[ q_c=\frac{\pi k_c}{4}\cdot d^2, \tag{9} \]

\[ l_y = 8k_y\left[(x_1+x_2)+0.7d+l_g\right], \tag{10} \]

\[ l_{st}=3(h+2l_y), \tag{11} \]

\[ Q_{st}=\gamma_c q l_{st}, \tag{12} \]

\[ Q_y=\gamma_c q l_y, \tag{13} \]

\[ Z_{st}=(A+DB^2+1.5\alpha_{rst})\,Q_{st}, \tag{14} \]

\[ Z_y=\left[A+D\left(\frac{B}{k_y}\right)^2+1.5\alpha_{ry}\right]Q_y, \tag{15} \]

\[ d_{2\,\mathrm{av}}=d+x_2+2\delta_{02}, \tag{16} \]

\[ d_{1\,\mathrm{av}}=d_{2\,\mathrm{av}}+x_2+x_1+2\delta_{12}, \tag{17} \]

\[ Q_{m2}= \frac{3\pi\gamma_m I_2 w_{k2}h}{y_2+\delta_2}\cdot \frac{d_{2\,\mathrm{av}}}{\Delta_2}, \tag{18} \]

\[ Q_{m1}= \frac{3\pi\gamma_m I_2 w_{k2}h}{y_2+\delta_2}\cdot \frac{d_{1\,\mathrm{av}}}{\Delta_1}, \tag{19} \]

\[ Q_m=Q_{m2}+Q_{m1}, \tag{20} \]

\[ Z_m=(\beta A+E\Delta_1^2)Q_{m1}+(\beta A+E\Delta_2^2)Q_{m2}, \tag{21} \]

\[ Z=Z_{st}+Z_y+Z_m, \tag{22} \]

\[ w_{k1}= \frac{x_1 y_1 \Delta_1} {I_1\left(1+\lambda_1 y_1 \Delta_1\right)}, \tag{23} \]

\[ \sigma_r= \frac{K_p I_2 w_{k2}}{y_2+\delta_2} \Delta_1(d+2x_2+l). \tag{24} \]

The independent variables chosen were \(x_1\) and \(y_1\). By formulas (11), (12), and (14), the length, weight, and costs for the cores were calculated; and by (10), (13), and (15), respectively, those for the transformer yoke. The minimum of the costs \(Z\) (22) was found by the Gauss--Seidel method [2]. In compiling the program, a shortage of internal machine memory was encountered.

memory. Therefore, an interpreting system was used. The calculation was carried out with a floating point. The initial steps in \(x_1, y_1\) were equal to 0.4; subsequently, as the minimum was approached, they were halved each time until the error in determining the final costs was less than 0.5%. The machine time for determining the optimum point for one transformer rating was approximately 2.5 minutes. The following quantities were printed out:

\[ \Delta_1,\ y_2,\ \Delta_2,\ x_2,\ d,\ h,\ Q_{\text{st}},\ Q_{\text{ya}},\ Z_{\text{st}},\ Z_{\text{ya}},\ Q_{\text{m}},\ Z_{\text{m}},\ Z,\ w_{k1},\ \sigma_{p1},\ x_1,\ y_1 . \]

The programming was carried out by M. Panikhina, a fifth-year student of Tomsk Polytechnic Institute.

Transformers with ratings of \(16 \div 63\) MVA were studied.

Beginning with ratings of \(40 \div 60\) MVA, the radial mechanical tensile force under short-circuit conditions becomes greater than the allowable value, taken as \(600 \div 800\ \text{kg}/\text{cm}^2\). Therefore, for these and higher transformer ratings, one of the initial equations in the investigation should be the equation of radial mechanical forces

\[ \sigma_{p1}=K_{\text{m}}\frac{I_2 w_{k2}\Delta_1}{y_2+\delta_2}(d+2x_2+l), \tag{25} \]

where

\[ K_{\text{m}}=\frac{2.04\cdot K_{\text{ud}}^2\cdot K_p}{u_k^2}\,\pi\cdot 10^{-8}. \]

Correspondingly, for the secondary winding one may write

\[ \sigma_{p2}=K_{\text{m}}\frac{I_2 w_{k2}\Delta_2}{y_2+\delta_2}(d+2x_2+l). \tag{26} \]

The simultaneous solution of (25) and (26), with \(\sigma_{p1}=\sigma_{p2}=[\sigma_p]\), gives equality of the current densities of the HV and LV windings,

\[ \Delta_1=\Delta_2 . \tag{27} \]

Equation (25), together with (1), gives a system of 7 equations with 3 unknowns, taking (27) into account (if the number of starts, and therefore also \(w_{k2}\) of the helical winding, is taken as a variable quantity). The number of turns in the LV winding coil \((w_{k2})\) was taken as the independent variable. Unfortunately, a simple analytical expression of the remaining variables through the adopted independent variable proved impossible because of the complexity of the resulting equations (of about the 10th degree). Therefore, it was decided to investigate the above system of equations on the “Minsk-1” digital computer as follows. All variables were expressed through \(y_2\) and \(w_{k2}\):

\[ x_2= \frac{-(y_2-I_2w_{k2}\lambda_2 k_{p2})+}{2k_{p2}} \]

\[ \frac{ \sqrt{(y_2-I_2w_{k2}\lambda_2 k_{p2})^2+4I_2w_{k2}k_{p2}\left(\frac{L_2I_2w_{k2}}{y_2}+\lambda_2y_2\right)} }{2k_{p2}}, \tag{28} \]

\[ \Delta=\frac{k_{p2}x_2+y_2}{L_2I_2w_{k2}}, \tag{29} \]

\[ d=\frac{c_p}{K_m l_2^2 w_{k2}}\cdot \frac{y_2+\delta_2}{\Delta}-2x_2-l, \tag{30} \]

\[ x_1=\frac{u_p K_m B}{K_p\cdot c_p}\,d^2\Delta-x_2-3\delta_{12}, \tag{31} \]

\[ y_1=\frac{k_{n1}(y_2+\delta_2)x_1-L_1 l_2 w_{k2}\cdot \delta_1\Delta}{L l_2 w_{k2}\Delta-y_2-\delta_2}, \tag{32} \]

\[ x_1'=\frac{l_2 w_{k2}}{y_2+\delta_2}\cdot \frac{(1+\lambda_1 y_1\Delta)(y_1+\delta_1)}{y_1\Delta}. \tag{33} \]

The remaining expressions for calculating the total costs are analogous to (8–24).

Fig. 1.

Next, by the method of successive approximations, for each adopted value of \(w_{k2}\) such a \(y_2\) was found at which \(x_1\) and \(x_1'\) would be equal. For the \(y_2\) found and the corresponding \(w_{k2}\), the costs and all intermediate quantities were calculated. The nature of the variation of \(x_1\) and \(x_1'\) is shown in Fig. 1.

The logical part of the program was compiled taking into account the fact that \(d\), \(x_1\), \(y_1\), or \(x_1'\) may take negative values. Therefore, if one of the indicated quantities became less than zero, the machine proceeded to the calculation of the next point. The point of intersection of the curves

\(x'_1\) and \(x_1=f(y_2)\) was detected when the sign of the difference \((x'_1-x_1)\) changed from “+” to “−”.

The search over \(y_2\) was carried out within the limits \(y_2=0 \div 4\), with an initial step of 0.1. Subsequently, the steps near the point of intersection were taken as: 0.01; 0.001; 0.0001, etc., until \(|x'_1-x_1|\leqslant 0.1\). The computation, as in the first case, was performed with a floating point and an interpreting system. The duration of a complete computation for one \(w_{k2}\) averaged one minute. The program did not provide for searching for the minimum of the costs (22) with respect to \(w_{k2}\). The following values of \(w_{k2}\) were specified: 0.125; 0.25; 0.5; 0.75; 1.0; 1.5; 2.0. The output was the same as in the first case.

Fig. 2.

It should be noted that, as the specific thermal load \((\delta)\) increases, the range of values of \(w_{k2}\) for which structural dimensions of the transformer exist decreases. Table 1 gives, as an example, values of \(w_{k2}\) for different \(\delta\) at \(S=80\) MVA. Only for these \(w_{k2}\), from the entire above-listed series of values adopted in the calculation, does a structural solution exist; one of the conditions for it is the intersection of the curves \(x'_1\) and \(x_1=f(y_2)\). Fig. 2 illustrates the absence of an intersection of the indicated curves for \(S=80\) MVA, \(\delta=0.25\), \(w_{k2}=1\), \(\sigma_{\mathrm{p}}=800\ \mathrm{kg}/\mathrm{cm}^2\).

The absence of an intersection point of \(x'_1\) and \(x_1=f(y_2)\) for certain values of \(\delta\) and \(w_{k2}\) is evidently explained by the presence of a condition determining-

mechanical load during a short circuit. The point is that, with increasing $\sigma$ and $w_{k2}$, the magnitude of the radial force also increases (Fig. 3, for $S=16000$ kVA). Therefore, quite definite combinations of $\sigma$ and $w_{k2}$ correspond to a quite definite value of $\sigma_p$. And the stopping of the machine for this reason occurred only for large values of $w_{k2}$. At small $w_{k2}$ the stopping occurred as a result of negative values of $y_1$, $x'_1$, or $d$.

The programming of the second case was carried out by engineer-programmer N. M. Bylino of TPI.

Table 1

Fig. 3.
Visible plot labels: $\sigma_p$ (kg/cm$^2$); $\sigma$ (t/cm$^2$); $w_{k2}$; $\sigma_p(w_{k2}),\ \sigma=0.1$; $\sigma_p(\sigma),\ w_{k2}=1$.

Conclusion

In techno-economic studies of three-phase transformers, it is not only desirable but in some cases even necessary to use computing and solving machines, owing to the complexity of the expressions, which rules out the possibility of investigation on a slide rule. For the indicated studies, a floating-point system is most commonly used, and it is desirable to employ a digital computer with a larger internal memory than the “Minsk-1.”

References

1. G. V. Del, V. P. Krasnov. Techno-economic studies of the optimal dimensions of power transformers. Izv. TBI, vol. 132, 1964.
2. A. A. Feldbaum. Computing Devices in Automatic Systems. State Publishing House of Physical and Mathematical Literature, Moscow, 1959.