ON THE LIMITING POWER OF A TRANSFORMER

I. D. KUTYAVIN and L. I. DEL

(Tomsk Polytechnic Institute)

Of undoubted interest is the determination of the limiting power of power transformers on the basis of transport restrictions or other conditions. This problem can be solved by the method of optimizing transformer geometry by minimizing the expression for the weight of the removable part under prescribed constraints. However, this method is associated with difficulties because of the complexity of the expressions under study. The problem is solved much more simply by maximizing the expression for transformer power when the necessary constraints are introduced. For convenience of investigation we shall use the expression for the power of a core in kVA:

\[ S_c = K \Delta B q_c q_m, \tag{1} \]

where \(\Delta\) is the current density in the winding;
\(B\) is the induction in the core steel;
\(q_c\) is the cross-sectional area of the core steel;
\(q_m\) is the total cross-sectional area of the winding material.

\[ K = 4.44 f \cdot 10^{-11}. \tag{2} \]

In this case the following constraints may be adopted:

1) the specific heat transfer from the winding surface, \(\varepsilon\), is limited;
2) the short-circuit voltage, \(U_k\), is limited;
3) the electrodynamic forces in the windings are limited;
4) the transport dimensions of the transformer are limited;
5) the transport weight is limited.

As a constraint with respect to winding heating we shall take the maximum permissible heat transfer from their surface, \(\varepsilon_m\), for the chosen method of cooling. Then the heat-balance equation for the coil surface (for continuous coil, disk, and helical windings) per one running centimeter of the mean turn [1] is

\[ 2 \varepsilon_m (k_b b + k_y y) = \rho k_r \Delta^2 \frac{bxy}{x + i}; \tag{3} \]

where \(x\) and \(y\) are the radial width and axial height of the copper of an elementary conductor (\(y\) is also the axial height of the coil);
\(i\) is the insulation of the elementary conductor on two sides;
\(b\) is the radial width of the cross-section of the coil side;
\(\rho\) is the resistivity of the winding material;

\(k_{\mathrm{в}}\) and \(k_y\) are coefficients that take into account the parts of the surfaces \(b\) and \(y\) covered by insulating spacers.

The expression for the coefficient of increase in the active resistance of the winding material due to the skin effect [2] is reduced to the following form:

\[ k_{\mathrm{г}}=1+\frac{c^2 b^2 x^4 y^2}{9(x+i)^2(y+b)^2}, \tag{4} \]

\[ c=\pi f \frac{\mu}{\rho}; \tag{5} \]

where \(\delta\) is the axial distance between the copper of adjacent coils, consisting of the axial width of the radial cooling duct and the insulation of the coil on two sides;

\(f\) is the number of current periods (Hz); \(\mu_0=4\pi\cdot 10^{-9}\).

Let us introduce the additional notations:

\[ \frac{k_{\mathrm{в}}}{k_y}b+y=k_{\mathrm{п}}b+y=P \quad \text{and} \quad \frac{2\varepsilon_{\mathrm{м}}k_y}{\rho}=a \tag{6} \]

and find from (3) the current density:

\[ \Delta=\sqrt{\frac{aP(x+i)}{k_{\mathrm{г}}bxy}}. \tag{7} \]

To simplify the solution of the problem, the following assumptions have been adopted: \(i_1=i_2=i\); \(\varepsilon_1=\varepsilon_2=\varepsilon\); \(h_1=h_2=h\), and the equalities following from these, \(x_1=x_2=x\); \(y_1=y_2=y\) and \(b_1=b_2=b\).

Additional losses due to circulating currents in the copper of the windings are not taken into account, under the assumption of ideal transposition.

Additional losses in steel structures are not taken into account.

The cross-sectional area of the copper of the winding is:

\[ q_{\mathrm{м}}=mxy\frac{h}{y+\delta} =\frac{bxyh}{(x+y)(y+\delta)}, \tag{8} \]

where \(m\) is the number of elementary conductors in the coil;

\(h\) is the axial height of the winding.

The cross-sectional area of the steel of the core is:

\[ q_{\mathrm{с}}=k_{\mathrm{с}}\frac{\pi}{4}d^2, \tag{9} \]

where \(d\) is the diameter of the circle circumscribing the core;

\(k_{\mathrm{с}}\) is the coefficient of filling with steel of the area of the circle with diameter \(d\).

Then the expression for the power (1), taking (7)—(9) into account, is:

\[ S_{\mathrm{с}}=KB\frac{k_{\mathrm{с}}\pi}{4}d^2h\sqrt{ab}\, \sqrt{\frac{Pxy}{k_{\mathrm{г}}(x+i)(y+\delta)^2}}. \tag{10} \]

Here \(d\) and \(h\) are determined by the transport dimensions of the transformer. Therefore, the core power depends only on three variables \(b\), \(x\), and \(y\). Preliminary investigations have shown that, as the variable \(b\) increases, the power \(S_{\mathrm{с}}\) increases for all constructive values of \(b\), while for certain values of \(x\) and \(y\) expression (10) has a maximum. This is seen from Figs. 1 and 2, which show the dependence

\[ \varphi=\sqrt{\frac{Pxy}{k_{\mathrm{г}}(x+i)(y+\delta)^2}} \tag{11} \]

for \(b=10;\ \delta=1;\ k_{\mathrm p}=0.78;\ i=0.01;\ c=0.92\). Figure 1 shows the dependence \(\varphi(x,y)\) for \(y=0.5;\ 1;\ 2;\ 4\), and Fig. 2 shows the dependence of the maxima of the functions on \(y\).

Fig. 1.

To determine \(x\) and \(y\) corresponding to the maximum of the function, we use the condition that

\[ \frac{\partial S_{\mathrm c}}{\partial x}=0 \quad \text{and} \quad \frac{\partial S_{\mathrm c}}{\partial y}=0, \tag{12} \]

As a result we obtain two equations:

\[ k_{\mathrm r} i - x(x+i) k'_{\mathrm r x}=0; \tag{13} \]

\[ k_{\mathrm r}\left[\delta(k_{\mathrm p}b+2y)-k_{\mathrm p}by\right]-P y(y+\delta)k'_{\mathrm r y}=0, \tag{14} \]

where \(k'_{\mathrm r x}\) and \(k'_{\mathrm r y}\) are the derivatives of \(k_{\mathrm r}\) with respect to the corresponding variables.

The simultaneous solution of these equations leads to the relations:

\[ \frac{3(k_{\mathrm p}b-\delta)}{k_{\mathrm p}cib^{2}} = U_{1} = \frac{v^{2}}{\sqrt{2v+3}}, \qquad \text{where } v=\frac{x}{i}. \tag{15} \]

\[ y= \frac{k_{\mathrm p}b\delta(x+i)} {i\delta+(x+2i)(k_{\mathrm p}b-2\delta)}. \tag{16} \]

Equation (15) is of the fourth degree with respect to \(x\) and cannot be solved directly. However, the critical value of \(x\) is easily determined from curve 1 in Fig. 3 if \(U_{1}\) is first found from the left-hand side of (15). Curve 1 is plotted for \(U_{1}(v)\) from the right-hand side of (15).

If the adopted value of \(v\) differs substantially from that found from (16), then the optimal \(x\) is determined from equation (13), or from curve 2 in Fig. 3, plotted from the right-hand side of the expression:

\[ \frac{3(y+\delta)}{cbiy} = U_{2} = \frac{v^{2}\sqrt{2v+3}}{v+1}, \tag{17} \]

\(U_{2}\) is found from the left-hand side of (17).

Fig. 2.

As noted above, the power \(S_c\) increases with increasing \(b\). At the same time, the short-circuit voltage also increases, as determined by the expression [1]:

\[ U_p=\frac{2k_R \Delta q_m l_m \delta'}{3.33\,B q_c h}= \]

\[ =\frac{8k_R(d+2\delta_{01}+2b+\delta_{12})(2b+3\delta_{12})}{3.33\,k_c B d^2} \sqrt{\frac{abPxy}{k_R(x+i)(y+\delta)^2}}, \tag{18} \]

where

\[ k_R \]
is the Rogowski coefficient;

\[ l_m \text{ and } \delta' \]
are the mean length and the reduced width of the leakage channel;

\[ \delta_{01} \text{ and } \delta_{12} \]
are the insulation distances from the core to the winding and between the windings.

Fig. 3.

In connection with this, the quantity \(b\) will be limited by the permissible voltage \(U_p\). The value of \(b\) could be determined from (18) by specifying \(U_p\), but this is practically impossible because of the complexity of (18), especially if one takes into account that \(P\), \(k_g\), \(x\), and \(y\) are functions of \(b\). However, the problem is solved comparatively simply by a search method, consisting in determining \(U_p\) from (18) for different \(b\) within the limits \((7 \div 12)\). Such a solution of the problem is shown in Tables 1 and 2.

Table 1

Limitation of the dimensions and power of transformers according to electrodynamic stability can be performed by rejecting variants that are electrodynamically unstable under short circuits. In this case, the calculated short-circuit current for determining the electrodynamic stability of step-up transformers should be determined taking into account the resistance of the generators, in view of their reliable connection with the transformers.

The limitation of the external dimensions of a transformer is determined by the permissible railway clearances, taking into account transportation on a special transporter [3]. If, in this case, the upper part of the tank is made trapezoidal in shape, then the limiting height of the load will be 500 cm and the width 325 cm. From the external limiting dimensions one can pass to the dimensions of the removable part of the transformer, which are related to the geometric dimensions by simple relations. The external diameter of the outer winding is:

\[ D_{\mathrm{v}} = d + 2\delta_{01} + 4b + 2\delta_{12}. \tag{19} \]

The height of the removable part:

\[ H_{\mathrm{v}} = h + 2l_{\mathrm{i}} + d, \tag{20} \]

where \(l_{\mathrm{i}}\) is the insulation distance from the winding to the yoke.

The diameter of the limb from (19):

\[ d = D_{\mathrm{v}} - 2\delta_{01} - 2\delta_{12} - 4b = A - 4b. \tag{21} \]

The height of the winding from (20):

\[ h = H_{\mathrm{v}} - 2l_{\mathrm{i}} - A + 4b = \mathrm{Б} + 4b. \tag{22} \]

The length of the mean turn of the limb windings:

\[ l_m = \pi (d + 2\delta_{01} + 2b + \delta_{12}) \tag{23} \]

Table 2

The length of the core steel, reduced to the rod cross section, for a transformer with one wound rod (shell type) is:

\[ l_{c1}=2(H_{\mathrm{B}}+a_1), \tag{24} \]

where \(a_1\) is the window width with one system of windings:

\[ a_1=\delta_{01}+2b+\delta_{12}+\delta_{20}. \tag{25} \]

For a transformer with two wound rods:

\[ l_{c2}=3H_{\mathrm{B}}+2a_1+a_2, \tag{26} \]

where \(a_2\) is the window width with two systems of windings:

\[ a_2=2\delta_{01}+4b+2\delta_{12}+\delta_{22}, \tag{27} \]

For a transformer with three wound rods:

\[ l_{c3}=4H_{\mathrm{B}}+2a_1+2a_2. \tag{28} \]

The same with four wound rods:

\[ l_{c4}=5H_{\mathrm{B}}+2a_1+3a_2. \tag{29} \]

Tables 1 and 2 present a procedure for determining the limiting power of a transformer for a specified winding height \(h=260\) and rod diameter \(d=120\) cm (all dimensions are in centimeters).

The insulation distances for a 20/750 kV transformer are taken as: \(\delta_{01}=5\); \(\delta_{12}=18\); \(\delta_{20}=e_i=38\). The remaining general data are \(E_M=0.30\); \(\rho=2.14\cdot10^{-6}\); \(k_y=0.9\); \(k_B=0.7\); \(k_c=0.8\); \(k_r=0.95\); \(B=17.5\) kG; \(f=50\) Hz.

Items 1–12 of Tables 1 and 2 determine the geometric dimensions and parameters of transformers of limiting power. Table 1 is compiled for \(i=0.01\) and \(\delta=0.8\), and Table 2 for \(i=0.1\) and \(\delta=1\). Item 8 shows the transformer power in megavolt-amperes. Items 11–14 show the weight of the core steel as a function of the number of wound rods (the index at \(Q_c\)) and the dimension \(b\). The weight of the steel was determined from the expression (with \(\gamma_c=7.6\)):

\[ Q_c=\gamma_c l_c q_c. \tag{30} \]

Items 15–18 give the total weight of active materials (all weights in tons):

\[ Q_H=Q_{cn}+Q_m2n, \tag{31} \]

where \(n\) is the number of wound rods;

\(Q_{\mathrm{M}}\) is the average weight of copper of one winding, determined from the expression:

\[ Q_{\mathrm{M}}=\gamma_{\mathrm{M}} q_{\mathrm{M}} l_{\mathrm{M}}. \tag{32} \]

If the specific heat removal from the winding surface \(\varepsilon_1\) differs from the value adopted for Tables 1 and 2, \(\varepsilon_{\mathrm{M}}=0.16\), then the current density, power, and voltage \(U_{\mathrm{p}}\) must be multiplied by the factor:

\[ \sqrt{\frac{\varepsilon_1}{\varepsilon_{\mathrm{M}}}} = \sqrt{\frac{\varepsilon_1}{0.16}}. \tag{33} \]

Comparison of the tables shows that reducing the conductor insulation to 0.01 (the use of enamel insulation) reduces the additional losses in the windings from \((11 \div 13)\%\) to \((3—4)\%\) and increases the transformer power by a factor of \((1.2 \div 1.3)\). The use of enamel insulation for the elementary conductors of a helical winding should present no difficulties. Enamel insulation may also be used in all disk windings for insulating parallel conductors. In this case, turn insulation may be made of strips of pressboard, or by using an outer conductor with full turn insulation. This leads to a sharp reduction in the average calculated insulation thickness. If the disk has 10 turns and 50 elementary conductors, then 4 conductors may be taken with enamel insulation \((i_1=0.01)\) and one with increased paper insulation \((i_2=0.2)\). Then the average calculated insulation thickness will be:

\[ i_{\mathrm{av}}=\frac{40\cdot 0.01+10\cdot 0.2}{50}=0.048. \]

In selecting the value of \(\delta\), it was assumed that, for the inter-disk insulation of most disks, an oil channel of \((0.8—0.9)\) cm is sufficient.

Investigations have shown that for transformers there exists an optimum coefficient of filling the winding cross-sectional area with copper:

\[ k_{\mathrm{MO}}=\frac{x_0 y_0}{(x_0+i)(y_0+\delta)}. \tag{34} \]

The value of this coefficient is shown in item 21 of Table 1 and in item 13 of Table 2.

Analysis of the data in Tables 1 and 2 shows that there is a practical possibility of increasing the limiting power of transformers to 600–900 MVA per unit.

The method presented may be used to determine the optimum geometry of a transformer for a specified power and short-circuit voltage.

REFERENCES

1. I. D. Kutyavin, G. V. Del’, V. P. Krasnov. On the techno-economic determination of the optimum dimensions of high-power substation three-phase two-winding transformers. Proceedings of the Tomsk Polytechnic Institute, vol. 130, 1964.

2. Dietrich W. Auslegung von Transformatoren-wicklungen mit kleinstem Wirkwiderstand. “Elektrotechn. Z,” No. 6, 1965.

3. V. Sh. Anshin, A. G. Kraiz. Assembly of Power Transformers. GEI, 1961.