Doctor of Technical Sciences I. D. KUTYAVIN, postgraduate student G. V. DEL,
postgraduate student V. P. KRASNOV.

ON THE TECHNICAL-ECONOMIC DETERMINATION OF THE OPTIMAL DIMENSIONS OF HIGH-POWER SUBSTATION THREE-PHASE TWO-WINDING TRANSFORMERS

The technical-economic determination of the optimal dimensions of transformers can be carried out by minimizing the expression for calculated costs [1], which takes into account the costs of manufacture and operation of the transformer, as well as capital investments for the additional capacity of the system with its fuel base.

\[ Z=(A_1+D B^2)Q_c+(\beta A_2+E\Delta_1^2)Q_{m1}+(\beta A_2+E\Delta_2^2)Q_{m2}, \tag{1} \]

where \(A_1, A_2, D, E, \beta\) are constants [1];
\[ \begin{aligned} B&\text{ is the induction in the core, gs;}\\ \Delta_1,\Delta_2&\text{ are the current densities of the primary and secondary windings, } a/cm^2;\\ Q_c&\text{ is the weight of the core steel, kg;}\\ Q_{m1}\text{ and }Q_{m2}&\text{ are the weights of the conducting material of the primary and secondary windings, kg.} \end{aligned} \]

In the present article the authors set forth an attempt to extend the method presented in [1] to high-power transformers with continuous disk and helical concentric windings [2], since both of these types of windings are described by the same mathematical relationships.

Figure 1 shows the mutual arrangement of the phase windings and the dimensions of the coils. In this case the low-voltage winding may be either helical or continuous disk. The following relationships between the elementary dimensions of the transformer are evident.

The height of the core window (all dimensions in centimeters) is

\[ H=h+2L_i, \tag{2} \]

where \(h\) is the height of the winding and \(L_i\) is the insulating distance from the windings to the yoke.

The width of the core window is

\[ a=2(x_1+x_2+l_r); \tag{3} \]

where \(x_1\) and \(x_2\) are the widths of the coils of the phase windings (Fig. 1);
\[ l_r\text{ is one half of the radial insulating distances in the window.} \]

Length of the mean turn of the phase windings

\[ l_{\mathrm{M}}=\pi(d+0.5x_1+1.5x_2+l), \tag{4} \]

where

\[ l=2\delta_{20}+\delta_{12}, \tag{5} \]

\(d\) is the diameter of the core limb.

Length of core steel, reduced to the cross-sectional area of the limb,

\[ l_c=3H+k_{\mathrm{y}}(4a+5.6d)3h+k_{\mathrm{y}}\left[\delta(x_1+x_2+l_r)+5.6d\right]+6l_{\mathrm{sh}}, \tag{6} \]

where \(k_{\mathrm{y}}\) is the coefficient of increase of the yoke cross-sectional area.

Fig. 1.

Cross-sectional areas of the material of the phase windings

\[ q_{\mathrm{m}1}=\frac{k_1y_1}{y_1+\delta_1}\,x_1h \quad \text{and} \quad q_{\mathrm{m}2}=\frac{k_2y_2}{y_2+\delta_2}\,x_2h, \tag{7} \]

where \(k_1\) and \(k_2\) are coefficients of filling, by conductor material, the cross-sectional areas of the coils of the primary and secondary phase windings;

\(y_1\) and \(y_2\) are the heights of the coils of the phase windings (Fig. 1);

\(\delta_1\) and \(\delta_2\) are the heights of the radial cooling ducts between coils.

Cross-sectional area of the core-limb steel

\[ q_c=k_c\frac{\pi d^2}{4}, \tag{8} \]

where \(k_c\) is the coefficient of filling with steel of the area of a circle of diameter \(d\).

The heat-balance equation of the winding-coil strand (per linear centimeter of the mean turn) is

\[ 2\sigma (k_x x + k_y y)=\rho \Delta^2 kxy, \tag{9} \]

where \(\sigma\) is the design heat-flux density from the coil surface, \(\mathrm{W/cm^2}\);

\(\rho\) is the resistivity of the winding material, \(\Omega\cdot\mathrm{cm}\);

\(k_x\) and \(k_y\) are coefficients allowing for the covering of parts of the surfaces \(x\) and \(y\) by insulating parts.

The permissible current densities in the phase windings \((\mathrm{A/cm^2})\), on the basis of (9), are:

\[ \left. \begin{aligned} \Delta_1 &= \sqrt{\frac{2\sigma_1\left(k_{x1}x_1+k_{y1}y_1\right)} {\rho k_1 x_1 y_1}} = a_1\sqrt{\frac{z_1}{x_1y_1}}, \\[6pt] \Delta_2 &= \sqrt{\frac{2\sigma_2\left(k_{x2}x_2+k_{y2}y_2\right)} {\rho k_2 x_2 y_2}} = a_2\sqrt{\frac{z_2}{x_2y_2}} . \end{aligned} \right\} \tag{10} \]

where

\[ z_1=k_{n1}x_1+y_1;\qquad z_2=k_{n2}x_2+y_2; \]

\[ a_1=\sqrt{\frac{2\sigma_1 k_{y1}}{\rho k_1}};\qquad a_2=\sqrt{\frac{2\sigma_2 k_{y2}}{\rho k_2}};\qquad k_{n1}=\frac{k_{x1}}{k_{y1}};\qquad k_{n2}=\frac{k_{x2}}{k_{y2}} . \tag{11} \]

The expression for the transformer power in kVA is

\[ S=K\Delta_1 q_{m1}q_c=K\Delta_2 q_{m2}q_c . \tag{12} \]

Substituting the variables into (12), we find the winding height

\[ h=\frac{N\left(y_2+\delta_2\right)} {d^2\sqrt{x_2y_2z_2}}, \tag{13} \]

where

\[ N=\frac{4S}{Kk_c k_2\pi a_2} \quad\text{and}\quad K=13.32 fB\cdot 10^{-11}. \tag{14} \]

The reactive component of the short-circuit voltage (in relative units) reduces to the following form:

\[ u_p= \frac{2k_r\Delta_2 q_{m2}l_m\left(x_1+x_2+3\delta_{12}\right)} {3.33\,bq_c k}, \tag{15} \]

where \(k_r\) is the Rogowski coefficient;

\(\delta_{12}\) is the insulating distance between the phase windings.

After substituting the variables, from (15) we find the core diameter:

\[ d= \frac{ U+\sqrt{U^2+4PUV\left(y_2+\delta_2\right)} } {2P\left(y_2+\delta_2\right)}, \tag{16} \]

where

\[ \left. \begin{aligned} U&=\sqrt{x_2y_2z_2}\,\left(x_1+x_2+3\delta_{12}\right),\\ V&=1.5x_2+0.5x_1+l,\\ P&=\frac{3.33\,B\,k_c\,u_p}{8k_r k_2 a_2}. \end{aligned} \right\} \tag{17} \]

We shall now determine the weights of the active materials.

Weight of the core steel, in kg:

\[ Q_{\mathrm c}=\gamma_{\mathrm c} q_{\mathrm c} l_{\mathrm c}\,10^{-3} = \frac{k_{\mathrm c}\pi\gamma_{\mathrm c}}{4}\cdot 10^{-3} \left\{ \frac{3N(y_2+\delta_2)}{\sqrt{x_2y_2z_2}} + d^2\left[8k_{\mathrm a}\left(x_1+x_2+l_{\mathrm r}\right)+5.6k_{\mathrm a}d+6l_{\mathrm sh}\right] \right\}. \tag{18} \]

Weights of the conducting material of the transformer windings, in kg.

\[ \begin{aligned} Q_{\mathrm m1}&\simeq 3\gamma_{\mathrm m}q_{\mathrm m1}l_{\mathrm m1}\,10^{-3} \simeq m_1\frac{y_1+\delta_1}{z_1(x_1+x_2+3\delta_{12})},\\ Q_{\mathrm m2}&\simeq 3\gamma_{\mathrm m}q_{\mathrm m2}l_{\mathrm m2}\,10^{-3} \simeq m_2\frac{y_2+\delta_2}{z_2(x_1+x_2+3\delta_{12})}, \end{aligned} \tag{19} \]

where

\[ \begin{aligned} m_1&\simeq \frac{10\gamma_{\mathrm m}B u_{\mathrm p} S\varphi\cdot10^{-3}} {4K k_{\mathrm r} k_{\mathrm y1}\sigma_1},\\ m_2&\simeq \frac{10\gamma_{\mathrm m}B u_{\mathrm p} S\varphi\cdot10^{-3}} {4K k_{\mathrm r} k_{\mathrm y2}\sigma_2}. \end{aligned} \tag{20} \]

It is interesting to emphasize here that, as is seen from (19), the weight of the conducting material of the winding (for constant initial data) is directly proportional to the coil pitch \((y+\delta)\) and inversely proportional to the half-perimeter of the coil \((z)\) and to the reduced leakage channel of the windings \((x_1+x_2+3\delta_{12})\).

As a result, the weights of the active materials and the principal dimensions of the transformer have been expressed through the dimensions of the winding coils \(x_1, y_1, x_2\), and \(y_2\). One of these variables can also be eliminated by using the condition of equality of the magnetomotive forces of the phase windings:

\[ \frac{\kappa_1x_1y_1}{y_1+\delta_1}\Delta_1 = \frac{\kappa_2x_2y_2}{y_2+\delta_2}\Delta_2. \tag{21} \]

Then three independent variables remain. Investigation for the minimum of the calculated costs (1), which are a function of three independent variables, is very difficult because of the complexity of the expressions for \(Q_{\mathrm c}\) and \(Q_{\mathrm m}\). In this connection we shall consider here the particular case in which \(\Delta_1=\Delta_2\). This case corresponds to the problem of the most rational use of the conducting material and to a considerable extent coincides with factory practice. Bearing in mind that \(\kappa_2\) is practically independent of the transformer rating, it is advantageous to take \(x_2\) and \(y_2\) as the independent variables. Then from (21), with \(k=\kappa_2:\kappa_1\),

\[ y_1= \frac{k\delta_1x_2y_2} {x_1(y_2+\delta_2)-kx_2y_2}. \tag{22} \]

Solving (10) and (22) jointly for \(k_{\mathrm p1}=k_{\mathrm p2}\), \(k_{\mathrm y1}=k_{\mathrm y2}\), \(\delta=\dfrac{\sigma_2}{\sigma_1}\), and \(\Delta_1=\Delta_2\), we find the expression for \(x_1\):

\[ x_1= \frac{ \left(k_{\mathrm p}kx_2y_2+\delta\delta_1z_2\right) + \sqrt{\left(k_{\mathrm p}kx_2y_2+\delta\delta_1z_2\right)^2 -4k_{\mathrm p}k\delta_1x_2y_2(y_2+\delta_2)} } {2k_{\mathrm y}(y_2+\delta_2)}. \tag{23} \]

The expression for the calculated costs (1) under the adopted assumptions is:

\[ Z=(A_1+D B^2)Q_c+(\beta A_2+E x_2^2)Q_m= \]

\[ =-\frac{k_c \pi \gamma_c}{4}\,10^{-3}(A_1+D B^2)\,\varphi(x_2,y_2); \tag{24} \]

where

\[ \varphi(x_2,y_2)=-3N(y_2+\delta_2) \left[ \frac{1}{x_2 y_2 z_2} + \frac{M\left(n+m-\dfrac{z_2}{x_2 y_2}\right)} {z_2(x_1+x_2+3\delta_{12})} \right] + \]

\[ +\,d^2\{k_q[8(x_1+x_2+l_r)+5.6d]+6l_u\}. \tag{25} \]

Here:

\[ M=\frac{8k_pP\gamma_m}{k_c\gamma_c}; \qquad n=\frac{\beta A_2}{A_1+D B^2}; \qquad m=\frac{E x_2^2}{A_1+D B^2}. \tag{26} \]

The determination of the critical values of \(x_2\) and \(y_2\) corresponding to the minimum calculated costs (24) can be carried out by minimizing function (25). However, a general analytical solution of this problem does not appear possible because of its complexity. Therefore, within the limits of the present article we shall consider the numerical determination of the critical values of \(x_2\) and \(y_2\) for a number of 110 kV substation two-winding transformers without voltage-regulating devices.

The numerical investigation of (25) for a minimum was carried out on a digital computer (Computing Center of the Academy of Sciences of the USSR) for transformers with cores made of cold-rolled steel, with the following initial data (dimensions as in the preceding text): \(\varkappa_c=0.82\); \(\varkappa_{x1}=\varkappa_{x2}=0.7\); \(k_{y1}=k_{y2}=0.9\); \(\varkappa_q=1.05\); \(\varkappa_z=0.8\); \(\varkappa_m=0.7\) [3]; \(\varkappa_n=0.8\) [3]; \(k_r=0.95\); \(k_t=0.7\); \(\delta_1=0.8\); \(\delta_2=0.6\); \(\delta_{02}=1.5\); \(\delta_{12}=5\); \(\delta_{11}=6\); \(l_{11}=9\); \(u_p=0.105\); \(B=16500\); \(a_c=0.8\cdot10^{-11}\ \mathrm{kW/kg}\); \(\gamma_c=7.65\); \(K=1.1\cdot10^{-4}\); \(p_n=0.125\); \(p_x=0.06\); \(C_1=124\ \mathrm{rub/kW}\); \(C'_2=0.007\ \mathrm{rub/kW\!-\!h}\); \(C''_2=0.009\ \mathrm{rub/kW\!-\!h}\); \(\tau=4000\) h; \(t=8700\) h;

\[ \left(\frac{k_n S_m}{S}\right)^2=0.4 \]

[3]; \(z_p=1.5\ \mathrm{rub/kvar}\); \(a_p=0.046\ \mathrm{kvar/kg}\).

The values of the adopted coefficient \(k_1\) for transformers of different power \(S\) with copper and aluminum windings are given in Table No. 1.

Table No. 1.

Adopted for copper: \(\rho_m=2.14\cdot10^{-6}\); \(a_m=2.4\cdot10^{-7}\ \mathrm{kW/kg}\); \(\gamma_m=8.9\); \(\beta=3.2\); \(\sigma_1=\sigma_2=0.15\).

Adopted for aluminum: \(\rho_a=3.6\cdot10^{-6}\); \(a_a=1.33\cdot10^{-6}\ \mathrm{kW/kg}\); \(\gamma_a=2.7\); \(\beta=4\); \(\sigma_1=\sigma_2=0.14\).

The investigation of (25) was carried out in the region \(x_2=3\text{–}10\) and \(y_2=0.5\text{–}10\), at intervals of 0.5. The results of the investigation are shown in Figs. 2–6. Figure 2 gives the dependence of \(\varphi(x_2,y_2)\) on \(x_2\) at \(y_2=1.5\) for transformers rated 5.6–60 MVA with copper windings, and Fig. 3 gives the same for transformers with aluminum windings at \(y_2=2\). As is evident from these figures, the quantity \(\varphi(x_2,y_2)\) has a minimum at a certain

Fig. 2. Curve 1 for a transformer of 5.6 MVA;
2 — 10; 3 — 20; 4 — 31.5; 5 — 40.5; 6 — 60 MVA (copper).

values of \(x_2\) depending on the transformer rating. In this case, the calculated value of \(x_2\) may be taken as differing from the critical value by at least \(\pm 0.5\) cm without any substantial increase in the calculated costs. But it must be borne in mind that reducing \(x_2\) for a given transformer leads to a decrease in the consumption of active materials and to a sharp increase in the core height, while increasing \(x_2\) leads to the opposite results.

The dependence of the critical values of \(x_2\) on transformer rating, for windings made of copper and aluminum, according to the data of Figs. 2 and 3, is given in Fig. 6 and can be used to determine the calculated \(x_2\).

Figures 4 and 5 show the dependence, on \(y_2\), of the minimum values of \(\varphi(x_2,y_2)\) found with respect to \(x_2\). This dependence also has a minimum, which for transformers of different ratings with copper windings lies within \(y_2 = 1.5 \div 2\). As can be seen from Fig. 4, the minimum with respect to \(y_2\) is so flat that the calculated value of \(y_2\) may differ greatly from the critical one, especially in the direction of increase. Thus, when the value of function (25) is increased above the smallest minimum by 2% for transformers with copper windings of 5.6–60 MVA, the calculated \(y_2\) may be taken within the range from 1 to 3 cm.

For transformers with aluminum windings rated 5.6–31.5 MVA (Fig. 5), the minimum value of function (25) is found near \(y_2 = 5\). When the minimum of (25) is increased above its smallest value by 2%, the calculated value \(y_2 = 2 \div 10\) may be adopted.

Fig. 3. Curve 1 for a 5.6 MVA transformer;
2 — 10; 3 — 30; 4 — 31.5 MVA (aluminum).

This property of function (25) is very important, since with a decrease in the calculated value of \(y_2\) the consumption of conductor material and the weight of the transformer decrease sharply.

Table 2 gives the principal dimensions of certain substation transformers with copper and aluminum windings, calculated according to the method set out above.

The calculated value of \(y_2\) was taken with calculated costs exceeding the minimum by no more than 2%, and for transformers with copper windings it is equal to 1 cm, while with aluminum windings it is 2 cm.

Fig. 4. Curve 1 for a 5.6 MVA transformer;
2 — 60 MVA (copper).

In item 9 of Table 2, the winding height \(h\) is increased by 5% to \(h'\) in order to accommodate the transposition of the low-voltage winding and to strengthen the insulation and reduce the current density in the end coils of the high-voltage winding. In a similar manner, the limb height can be allowed for in order to accommodate the regulating turns of the winding.

Fig. 5. Curve 1 for a 5.6 MVA transformer;
2 — 31.5 MVA (aluminum).

In item 18 the mechanical stress in the winding material is indicated, caused by radial forces arising during a short circuit beyond the transformer. This stress was determined from expression [3], reduced to the following form:

\[ \sigma_{\mathrm{p}} = C\,\frac{\Delta_1^2 q_{\mathrm{m}1} l_{\mathrm{m}}}{h}, \tag{27} \]

where \(C \simeq 5.7 \cdot 10^{-6}\) is a constant.*)

As can be seen from this table, the stresses in the winding material for the transformer ratings investigated proved to be within permissible limits.

Table 2.

*) Ratio of stresses in the winding material: \(\sigma_{\mathrm{p}1}:\sigma_{\mathrm{p}2}=\Delta_1:\Delta_2\).

It is also of interest to compare modern factory-built transformers with transformers whose dimensions are determined by the proposed method. This comparison is possible only for transformers with cores of hot-rolled steel and with copper windings, for which data are available in the literature.

The weights of the active materials of transformers with cores of hot-rolled steel and with copper windings were determined for \(k_t = 0.5\); \(B = 14500\) gauss; \(\gamma_c = 1.2 \cdot 10^{-11}\ \text{kW}/\text{kg}\); \(\beta = 3.7\), and are given in item 2 of Table 3. Item 1 gives the weights of the active materials of factory-built transformers, and item 3—for transformers with cold-rolled steel.

Table 3.

Fig. 6. Curve 1 for copper and 2 for aluminum.

In conclusion, let us note that the proposed method, despite its complexity, may find application in factory design and in the general study of transformers. The complexity of the method is compensated by its universality and sufficient accuracy.

References

1. Kutyavin I. D., On the determination of the optimal sizes of three-phase two-winding transformers, see above.

2. Postnikov I. M., Design of Electrical Machines. Gostekhizdat, Ukrainian SSR, 1960.

3. Tikhomirov P. M., Calculation of Transformers. Gosenergoizdat, 1962.