Toward a Techno-Economic Study of the Optimal Dimensions of Medium-Power Three-Phase Two-Winding Power Transformers

I. D. Kutyavin, G. V. Del

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

In [1] a procedure is presented for the techno-economic determination of the optimal dimensions of transformers at constant coefficients of filling, by conductor material, of the cross-sectional area of the coils of the primary and secondary windings \((K_1\) and \(K_2)\). In reality these coefficients are variables, depending on the ratio of the dimensions of the elementary conductors and their insulation. The solution of the problem with allowance for the variability of the coefficients \(K_1\) and \(K_2\) increases the accuracy of the result and is given below.

The equations of transformer power, short-circuit voltage, heat balance of the coils of the primary (HV) and secondary (LV) windings, equality of magnetomotive forces, and voltage in the material of the outer winding under a short circuit beyond the transformer may be represented, respectively, in the following form:

\[ \left. \begin{aligned} S &= KB\,k_1 x_1 y_1 \Delta_1 \cdot \frac{d^2 h}{y_1+\delta_1},\\[4pt] u_p &= \frac{K_p k_1 x_1 y_1 \Delta_1}{B d^2 (y_1+\delta_1)} \,(d+2x_2+l)(x_1+x_2+3\delta_{12}),\\[4pt] 2\sigma_1 k_{y1}(k_{n1}x_1+y_1) &= \rho k_1 x_1 y_1 \Delta_1^2,\\[4pt] 2\sigma_2 k_{y2}(k_{n2}x_2+y_2) &= \rho k_2 x_2 y_2 \Delta_2^2,\\[4pt] \frac{k_1 x_1 y_1 \Delta_1}{y_1+\delta_1} &= \frac{k_2 x_2 y_2 \Delta_2}{y_2+\delta_2},\\[4pt] \sigma_p &= K_m \frac{k_1 x_1 y_1 \Delta_1^2}{y_1+\delta_1}(d+2x_2+l). \end{aligned} \right\} \tag{1} \]

In the system of equations (1) it is assumed that:

\[ \left. \begin{aligned} K &= 3{,}33 \cdot \pi \cdot f \cdot k_c \cdot 10^{-11},\\[4pt] K_p &= \frac{8\pi^2 \cdot k_r}{3{,}33 \cdot k_c},\\[4pt] K_m &= 2{,}04\pi \frac{k_r \cdot k_{yd}^2}{u_k^2}\cdot 10^{-8},\\[4pt] l &= 2\delta_{02}+\delta_{12},\\[4pt] k_{n1} &= k_{x1}:k_{y1};\qquad k_{n2}=k_{x2}:k_{y2}. \end{aligned} \right\} \tag{2} \]

All the notation in (1) and (2) corresponds to [1], with the exception of \(y\), \(\delta\), and \(k_{\mathrm{ud}}\). Here \(y\) is the axial dimension of the bare conductor of the winding coil (Fig. 1); \(\delta\) is the axial dimension of the radial oil duct between coils plus the insulation of the elementary conductor on two sides \((i)\); \(k_{\mathrm{ud}}=1.8\) is the surge factor.

Fig. 1.

The number of turns in the winding is

\[ W_k=\frac{kxy}{q_{\mathrm{v}}}=\frac{kxy\Delta}{I_{\mathrm{n}}}, \tag{3} \]

where \(q_{\mathrm{v}}\) is the cross section of one turn, and \(I_{\mathrm{n}}\) is the rated current of the winding.

The cross section of the conducting material in the winding is

\[ kxy=(x-miw_k)y. \tag{4} \]

The number of parallel branches of the winding \((m)\) may be taken as constant, determined by the magnitude of the rated current of the winding \((I_{\mathrm{n}})\) and the current of one parallel branch \((I_{\mathrm{v}})\).

Substituting (3) into (4), we obtain

\[ k=\frac{1}{1+\lambda\Delta y}, \tag{5} \]

where

\[ \lambda=\frac{im}{I_{\mathrm{n}}}=\frac{i}{I_{\mathrm{v}}}. \tag{6} \]

Then the coefficients of filling the cross-sectional area of the coils with conducting material are

\[ k_1=\frac{1}{1+\lambda_1\Delta_1 y_1};\qquad k_2=\frac{1}{1+\lambda_2\Delta_2 y_2}. \tag{7} \]

The coefficients \(k_1\) and \(k_2\) have thus been expressed through the adopted variables, whose number has not increased. Introducing the numbers of parallel branches \((m_1, m_2)\) as initial quantities in the calculation makes this number integral.

Substituting (7) into the system of equations (1), we obtain

\[ \left\{ \begin{aligned} S&=KB\,\frac{x_1y_1\Delta_1hd^2} {(1+\lambda_1\Delta_1y_1)(y_1+\delta_1)},\\[6pt] u_{\mathrm{p}}&=K_{\mathrm{p}}\, \frac{x_1y_1\Delta_1(x_1+x_2+3\delta_{12})} {Bd^2(1+\lambda_1\Delta_1y_1)(y_1+\delta_1)} \,(d+2x_2+l),\\[6pt] k_{\mathrm{p}1}x_1+y_1&=L_1\Delta_1^2\, \frac{x_1y_1}{1+\lambda_1\Delta_1y_1},\\[6pt] k_{\mathrm{p}2}x_2+y_2&=L_2\Delta_2^2\, \frac{x_2y_2}{1+\lambda_2\Delta_2y_2},\\[6pt] \frac{x_1y_1\Delta_1}{(1+\lambda_1\Delta_1y_1)(y_1+\delta_1)} &= \frac{x_2y_2\Delta_2}{(1+\lambda_2\Delta_2y_2)(y_2+\delta_2)}. \end{aligned} \right. \tag{8} \]

\[ \sigma_p = K_m \frac{x_1 y_1 \Delta_1^2}{(1+k_1\Delta_1 y_1)(y_1+\delta_1)}(d+2x_2+l), \tag{9} \]

where

\[ L_1=\frac{\rho}{2\sigma_2 k_{y1}};\qquad L_2=\frac{\rho}{2\sigma_2 k_{y2}}. \tag{10} \]

In medium-power 110-kV transformers, mainly two types of windings are used: continuous disk and helical. Both types of windings have the same mathematical description, used for disk windings. In this case it must be taken into account that the number of turns in the equivalent coil of a helical winding is inversely proportional to the number of starts \((n_3)\).

\[ w_{k2}=\frac{1}{n_3}. \tag{11} \]

The latter circumstance makes it necessary to divide the transformers under consideration, depending on the windings used in them, into two groups:

1. Transformers with the winding combination—continuous disk (HV)—continuous disk (LV) [Cont.—Cont.].
2. Transformers with the winding combination—continuous disk (HV) and helical (LV) [Cont.—Hel.].

The first group may include transformers with ratings of 5.6–15 MVA at 110 kV. The second includes transformers of higher rating.

In the event of a short circuit beyond a transformer of the first group, the mechanical stress in the winding material will be less than the permissible value; therefore, to investigate the parameters and dimensions of these transformers, the system of equations (8) is sufficient. To investigate transformers of the second group, starting from a certain rating, when the mechanical stresses in the winding material will exceed the permissible values, allowance for equation (9) will be required.

This article presents the results of investigating transformers of the first group by means of system (8), which contains five equations with eight unknowns—\(x_1, x_2, y_1, y_2, \Delta_1, \Delta_2, d\), and \(h\).

The lowest calculated costs for the manufacture and operation of the transformer will occur when the current densities of windings made of the same material are equal. This gives us one more equation, \(\Delta_1=\Delta_2\). A preliminary analysis showed that, as independent variables, it is most rational to use the variables \(x_1\) and \(y_1\). In this case, the system of equations (8) and the condition \(\Delta_1=\Delta_2=\Delta\) give the following expressions for the remaining variables:

\[ \Delta=\frac{\lambda_1+\sqrt{\lambda_1^2+4L_1\frac{z_1}{y_1}}}{2L_1 z_1}, \tag{12} \]

where

\[ z_1=\frac{x_1}{k_{p1}x_1+y_1}; \]

\[ y_2=\frac{U_1+\sqrt{U_1^2+4k_{p2}\delta_2 U_2}}{2U_2}, \tag{13} \]

where

\[ U_1=k_{p2}-\delta_2\Delta(L_2\Delta-k_{p2}\lambda_2), \]

\[ U_2=\Delta(L_2\Delta-k_{p2}\lambda_2)-\frac{(1+k_1\Delta_1 y_1)(y_1+\delta_1)}{x_1 y_1}; \]

\[ x_2=\frac{x_1y_1(y_2+\delta_2)(1+\lambda_2\Delta_2y_2)} {y_2(y_1+\delta_1)(1+\lambda_1\Delta_1y_1)}, \tag{14} \]

\[ d=\frac{V_1+\sqrt{V_1^2+4V_1V_2}}{2}, \tag{15} \]

where

\[ V_2=2x_2+l; \]

\[ V_1=\frac{K_p x_1y_1\Delta_1(x_1+x_2+3\delta_{12})} {u_p(y_1+\delta_1)(1+\lambda_1\Delta_1y_1)}, \]

\[ h=\frac{S}{K}\cdot \frac{(y_1+\delta_1)(1+\lambda_1\Delta_1y_1)} {x_1y_1\Delta_1d^2}. \tag{16} \]

The expression for the calculated costs of the transformer is

\[ З=\frac{\pi\gamma_c k_c}{4}\left(A+ДB^2+1{,}5\alpha_p\right)\cdot\varphi(x_1,y_1), \tag{17} \]

\[ \begin{aligned} \varphi(x_1,y_1)=& \,N\frac{(y_1+\delta_1)(1+\lambda_1\Delta_1y_1)} {x_1y_1\Delta_1} \left[ 1+M(\beta A+E\Delta^2) \frac{d_{\mathrm{cp}}}{d_{12}\Delta(x_1+x_2+3\delta_{12})} \right] \\ &+d^2\left[8k_a(x_1+x_2)+5{,}6k_ad+8k_al_r+6l_u\right]\cdot10^{-3}, \end{aligned} \tag{18} \]

where

\[ M=\frac{8\gamma_m\cdot u_p} {K_p\gamma_c k_c\left(A+ДB^2+1{,}5\alpha_p\right)}. \tag{19} \]

\[ N=\frac{3S}{K}\cdot10^{-3};\qquad l_r=\delta_{02}+\delta_{12}+0{,}5\delta_{11}. \tag{20} \]

Diameter of the mean turn

\[ d_{\mathrm{cp}}=d+0{,}5x_1+1{,}5x_2+l. \tag{21} \]

Diameter of the leakage channel

\[ d_{12}=d+2x_2+l. \tag{22} \]

The function \(\varphi(x_1,y_1)\), having the expression (18), was investigated for a minimum for 110/11 kV transformers of ratings 5.6; 7.5; 10 and 15 MVA. The following initial data were adopted: winding connection \(\lambda/\Delta\), material—copper, \(u_p=0{,}105\); \(B=16500\) gs; \(\rho=2{,}14\cdot10^{-6}\) ohm·cm; \(\sigma_1=\sigma_2=0{,}15\) W/cm\(^2\); \(m_1=1\); \(i_1=0{,}14\); \(i_2=0{,}05\); \(\delta_1=0{,}94\); \(\delta_2=0{,}66\), \(l=8\); \(l_r=9{,}5\); \(k_{x1}=k_{x2}=0{,}7\); \(k_{y1}=k_{y2}=0{,}91\) (all linear dimensions in cm).

Calculated: \(L_1=L_2=11{,}8\cdot10^{-6}\); \(K=4{,}3\cdot10^{-9}\); \(K_p=2{,}75\); \(M=1{,}74\cdot10^4\). Parameters depending on the rating are given in Table 1.

Table 1

Constants for the economic calculation: \(K_{\mathrm{t}}=0{,}96\) and \(A=0{,}1775\) for \(S=5{,}6\) and \(7{,}5\) MVA; \(k_{\mathrm{t}}=0{,}9\) and \(A=0{,}1665\) for \(S=10\) and \(15\) MVA; \(\beta=3{,}2\); \(D=45{,}8\cdot10^{-11}\), \(E=47\cdot10^{-7}\); \(a_{\mathrm{p}}=19{,}3\) VA/kg. The study was carried out in the range \(y_1=0{,}5—2{,}0\) cm. The main results of this study are shown in Table 2.

Analysis of the results of the study (Table 2) indicates a weak dependence of the calculated costs on the axial dimension of the coil, \(y_1\). The optimal values of \(y_1\) for transformers of the ratings investigated remain approximately constant and equal to 1.1 cm. This makes it possible, in investigations, to assume \(y_1=\mathrm{const}\). The recommended range is \(y_1=0{,}8\div1{,}6\) cm, without any substantial increase in costs compared with the minimum (Table 2).

The optimal values of \(y_2\) for almost the entire range of ratings of the transformers investigated proved to be structurally too small (Table 2). This is explained by the fact that in the study it was assumed that \(\lambda_2\ll\lambda_1\), \(\delta_1=\delta_2\), and \(\Delta_1=\Delta_2\).

Table 2

An increase of the optimal \(y_2\) to an acceptable structural value can be obtained by increasing \(\sigma_2\) and \(\lambda_2\) while maintaining \(\Delta_1=\Delta_2\). This can be shown by using the heat-balance equation of the secondary-winding coil (8). If the value \(y_2\) in the left-hand side of (8) is neglected in comparison with \(\kappa_{\mathrm{p}2}x_2\), and this equation is solved with respect to \(y_2\), we obtain

\[ y_2=\frac{\kappa_{\mathrm{p}2}}{\Delta\left(\dfrac{\varphi\Delta}{2\kappa_{\mathrm{uy}}\sigma_2}-\kappa_{\mathrm{p}2}\lambda_2\right)}. \tag{23} \]

Increasing \(\sigma_2\) to a value \(\sigma_2>\sigma_1\) is entirely justified despite the poorer cooling conditions of the secondary winding, since the thickness of the conductor insulation of this winding \(i_2\) is much smaller than \(i_1\). Consequently, with an insignificant increase in the temperature drop in the insulation of the secondary-winding conductor, the specific heat load on its surface can be increased to the level \(\sigma_2>\sigma_1\).

The determination of \(\sigma_2\) required to obtain the constructive value \(y_2=\kappa' y_1\) can be carried out if (23) is used under the condition \(k_{\mathrm{p}1}=k_{\mathrm{p}2}=k_{\mathrm{p}}\) and \(k_{\mathrm{y}1}=k_{\mathrm{y}2}=k_{\mathrm{y}}\):

\[ \sigma_2 = \frac{\kappa'} {\dfrac{1}{\sigma_1}+\dfrac{2k_{\mathrm{y}}k_{\mathrm{p}}}{r\Delta}\left(\kappa'\lambda_2-\lambda_1\right)} . \tag{24} \]

For a preliminary determination of \(\sigma_2\), one may take \(\Delta=400\ \mathrm{A/cm^2}\) and \(\kappa'=0.7\div 1\).

However, in most cases merely increasing \(\sigma_2\) in order to obtain the constructive value \(y_2\) may prove insufficient, owing to \(\lambda_2 \ll \lambda_1\). Therefore \(\lambda_2\) should also be increased simultaneously. Ideally, one would achieve the condition \(\kappa'\lambda_2-\lambda_1=0\), which is satisfied at the branch current of the secondary winding

\[ I_{\mathrm{v}2}=I_1\,\frac{\kappa'i_2}{i_1}. \tag{25} \]

But in all this it must be borne in mind that any artificial increase in \(y_2\) will lead to an increase in the calculated costs. The method described for increasing \(y_2\) to a constructive value may prove insufficient for transformers rated 1.8–5.6 MVA, 110 kV. In this case it is simplest to abandon the equality of current densities in the windings.

When \(\Delta_1\ne\Delta_2\), the number of independent variables can be reduced to two by taking \(y_1=\mathrm{const}\) on the basis of previous studies. Taking \(x_2\) and \(y_2\) as the independent variables, we obtain from the system of equations (8) expressions for the remaining variables:

\[ \Delta_2= \frac{\lambda_2+\sqrt{\lambda_2^2+4L_2\dfrac{z_2}{y_2}}}{\,}, \qquad \text{where}\quad z_2=\frac{x_2}{k_{\mathrm{p}2}x_2+y_2}; \tag{26} \]

\[ \Delta_1= \frac{ \left(k_{\mathrm{p}1}\lambda_1+\dfrac{y_1X_1}{y_1+\delta_1}\right) + \sqrt{ \left(k_{\mathrm{p}1}\lambda_1+\dfrac{y_1X}{y_1+\delta_1}\right)^2 +4L_1\cdot\dfrac{k_{\mathrm{p}1}}{y_1} } } {2L_1}, \tag{27} \]

where

\[ X=\frac{(y_2+\delta_2)(1+\lambda_2\Delta_2y_2)}{x_2y_2\Delta_2}; \]

\[ x_1=\frac{(y_1+\delta_1)(1+\lambda_1\Delta_1y_1)}{Xy_1\Delta_1}. \tag{28} \]

The expressions for \(d\) and \(h\) are analogous to (15) and (16). The expression for the calculated costs corresponds to (17), but the function \(\varphi\) has the following form:

\[ \varphi(x_2,y_2)=NX+d^2\left\{\kappa_{\mathrm{a}}\left[5.6d+8(x_1+x_2)+8l_{\mathrm{r}}\right]+6l_{\mathrm{i}}\right\}\cdot 10^{-3} + \frac{M_3}{d^2} \left[ (\beta A+E\Delta_1^2)\frac{d_{\mathrm{av}.1}}{\Delta_1} + (\beta A+E\Delta_2^2)\frac{d_{\mathrm{av}.2}}{\Delta_2} \right], \tag{29} \]

where

\[ M_3=\frac{4\gamma_{\mathrm{m}}N}{\gamma_{\mathrm{c}}k_{\mathrm{c}}\left(A+DB^2+1.5a_{\mathrm{p}}\right)}; \tag{30} \]

\[ d_{\mathrm{av}.2}=d+x_2+2\delta_{02}; \qquad d_{\mathrm{av}.1}=d+x_1+2\left(x_2+\delta_{02}+\delta_{12}\right). \tag{31} \]

Function (29) was investigated for a minimum for a 10-MVA transformer, with the initial data adopted above and with \(y_1=0.7;\ 1.0;\ 1.5\). The results of the investigation are given in Table 3, where the absolute minimum of the function is taken as \(\varphi=3060\) at \(y_1=y_2=1\). Within the limits of \(y_1\) and \(y_2\) given in Table 3, the calculated-cost function may practically be considered constant.

Table 3

The character of the dependence of function (29) on \(x_2\) is shown in Fig. 2, and Fig. 3 gives the dependences \(x_1(x_2)\) for three ratings at

Fig. 2.

Fig. 3.

\(y_1=y_2=1.0\) cm and \(d(x_2)\) for \(S=10\) MVA and the same \(y_1\) and \(y_2\). The curves \(d(x_2)\) for 5-, 6-, and 15-MVA transformers practically coincide with the curve for \(S=10\) MVA. For this same reason it was not possible to reflect the dependence

$\hat d(x_2)$ for various $y_2$. It should be noted that the minimum of function (29) proved to be 1.4% greater than the minimum of function (18).

The conclusion made above about the weak dependence of the calculated costs on the quantities $y_1$ and $y_2$ makes it possible to reduce the number of independent variables to one, for example, $x_2$. This makes it possible to investigate the behavior of the costs (29) when any of the parameters changes: $B$, $\delta$, $U_k$, $K_c$, etc. The dependence of the costs (29) on $B$ and $\delta$ was investigated for $y_1 = y_2 = 1\ \text{cm}$ and for the initial data adopted above. It turned out that the calculated costs have a minimum both with respect to $\delta$ (Fig. 4) and with respect to $B$ (Fig. 5). The optimal

Fig. 4.                  Fig. 5.

Table 4

the quantity \(\delta\) depends on power, increasing as the latter increases. The calculated costs \(Z(B)\) have a very weak dependence on transformer power (the \(Z(B)\) curves merge for powers of \(5.6 \div 15\) MVA). The optimum induction for E-330 steel \(0.35\) mm thick according to GOST 802-58 (Fig. 5) was found to be 16,500 G. Deterioration of the steel properties lowers the optimum induction and increases the minimum costs.

The study was carried out without taking account of additional losses, on the assumption that their magnitude for transformers of \(5.6 \div 15\) MVA is comparatively small and has little effect on the position of the minimum of \(Z(x_2)\).

Table 4 gives the results of calculating 110/10 kV transformers. For a power of 10 MVA the calculation was performed for several values of \(y_1\) and \(y_2\). In this case the discrepancy between the minimum calculated costs does not exceed 1.6%. Of particular interest is the fact that, with an increase in the axial dimensions of the coils, the copper losses decrease appreciably, while the weight of copper remains practically constant. This leads to a certain increase in the losses and weight of steel.

Table 5

Table 5 gives the specific weights and specific losses per 1 kVA of power of the transformers considered, as well as of 110 kV transformers of domestic and foreign manufacture of comparable ratings.

Conclusions

1. A techno-economic method is given for studying the optimum dimensions of transformers with variable filling factors of the winding-coil area by conductor material.

2. For transformers with a continuous—interleaved winding combination, the dependence of calculated costs on the axial dimension of the HV and LV winding coils is very weak.

3. The existence of a cost minimum with respect to the specific thermal loading of the windings (\(\sigma_{\mathrm{opt}}\)) and with respect to induction (\(B_{\mathrm{opt}}\)) is noted.

4. The dependences \(d(x_2)\) and \(x_1(x_2)\) are rectilinear in character.

5. For all transformer powers considered above, the tensile stress in the winding conductor under short-circuit conditions is less than the permissible value.

1. As the axial dimension of the windings increases, \(Q_c\) and \(P_c\) increase only slightly, while \(P_m\) decreases.

2. For the same \(y\), the winding heights of the transformers considered, of all power ratings, are equal.

REFERENCES

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

2. P. M. Tikhomirov, Calculation of Transformers, GEI, 1962.