A Simplified Method for the Techno-Economic Calculation of Three-Phase Two-Winding Power Transformers

G. V. DEL

Recommended by the scientific seminar of the Department of Electric Power Stations

In [1] and [2] a method is set forth for techno-economic studies of power transformers, proposed by I. D. Kutyavin, V. P. Krasnov, and the author of the present article. In the form presented, this method is rather complicated and is suitable mainly for comprehensive studies, especially of newly designed transformers. The method does not exclude the use of electronic computing machines.

In the present work it is shown that the named method, by introducing a number of assumptions (without a significant reduction in the accuracy of the calculation), can be simplified and reduced to an engineering procedure for calculating and studying transformers. Three-phase two-winding power transformers with coil-type windings are considered (it is assumed that the expression for the mechanical stress in the winding during a short circuit beyond the transformer is not included in the system of equations of the “physical constraint”).

We have previously shown the weak dependence of the magnitude of the calculated costs on the axial size of the coils (without insulation) of the primary and secondary windings, \(y_1\) and \(y_2\) (see [2]). The value of this conclusion is increased by the fact that the minimum of the calculated costs always lies within the range of design values of \(y_1\) and \(y_2\) \((0.7 \div 2.0\ \text{cm})\). For these limits the possible deviation of the total calculated costs from the minimum does not exceed \(2\text{–}3\%\). On the basis of these considerations we adopt the first assumption

\[ \begin{aligned} y_1 &= \mathrm{const} \\ y_2 &= \mathrm{const} \end{aligned} \tag{1} \]

This assumption means that it is possible in advance to choose (specify) the elementary conductor, by its largest side, from the standard series of rectangular wires.

From [2] the equation of the heat balance of the windings is written as

\[ k_{\mathrm{ц}}\, x^{-\frac{1}{2}} y = L \frac{x y \Delta^2}{1 + \lambda \Delta y}. \tag{2} \]

(the designations of the quantities correspond to [2]),

Analysis showed the possibility of accepting the assumption

\[ k_{\mathrm{п}} x + y \simeq x . \tag{3} \]

The latter is close to the truth, since \(0.5 < k_{\mathrm{п}} < 1\), and \(y < 2\) (usually), therefore

\[ (1-k_{\mathrm{п}})x \simeq y . \tag{3'} \]

Taking 3 into account in 2, we have

\[ \left. \begin{aligned} \Delta_1 &= \frac{\lambda_1+\sqrt{\lambda_1^2+\dfrac{4L_1}{y_1}}}{2L_1},\\[6pt] \text{and }\ \Delta_2 &= \frac{\lambda_2+\sqrt{\lambda_2^2+\dfrac{4L_2}{y_2}}}{2L_2}. \end{aligned} \right\} \tag{4} \]

Taking (1) into account, expression (4) means

\[ \left. \begin{aligned} \Delta_1 &= \text{const}\\ \Delta_2 &= \text{const} \end{aligned} \right\} \tag{4'} \]

Assumption (4′), which is a consequence of (3), is confirmed by calculation. Having accepted these assumptions, we must accept the necessity of constancy of the expressions

\[ \frac{y_1\Delta_1}{(1+\lambda_1\Delta_1 y_1)(y_1+\delta_1)} = Y_1 = \text{const}, \tag{5} \]

\[ \frac{y_2\Delta_2}{(1+\lambda_2\Delta_2 y_2)(y_2+\delta_2)} = Y_2 = \text{const}. \tag{6} \]

On the basis of equality of the magnetizing forces of the primary and secondary windings [2], taking (5) and (6) into account, we have

\[ x_2 = b x_1, \tag{7} \]

where

\[ b=\frac{Y_1}{Y_2}. \tag{8} \]

Dependence (7) also agrees with the calculated data. By calculation we have found a directly proportional dependence of the core diameter \(d\) on the radial width of the primary winding coil \(x_1\). On this basis, for values \(x_1 > 3 \div 4\ \text{cm}\), we adopt the third assumption

\[ d \simeq x_1, \tag{9} \]

or

\[ d = ax_1 - c. \tag{9'} \]

To determine “\(a\)” and “\(c\)” we shall use Lagrange’s theorem, as a special case of Cauchy’s generalized mean-value theorem [3]. Representing the rectified function of the diameter, (9′), as a chord drawn through two points (let us assume \(A\) and \(B\)) of the curve \(d(x_1)\), by Lagrange’s formula we find the slope of this chord

\[ a=\frac{d_{в}-d_{a}}{x_{1в}-x_{1a}} . \tag{10} \]

Then the equation of the straight line is written as follows:

\[ (d_{\mathrm{B}}-d_a)=a(x_{1\mathrm{B}}-x_{1a}). \tag{11} \]

where \(d_a\) and \(d_{\mathrm{B}}\) are the values of the core diameter calculated for \(x_1=x_{1a}\) and \(x_1=x_{1\mathrm{B}}\) at points A and B (by formula (13) of work [2]).

Expression (11), taking (10) into account, is easily transformed to the form

\[ d=\frac{d_{\mathrm{B}}-d_a}{x_{1\mathrm{B}}-x_{1a}}\,x_1 -\frac{d_{\mathrm{B}}x_{1a}-d_a x_{1\mathrm{B}}}{x_{1\mathrm{B}}-x_{1a}}. \]

Thus,

\[ a=\frac{d_a}{x_{1\mathrm{B}}-x_{1a}}\left(\frac{d_{\mathrm{B}}}{d_a}-1\right), \tag{12} \]

\[ c=\frac{d_a}{x_{1\mathrm{B}}-x_{1a}}\left(\frac{d_{\mathrm{B}}}{d_a}x_{1a}-x_{1\mathrm{B}}\right). \tag{13} \]

According to preliminary calculations, it may be recommended that the following restrictions be observed when choosing the values \(x_{1a}\) and \(x_{1\mathrm{B}}\): \(x_{1a}>5\), and \(x_{1\mathrm{B}}<10\) for copper windings, and \(x_{1a}>7\), \(x_{1\mathrm{B}}<12\) for aluminum windings.

Using formulas (15) and (6) of work [2] and taking into account (1), (4′), (7) and (9′), one can express the total annual calculated costs for the manufacture and operation of the transformer as a function of one independent variable, namely \(d\). If, after this, we take the first derivative of the calculated costs with respect to \(d\) and set it equal to zero in order to find the optimum value of the core diameter corresponding to the minimum calculated costs, then we obtain the following equation with respect to \(d\) (for the abbreviated derivation see the Appendix):

\[ \frac{N}{S}\,d^4(3b_1d+2c_1)=M(Rd+2L)+\frac{a^2d^3}{Y_1(d+c)^2}, \tag{14} \]

where, for brevity of notation, the following has been adopted:

\[ N=\frac{8}{3}Kk_{\mathrm{n}}\, \frac{A+DB^2-\dfrac{1}{k_{\mathrm{y}}^2}+1.5\,q_{\mathrm{ya}}} {A+DB^2+1.5\,q_{\mathrm{rst}}}; \tag{15} \]

\[ M=\frac{4\gamma_{\mathrm{m}}}{\Delta_1\gamma_{\mathrm{c}}k_{\mathrm{c}}}\cdot \frac{3A+E\Delta_1^2}{A+DB^2+1.5\,q_{\mathrm{rst}}}; \tag{16} \]

\[ r=\frac{\beta A+E\Delta_2^2}{\beta A+E\Delta_1^2}\cdot\frac{\Delta_1}{\Delta_2}; \tag{17} \]

\[ R=1+b+(a+b)(1+r); \tag{18} \]

\[ L=c[1+b(2+r)]+a[2\delta_{12}+2\delta_{02}(1+r)]; \tag{19} \]

\[ b_1=1+b+0.7a; \tag{20} \]

\[ c_1=c(1+b)+a\left(l_{\mathrm{r}}+\frac{3}{4}l_{\mathrm{u}}\right). \tag{21} \]

Equation (14) is an equation of the seventh degree with respect to \(d\), and therefore is not solvable in a strictly analytical way. However, for \(d\) one may adopt the following form of notation:

\[ d=\sqrt[4]{\frac{S}{N}}\cdot \sqrt[4]{\,M\cdot\frac{Rd+2L}{3b_1d+2c_1} +\frac{a^2d^3}{Y_1(d+c)^2(3b_1d+2c_1)}\,}. \tag{22} \]

Table 1 gives the variation of \(d\), calculated from (22), for different values of \(d\) adopted in the radical expression of the second of the factors in (22). (For \(S=10\) MVA; \(\sigma_1=\sigma_2=0.1\ \text{W}/\text{cm}^2\).)

Table 1

On the basis of the data in this table, it is possible not to solve equation (22) with respect to \(d\), but to assume that its second factor does not depend on \(d\).

Therefore the optimum value of the diameter of the circle circumscribed about the stepped core can be found from the formula [taking \(d=40\ \text{cm}\) in the radical expression of (22)]:

\[ d=\sqrt[4]{\frac{S}{N}}\cdot \sqrt[4]{\,\frac{20R+L}{60b_1+c_1}\cdot M +\frac{8a^2\cdot 10^3}{Y_1\left(20+\frac{c}{2}\right)^2(60b_1+c_1)}\,}. \tag{23} \]

This expression for \(d\) is indicative to a certain degree, since it shows that \(S=d^4\). We obtained such a conclusion only because it was assumed that \(B=\mathrm{const}\) and \(\Delta=\mathrm{const}\).

After determining \(d\), from (23) or (22) we find \(x_1\) from the expression

\[ x_1=\frac{d+c}{a}. \tag{24} \]

The remaining dimensions and parameters of the transformer are determined from the formulas:

\[ \Delta_1= \frac{\lambda_1+\sqrt{\lambda_1^2+\dfrac{2z_1}{y_1}}}{z_1}, \tag{25} \]

where

\[ z_1=\frac{2l_1x_1}{k_{\mathrm{p}1}x_1+y_1}; \]

\[ \Delta_2= \frac{ k_{\mathrm{n}2}\lambda_2+\dfrac{y_2}{X(y_2+\delta_2)} + \sqrt{ \left[ k_{\mathrm{n}2}\lambda_2+\dfrac{y_2}{X(y_2+\delta_2)} \right]^2 +4L_2\dfrac{k_{\mathrm{n}2}}{y_2} } }{2L_2}, \tag{26} \]

where

\[ X=\frac{x_1y_1\Delta_1}{(1+\lambda_1y_1\Delta_1)(y_1+\delta_1)}; \]

\[ x_2=\frac{X}{\Delta_2y_2}(1+\lambda_2y_2\Delta_2)(y_2+\delta_2); \tag{27} \]

\[ h=\frac{S}{K}\cdot\frac{1}{Xd^2}. \tag{28} \]

Table 2 gives data for several transformers calculated by the exact method [2] and by the proposed simplified procedure.

Table 2

Note. The notation of the quantities in the table corresponds to [2].

General conclusion: there are real ways of simplifying the previously proposed method of techno-economic studies of transformers (see [1] and [2]), one of which, as applied to three-phase transformers, is shown above. At the same time, the accuracy of the calculation, associated with the adoption of certain assumptions, remains quite satisfactory in comparison with the calculation according to [2]. In the present case (according to Table 2) the error of calculation by the proposed procedure does not exceed 1.5% when compared in terms of the calculated costs.

Appendix: after substituting the variables in simplified form and transforming formulas (15) and (16) of [2], we obtain:

\[ Z=\frac{3}{4}\pi\gamma_{\mathrm{c}}k_{\mathrm{c}}\cdot\frac{S}{K} \left(A+DB^2+1.5z_{\mathrm{pc}}\right)\varphi(x_1). \]

where

\[ \varphi(x_1)=\frac{1}{Y_1x_1}+\frac{N}{S}(ax_1-c)^2\left(b_1x_1-0.7c+l_r+\frac{3}{4}l_u\right)+ \]

\[ +\frac{M}{(ax_1-c)^2}\left[(a+2b+1+ra+rb)x_1+(l_i-2c+r\cdot\delta_{02})\right]. \]

Here \(N\), \(M\), and \(r\) correspond to (15)–(17). From \(\varphi(x_1)\) one can pass to \(\varphi(d)\) if (24) is substituted in place of \(x_1\).

\[ \varphi(d)=\frac{a}{Y(d+c)}+\frac{N}{S\cdot a}d^2(b_1d+c_1)+M\frac{(Rd+L)}{a\cdot d^2}; \]

where \(b_1\), \(c_1\), \(R\), and \(L\) are according to (18)–(21).

Taking the first derivative \(\varphi(d)\) with respect to \(d\), we obtain:

\[ \varphi'(d)=-\frac{a}{Y_1(d+c)^2}+\frac{N}{a\cdot S}\left[2d(b_1d+c_1)+b_1d^2\right]+ \]

\[ +\frac{M}{a}\cdot\frac{Rd^2-2d(Rd+L)}{d^4}, \]

which, after transformation, gives equation (14).

REFERENCES

1. I. D. Kutyavin, G. V. Del, V. P. Krasnov, On the techno-economic determination of the optimal dimensions of substation three-phase two-winding transformers of high power, Izvestiya TPI, vol. 130, 1964.

2. G. V. Del, V. P. Krasnov. Techno-economic studies of the optimal dimensions of power transformers, Izvestiya TPI, vol. 132, 1965.

3. V. I. Smirnov. Course of Higher Mathematics, GITTL, vol. 1, 1953.