SOME TECHNICAL AND ECONOMIC STUDIES OF STATIONARY SINGLE-PHASE TWO-WINDING TRANSFORMERS

I. D. Kutyavin, V. P. Krasov, and G. V. Del

Technical and economic studies of optimal dimensions and parameters can be carried out by minimizing the cost expression [1], which takes into account the costs of manufacturing and operating the transformer and the capital investments for creating additional capacity in the system (with allowance for the fuel base) to cover the losses in the transformer.

\[ Z = (A_1 + D B^2)Q_c + (\beta A_2 + \varepsilon A_1^2)Q_{\mathrm{M}1} + (\beta A_2 + \varepsilon A_2^2)Q_{\mathrm{M}2}. \tag{1} \]

The study of costs in general form is very complex; therefore, in the present article the results are given of a numerical study obtained on a digital computer for transformers with core-type and shell-type cores (Fig. 1).

The notation adopted is:

\(h\) — winding height, cm,

\(l_u\) — insulation distance from the winding to the yoke, cm,

\(H\) — height of the core window, cm,

\(d\) — diameter of the core limb, cm,

\(x_1\) and \(x_2\) — radial width of the high-voltage (HV) and low-voltage (LV) windings, cm,

\(y_1\) and \(y_2\) — height of the coils of the HV and LV windings, cm,

\(\delta_1\) and \(\delta_2\) — height of the radial cooling ducts between the coils of the HV and LV windings, allowing for conductor insulation, cm,

\(\sigma_1\) and \(\sigma_2\) — heat-flux density from the surface of the HV and LV windings, W/cm\(^2\),

\(\rho_{\mathrm{M}}\) — specific resistance of the conductor, ohm·cm,

\(\delta_{02}\) — insulation distance between the LV winding and the core limb, cm,

\(\delta_{12}\) — insulation distance between the HV and LV windings, cm,

\(\delta_{11}\) — insulation distance between the HV windings, cm,

\(\delta_{10}\) — insulation distance between the HV winding and the core limb, cm,

\(k_1\) and \(k_2\) — copper fill factor of the cross section of the HV and LV windings,

\(\Delta_1\) and \(\Delta_2\) are the current densities in the HV and LV windings, A/cm\(^2\),

\(q_{\mathrm{M}1}\) and \(q_{\mathrm{M}2}\) are the cross-sectional areas of the HV and LV windings, cm\(^2\),

\(q_{\mathrm{c}}\) is the cross-sectional area of the core steel, cm\(^2\),

\(k_{\mathrm{c}}\) is the coefficient of filling the core cross-sectional area with steel,

\(m_1\) and \(m_2\) are the number of parallel conductors in the HV winding and the number of parallel conductors per turn of the LV winding,

\(i_1\) and \(i_2\) are the thicknesses of insulation on two sides of the conductors of the HV and LV windings, cm,

\(n_2\) is the number of turns of the LV winding,

\(k_{\mathrm{y}}\) is the coefficient of increase of the yoke cross section,

\(n_{\mathrm{c}}\), \(n_{\mathrm{d}}\), \(n_{\mathrm{e}}\) are transition coefficients from a core-type core to a shell-type core,

\(l_{\mathrm{c}}\) is the length of core steel referred to the cross-sectional area of the limb,

\(l_{\mathrm{M}1}\) and \(l_{\mathrm{M}2}\) are the lengths of the mean turn of the HV and LV windings, cm,

\(Q_{\mathrm{M}}\) is the weight of winding material, kg,

\(Q_{\mathrm{C}}\) is the weight of core steel, kg,

\(\gamma_{\mathrm{M}}\) and \(\gamma_{\mathrm{C}}\) are the densities of the conductor material and of the core steel, kg/cm\(^3\),

\(k_x\) and \(k_y\) are coefficients accounting for the coverage of the winding surfaces by insulating spacers,

\(f\) is the network frequency, Hz,

\(B\) is the induction in the limb,

\(I_{\mathrm{H}1}\) and \(I_{\mathrm{H}2}\) are the rated currents of the HV and LV windings,

\(C\) is a constant.

The designations of the remaining coefficients are taken from [1].

To compile the calculation equations, the following dependencies were obtained [1; 2; 4]

\[ H = h + 2l_{\mathrm{u}}; \tag{2} \]

\[ l_{\mathrm{M}1} = \pi(d + x_2 + 2x_2 + 2\delta_{02} + 2\delta_{12}); \tag{3} \]

\[ l_{\mathrm{M}2} = \pi(d + x_2 + 2\delta_{02}); \tag{4} \]

\[ q_{\mathrm{M}1} = \frac{k_1 x_1 y_1 h}{y_1 + \delta_1}; \qquad q_{\mathrm{M}2} = \frac{k_2 x_2 y_2 h}{y_2 + \delta_2}, \tag{5} \]

where

\[ k_1 = \frac{I_{\mathrm{H}1}}{I_{\mathrm{H}1} + m_1 i_1 y_1' \Delta_1}; \qquad k_2 = \frac{I_{\mathrm{H}2}}{I_{\mathrm{H}2} + m_2 i_1 y_2 \Delta_2}; \tag{6} \]

\[ l_{\mathrm{c}} = n_{\mathrm{c}}(h + 2l_{\mathrm{u}}) + k_{\mathrm{y}} \left[ 2n_{\mathrm{e}}(x_1 + x_2 + \delta_{02} + \delta_{12} + \delta_{10} + 0.5\delta_{11}) + n_{\mathrm{d}} d \right]; \tag{7} \]

\[ q_{\mathrm{c}} = k_{\mathrm{c}} \frac{\pi d^2}{4}; \tag{8} \]

\[ Q_{\mathrm{M}1} = n_{\mathrm{e}} \gamma_{\mathrm{M}} q_{\mathrm{M}1} l_{\mathrm{M}1} \cdot 10^{-3}; \qquad Q_{\mathrm{M}2} = n_{\mathrm{e}} \gamma_{\mathrm{M}} q_{\mathrm{M}2} l_{\mathrm{M}2} \cdot 10^{-3}; \tag{9} \]

\[ Q_{\mathrm{M}} = Q_{\mathrm{M}1} + Q_{\mathrm{M}2}; \tag{10} \]

\[ Q_{\mathrm{C}} = \gamma_{\mathrm{C}} l_{\mathrm{c}} q_{\mathrm{c}} \cdot 10^{-3}. \tag{11} \]

Fig. 1.

Equations of the heat balance of the coil per linear centimeter of the mean turn of the windings

\[ \begin{aligned} x_1 k_{\Pi 1} + y_1 &= \alpha_1 \Delta_1^2 x_1 y_1 k_1 \\ x_2 k_{\Pi 2} + y_2 &= \alpha_2 \Delta_2^2 x_2 y_2 k_2 \end{aligned} \tag{12} \]

where

\[ k_{\Pi 1}=\frac{k_{x_1}}{k_{y_1}};\quad k_{\Pi 2}=\frac{k_{x_2}}{k_{y_2}};\quad \alpha_1=\frac{\rho_{\mathrm{M}}}{2\sigma_1 k_{y_1}};\quad \alpha_2=\frac{\rho_{\mathrm{M}}}{2\sigma_2 k_{y_2}}. \tag{13} \]

Transformer power

\[ S = K\Delta_1 q_{\mathrm{M}_1} q_c = K\Delta_2 q_{\mathrm{M}_2} q_c, \tag{14} \]

where

\[ K = 4.44 f B n_{\mathrm{э}}\cdot 10^{-7}. \tag{15} \]

Reactive component of the short-circuit voltage, in relative units

\[ u_p=\frac{6.8\Delta_2 q_{\mathrm{M}_2} d_{12} a_p}{k_c B h d^2}, \tag{16} \]

where

\[ d_{12}=d+2x_2+2\delta_{02}+\delta_{12}; \tag{17} \]

\[ a_{\mathrm{p}}=\frac{x_1+x_2+3\delta_{12}}{3}. \tag{18} \]

Equality of the magnetizing forces of the windings

\[ \frac{k_1 x_1 y_1 \Delta_1}{y_1+\delta_1} = \frac{k_2 x_2 y_2 \Delta_2}{y_2+\delta_2}. \tag{19} \]

Since the number of layers of the LV winding is an integer, one may use the equation for the magnetizing force of a turn

\[ I_{\mathrm{n}2} W_{k_2}=k_2 x_2 y_2 \Delta_2,\quad \text{where } W_{k_2}=\frac{1}{n_2}. \tag{20} \]

Thus, there are six basic equations (12; 14; 16; 19 and 20), connecting nine unknowns \((W_{k_2};\ x_1;\ x_2;\ y_1;\ y_2;\ \Delta_1;\ \Delta_2;\ d;\ h)\). It is convenient to carry out the investigation of the costs (1) by taking as independent variables the dimension \(y_2\), the current density \(\Delta_1\), and the number of turns in the coil of the LV winding.

As a result of the simultaneous solution of the indicated system of equations, for determining the dimensions of the transformer we obtain the following formulas:

\[ M=\frac{k_{\Pi_1} m_1 i_1}{I_{\mathrm{n}1}}-a_1\Delta_1; \tag{21} \]

\[ y_1= \frac{ \left(\dfrac{k_{\Pi_1}}{\Delta_1}+\delta_1 M\right) + \sqrt{ \left(\dfrac{k_{\Pi_1}}{\Delta_1}+\delta_1 M\right)^2 + \dfrac{4k_{\Pi_1}\delta_1}{\Delta_1} \left(-M-\dfrac{y_2+\delta_2}{I_{\mathrm{n}2}W_{k_2}}\right) } }{ 2\left(-M-\dfrac{y_2+\delta_2}{I_{\mathrm{n}2}W_{k_2}}\right) }; \tag{22} \]

\[ x_1= \left[ \frac{a_1\Delta_1 I_{\mathrm{n}2}W_{k_2}(y_1+\delta_1)}{y_2+\delta_2} -y_1 \right]\frac{1}{k_{\Pi_1}}; \tag{23} \]

\[ N=\frac{y_2}{k_{\Pi_2}W_{k_2}}+m_2 i_2; \tag{24} \]

\[ \Delta_2= \frac{ N+\sqrt{N^2+\dfrac{4a_2 I_{\mathrm{n}2}^2}{k_{\Pi_2}y_2}} }{ \dfrac{2a_2 I_{\mathrm{n}2}}{k_{\Pi_2}} }; \tag{25} \]

\[ x_2=\frac{a_2\Delta_2 I_{\mathrm{n}2}W_{k_2}-y_2}{k_{\Pi_2}}; \tag{26} \]

\[ a'_{\mathrm{p}}=x_1+x_2+3\delta_{12}; \tag{27} \]

\[ v=2x_2+2\delta_{02}+\delta_{12}; \tag{28} \]

\[ d= \frac{ a'_{\mathrm{p}} - \sqrt{ (a'_{\mathrm{p}})^2 + \dfrac{4Bk_c}{2.26 I_{\mathrm{n}2}W_{k_2}}\, u_{\mathrm{p}} - (y_2+\delta_2)a'_{\mathrm{p}}v } }{ 2\dfrac{Bk_c}{2.26 I_{\mathrm{n}2}W_{k_2}}\,u_{\mathrm{p}}-(y_2+\delta_2) }. \tag{29} \]

\[ k=\frac{S(y_2+\delta_2)}{0.17436\cdot 10^{-8} I_{\mathrm{n}2} W_{k2} k_C B d^2 n_3}. \tag{30} \]

For carrying out the studies, the following were taken:

\[ \begin{gathered} k_{x1}=k_{x2}=0.7;\qquad k_{y1}=k_{y2}=0.9;\qquad \sigma_1=\sigma_2=0.15;\qquad k_M=k_H=1;\\ \rho=2.14\cdot 10^{-6};\qquad \gamma_C=7.65;\qquad \gamma_M=8.9;\qquad \delta_M=0.24\cdot 10^{-11};\qquad \alpha_C=0.6\cdot 10^{-11};\\ P_H=0.125;\qquad P_d=0.06;\qquad A=0.1295;\qquad x_p=0.0193;\qquad k_C=0.8;\\ D=0.159\cdot 10^{-9};\qquad i_2=0.055;\qquad k_{\mathrm{я}}=1.05;\qquad \delta_2=0.655;\qquad \delta_{02}=1.5;\\ \varepsilon=0.588\cdot 10^{[[unclear: exponent]]};\qquad u_p=0.105;\qquad k_T=0.7; \end{gathered} \]

for 110 kV:

\[ i_1=0.14;\qquad \delta_1=0.94;\qquad \delta_{12}=5;\qquad \delta_{10}=9;\qquad \delta_{11}=6;\qquad \beta=2.8; \]

for 220 kV:

\[ i_1=0.2;\qquad \delta_1=1.4;\qquad \delta_{12}=11;\qquad \delta_{10}=12;\qquad \delta_{11}=14;\qquad \beta=3; \]

for transformers with a core-type magnetic circuit,

\[ n_C=2;\qquad n_3=2;\qquad n_d=3.6; \]

Fig. 2.

for transformers with a shell-type core, \(n_{\mathrm{c}} = 2.05;\; n_{\sigma} = 1;\; n_{\mathrm{d}} = 1.8\); the remaining data are given in Table 1.

Fig. 3.

The investigations were carried out for varying $\Delta_1$ with a step of $2.5\ \mathrm{A/cm^2}$ and $y_1$ with a step of $0.05\ \mathrm{cm}$. Some results of the investigation are given in Figs. 2–7.

Figure 2 shows the dependence of costs (1) on the value of $y_1$ at a constant current density $\Delta_1$, for transformers with a core-type (curves 1, 2, and 3) and shell-type (curves 4, 5, and 6) core, with ratings of 40, 60, and 80 MVA. The nature of the change in costs depends almost not at all on the form of the core. The minimum costs for a transformer with a core-type core are somewhat lower, and the difference in costs increases with increasing transformer rating. It should be noted that the value of $y_1$ at the minimum point for transformers with a core-type core is greater than the standard value ($2.2\ \mathrm{cm}$). Reducing $y_1$ to the standard value leads to an increase in costs up to the level of those for a transformer with a shell-type core.

Fig. 4.

Figures 3 and 4 give the variation of the winding height $(h)$ and the limb diameter $(d)$ as functions of the dimension $y_1$.

One of the principal requirements imposed on a transformer is the mechanical strength of the windings under short-circuit conditions.

For calculating the mechanical stresses arising in the windings from the action of radial forces under short-circuit conditions, one may use the expressions [2, 3]:

\[ \sigma_{\mathrm{p}}=\frac{C\Delta^{2}q_{\mathrm{M}}d_{12}}{h u_{\mathrm{p}}^{2}}, \tag{31} \]

where

\[ C=19.7\cdot 10^{-8}. \tag{32} \]

Figure 5 shows the result of calculating \(\sigma_{\mathrm{p}}\) for various values of \(y_1\) at the constant current density \(\Delta_1\) corresponding to the point of minimum costs; the designations of the curves in Figs. 3, 4, and 5 are the same as in Fig. 2. For an 80 MVA transformer with a core-type magnetic circuit, Fig. 6 gives the result of studying the minimum costs for different coefficients of filling the core cross-sectional area with steel \((k_{\mathrm{c}})\), the induction in the core \((B)\), and the number of starts of the helical winding

\[ \left(\frac{1}{W_{\mathrm{k}}^{2}}\right). \]

Fig. 5.

Of interest is the existence of a minimum of costs as a function of the number of starts of the helical winding \(\left(\frac{1}{W_{\mathrm{k}}^{2}}\right)\). It is seen from the figure that when the number of starts is varied from 1 to 2, the costs change only slightly; however, it should be noted here that at a smaller value of \(\left(\frac{1}{W_{\mathrm{k}}^{2}}\right)\), a larg-

smallest value of \(y_2\) at the point of minimum costs. For example, for the case considered, at \(\frac{1}{W_{k2}} = 2\), \(y_2 = 2.25\) cm, while at \(\frac{1}{W_{k2}} = 1\), \(y_2 = 3.9\) cm.

In the study of transformers with \(VН\) 220 kV, the dependences are analogous.

Table 2 gives the principal quantities at the point of minimum costs for transformers with core-type (I) and shell-type (II) cores, voltage \(VН\) 220 kV.

Table 2

Of particular interest is a techno-economic study of transformers with windings made of aluminum conductor. Below are given the results of such a study for the same ratings as for transformers with copper windings. It is assumed: \(\beta = 3.8\); \(\rho = 3.44 \cdot 10^{-6}\); \(\gamma_m = 2.7\); \(a = 0.1275 \cdot 10^{-5}\); \(\varepsilon = 0.312 \cdot 10^{-4}\); the quantities \(m_1\), \(m_2\), and \(n_2\) are given in Table 3.

Table 3

The character of the change in the costs and dimensions of transformers with aluminum windings is the same as for transformers with copper windings.

Figure 7 shows the dependence of \(Э_p\) on \(y_2\) at the optimum value of \(\Delta_2\), for transformers with \(VН\) 110 kV (curves 1, 2, and 3) and 220 kV

Fig. 6.

Table 4

(curves 3, 4, and 5) with a core of the core-type construction and ratings of 40, 60, and 80 MVA.

Table 4 gives the principal quantities of the transformers at the point of least cost for HV 220 kV. In column 1 the data are for transformers with a core-type core, and in column 11 for those with a shell-type core.

Fig. 7.

REFERENCES

1. I. D. Kutyavin, “On the determination of the optimal dimensions of three-phase two-winding transformers,” Vol. 130, Izvestiya TPI, 1964.

2. P. M. Tikhomirov, Calculation of a Transformer. Gosenergoizdat, 1962.

3. G. N. Petrov, Transformers. ONTI, Energoizdat, 1934.

4. I. D. Kutyavin, G. V. Del, V. P. Krasnov, “On the technical-economic determination of the dimensions of single-phase two-winding transformers,” Vol. 130, Izvestiya TPI, 1964.