TECHNICAL-ECONOMIC SELECTION OF THE OPTIMAL GEOMETRY OF MINE TRANSFORMERS

V. A. ZORIN

(Presented by the scientific seminar of the Department of Electric Stations)

One of the most important directions in improving the electric power supply of mining sections of coal mines at the present time is the introduction of mobile transformer substations and transformers in explosion-proof design instead of stationary transformer substations with oil transformers of the TMSh type, which leads to a further increase in labor productivity and an improvement in the technical-economic characteristics of mine operation.

Experience in the use of domestic explosion-proof substations of the TSShVP type shows that, in this case, the annual savings for a medium-power section are no less than 10–15 thousand rubles solely due to the elimination of fire-hazardous chambers [1].

However, what has been said above does not remove from the agenda the question of a technical-economic approach to the selection of the geometric dimensions of transformers for mine mobile explosion-proof substations, the volume of production of which is increasing from year to year.

Domestic industry produces two types of such substations: TKShVP—with filling by quartz sand and insulation of class \(E\); TSShVP—with a dry-type air transformer and silicon-organic insulation of class \(H\). In the present article an attempt is made to apply the method of technical-economic selection of the optimal geometry of power transformers of the 1st and 2nd frame sizes [2] to transformers of substations of the TSShVP type*).

The formula for calculating the total annual calculated costs [2]:

\[ \begin{aligned} Z &= (A + DB^2 + Z_{\mathrm{p}}\alpha_{\mathrm{st}})Q_{\mathrm{st}} + \left[A + D\left(\frac{B}{K_{\mathrm{y}}}\right)^2 + Z_{\mathrm{p}}\alpha_{\mathrm{ya}}\right]Q_{\mathrm{ya}} \\ &\quad + (\beta A k_{\mathrm{p}} + E\Delta_1^2)Q_{m1} + (\beta A + E\Delta_2^2)Q_{m2} = Z_{\mathrm{st}} + Z_{\mathrm{ya}} + Z_{m1} + Z_{m2}, \end{aligned} \tag{1} \]

where

\[ Z_{\mathrm{st}} \text{ and } Z_{\mathrm{ya}} \]

are the costs caused by the presence of steel in the limbs and yoke of the transformer;

\[ Z_{m1} \text{ and } Z_{m2} \]

are the costs caused by the presence of copper in the HV and LV windings of the transformer.

By analogy with oil transformers, it may be concluded that a transformer will be designed rationally if the following are found

*) The work is being carried out under the supervision of Doctor of Technical Sciences I. D. Kutyavin.

such a set of values of the variables, from variant to variant, of the quantities \(x_1, x_2, y_1, y_2, d, h\) (Fig. 1), \(\Delta_1, \Delta_2, w_k\), for which the total calculated costs (1) are minimal.

Fig. 1

a—sketch of the axial section of a phase; b—sketch of the transformer core section.

In the thermal calculation of the transformers under consideration, unlike oil-filled ones, the mutual thermal influence of heated bodies is taken into account by including, among the “physical constraints,” the equations of the heat balances of the windings and the magnetic core.

The method of thermal calculation developed at the Moscow Electrical Plant for dry-type transformers with natural air cooling and class \(B\) insulation [3], which was supplemented and refined in the design of transformers for the TSShVP at the Zaporozhye Transformer Plant, gives, for the densities of heat fluxes from the surfaces of the windings and the magnetic core, very cumbersome multinomial expressions as functions of the variables. In addition, the literature lacks the necessary recommendations for choosing permissible heat-flux densities. Therefore G. B. Fridman’s conclusion becomes understandable that “... in calculations for selecting the basic dimensions of such a (mine air) transformer, it is difficult to introduce constraints associated with heating of the active parts” [3].

In order to obtain the simplest possible calculation expressions for the variables, as functions of which all parameters and costs are found in one way or another, we introduce the calculated heat-flux densities for the windings and the magnetic core \(\sigma_1, \sigma_2, \sigma_0\).

HV winding (continuous helical coil winding):
In the coil, the mean conductor is considered; we neglect its heat transfer in the radial directions. The effective cooling surface and losses for this conductor*) are:

\[ S_1=\pi D_{\mathrm{cp}}(x_1+i_1)2\beta_{1\mathrm{k}}k_{\mathrm{П}1}, \tag{2} \]

\[ P_{m1}=I_{\mathrm{в}1}^{2}r_{\mathrm{в}}=\rho\pi D_{\mathrm{cp}}l_{\mathrm{в}1}\Delta_1 . \tag{3} \]

The expression for the coefficient of heat-transfer efficiency by convection in horizontal ducts \(\beta_{1\mathrm{k}}\) [4], without appreciable effect on the accuracy of the calculation, is approximated by a straight line:

\[ \beta_{1\mathrm{k}}=1.64\,\frac{\delta_1}{b_1}, \tag{4} \]

where

\[ b_1=(x_1+i_1)m_1 w_{\mathrm{k}}. \tag{5} \]

Then the calculated heat-flux density is:

\[ \sigma_1=\frac{P_{m1}}{S_1} = \frac{\Delta_1\rho l_1 w_{\mathrm{k}}}{3.28\delta_1 k_{\mathrm{П}1}}, \quad \frac{\mathrm{W}}{\mathrm{cm}^2}. \tag{6} \]

Cross section of the active copper of the coil:

\[ q'_m=x_1y_1m_1w_{\mathrm{k}}, \quad \mathrm{cm}^2. \tag{7} \]

Cross-sectional area of the coil:

\[ q_m=(x_1+i_1)(y_1+i_1)m_1w_{\mathrm{k}}, \quad \mathrm{cm}^2. \tag{8} \]

Coefficient of filling of the coil cross-sectional area with copper:

\[ K_1=\frac{q'_m}{q_m} = \frac{l_{\mathrm{в}1}}{\Delta_1(x_1+i_1)(y_1+i_1)}. \tag{9} \]

LV winding (layer winding):
We consider the layer of greatest diameter \(D\). The coefficient of filling of the layer area with copper, \(K\), as for oil transformers [2], is found from the relation:

\[ K_2= \frac{l_{\mathrm{в}2}} {\Delta_2(x_2+i_2)(y_2+i_2)\kappa''}. \tag{10} \]

The number of turns in the layer, taking (10) into account:

\[ w_{c2} = \frac{(x_2+i_2)m_2hK_2}{q_{w2}} = \frac{m_2l_{\mathrm{в}2}h} {q_{w2}\Delta_2(y_2+i_2)\kappa''}. \tag{11} \]

The effective cooling surface and losses for the layer:

\[ S'_2=p\pi DhK_{\mathrm{П}2}\beta_{2-1}, \quad \mathrm{cm}^2, \tag{12} \]

\[ P_{m2}=I_2^{2}r_c=\Delta_2^{2}q_{w2}\rho\pi D w_{c2}, \quad \mathrm{kW}. \tag{13} \]

The coefficient of heat-transfer efficiency by convection in vertical ducts \(\beta_{2-1}\), according to [4]:

\[ \beta_{2-1} = 0.56\sqrt[4]{\frac{(10i_{2-1})^{1.6}}{10h}}. \tag{14} \]

Then the calculated heat-flux density \(\sigma_2\) is:

\[ \sigma_2 = \frac{P'_{m2}}{S'_2} = \frac{\rho l_{\mathrm{в}2}m_2\Delta_2\sqrt[4]{10h}} {(y_2+i_2)\,0.56h\sqrt[4]{(10i_{2-1})^{1.6}}\,p\,\varepsilon_{\mathrm{П}}\kappa''}, \quad \frac{\mathrm{W}}{\mathrm{cm}^2}. \tag{15} \]

*) The notation is given in [3] and partially at the end of the article.

Formula (15) is valid for \(p=1\). If \(p=2\), then instead of \(\delta_{2-1}\) in (15) one should substitute \(\delta'_{2-1}\):

\[ \delta'_{2-1}=\frac{\delta_2+\delta_{2-1}}{2},\ \text{cm}. \tag{16} \]

Magnetic core:

We consider only the limbs over the window height. Losses in the limbs:

\[ P_{\text{st}}=3\lambda_c B^2\gamma_c\frac{\pi d^2 k_c}{4}(h+2l_u),\ \text{kW}. \tag{17} \]

Effective cooling surface of the limbs:

\[ S_{\text{st}}=S'_{0-1}\beta'_{0-2}+S'_{0-2}\beta_0,\ \text{cm}^2. \tag{18} \]

where

\[ \beta'_{0-2}=0.56\sqrt[4]{\frac{(5\delta_{0-2})^{1.6}}{10(h+2l_u)}} ; \tag{19} \]

\[ \beta_0=0.56\sqrt[4]{\frac{(10\delta_0)^{1.6}}{10(h+2l_u)}} ; \tag{20} \]

\[ S'_{0-1}=6(h+2l_u)(e'+e'')\,d k_{\text{po}},\ \text{cm}^2; \tag{21} \]

\[ S'_{0-2}=6(h+2l_u)n_0\,d e',\ \text{cm}^2. \tag{22} \]

The calculated heat-flux density \(\sigma_0\):

\[ \sigma_0=\frac{P_{\text{st}}}{S_{\text{st}}} = \frac{ 0.7\alpha_c\gamma_c k_c B^2 d\sqrt[4]{10(h+2l_u)} }{ \left[(e'+e'')k_{\text{po}}\sqrt[4]{(5\delta_{0-2})^{1.6}} +e'n_0\sqrt[4]{(10\delta_0)^{1.6}}\right] }, \ \frac{\text{W}}{\text{cm}^2}. \tag{23} \]

For the copper cross sections of the HV and LV windings, taking (9) and (10) into account, we shall have the expressions:

\[ q_{m1}=\frac{I_1 w_k h}{\Delta_1 k_p\left(y_1+i_1+\delta_1 k'\right)},\ \text{cm}^2; \tag{24} \]

\[ q_{m2}=\frac{m'_2 n_2 I_{\text{v}1} h}{\Delta_2 (y_2+i_2)k''},\ \text{cm}^2. \tag{25} \]

In addition to the formulas obtained for the conditional heat-flux densities (6), (15), (23), the system of equations of “physical constraints” includes the following relations, common also for oil transformers [2], which for dry mine transformers may be written, taking (24) and (25) into account, in the following form:

\[ \left\{ \begin{aligned} \frac{I_{\text{v}1}m_1 w_k}{(y_1+i_1+\delta_1 k')k_p} &= \frac{I_{\text{v}2}n_2 m'_2}{(y_2+i_2)k''}; \tag{26} \\[6pt] \Delta_1&=\frac{I_{\text{v}1}}{x_1y_1}; \tag{27} \\[6pt] \Delta_2&=\frac{I_{\text{v}2}}{x_2y_2}; \tag{28} \\[6pt] S&=K\frac{I_{\text{v}2}n_2m'_2Bhq_c}{(y_2+i_2)k''}; \tag{29} \\[6pt] u_p&=\frac{2\pi^2 k_r\Delta_2 q_{m2}l_m(b_1+b_2+3\delta_{12})}{33.3\,B h q_c}. \tag{30} \end{aligned} \right. \]

If the values of the induction in the leg, the conventional heat-flux densities, and the numbers of parallel conductors and layers are regarded as specified, the indicated system of equations contains nine unknown quantities, one of which must be taken as independent.

In principle, any of the variables may be taken as independent; however, the simplest and most convenient calculation formulas are obtained when the winding height \(h\) is used as the independent variable.

In this case, directly from (23) we obtain, for the leg diameter:

\[ d=\frac{\alpha_0}{B^2\sqrt[4]{10(h+2l_u)}},\ \text{cm}, \tag{31} \]

\[ \alpha_0= \frac{\sigma_0\left[(e'+e'')k_{\mathrm{p}0}\sqrt[4]{(5\delta_0-2)^{1.6}} +n_0e'\sqrt[4]{(10\delta_0)^{1.6}}\right]} {0.7a_c\gamma_c k_c}. \tag{32} \]

From (29) we find \(y_2\):

\[ y_2=\frac{BKI_{\mathrm{v}2}n_2m'_2hq_c}{Sk''}-i_2,\ \text{cm}. \tag{33} \]

Substitute (33) into (15) and solve the resulting expression with respect to \(\Delta_2\):

\[ \Delta_2= \frac{BKI_{\mathrm{v}2}n_2m'_2a_2\sqrt[4]{h^3}\,q_c} {\sqrt[4]{10}\,Sk''},\ \frac{\text{A}}{\text{cm}^2}, \tag{34} \]

\[ a_2= \frac{\sigma_2 k''\,p k_{\mathrm{p}2}\,0.56\sqrt[4]{(10\delta_2-1)^{1.6}}} {\rho m'_2 I_{\mathrm{v}2}}. \tag{35} \]

Taking into account the obtained relations for \(y_2\) (33) and \(\Delta_2\) (34), from (28) we determine \(x_2\):

\[ x_2=\frac{I_{\mathrm{v}2}}{\Delta_2y_2},\ \text{cm}. \tag{36} \]

From (6), for the number of turns in a coil \(w_k\):

\[ w_k=\frac{\alpha_1}{\Delta_1}, \tag{37} \]

\[ \alpha_1=\frac{3.28\,\sigma_1 k_{\mathrm{p}1}\delta_1}{I_1\rho}. \tag{38} \]

Substitute (33) and (38) into (26) and find \(y_1\):

\[ y_1= \frac{\alpha_1 I_{\mathrm{v}1}m_1KBhq_c}{S\Delta_1k_p} -\left(i_1+\delta_1k'\right),\ \text{cm}. \tag{39} \]

Then from (27) \(x_1\) is determined:

\[ x_1=\frac{I_{\mathrm{v}1}}{\Delta_1y_1},\ \text{cm}. \tag{40} \]

From the remaining “physical constraint” (30) it is necessary to obtain a calculation expression for \(\Delta_1\). For this purpose, into (30) we substitute (34), (25), (33), and obtain the equation:

\[ \frac{33.3\,B^2q_c^2hK}{2\pi^3k_rS} = \left(d+2a_{02}+2b_2+\delta_{12}\right) \left(b_1+b_2+3\delta_{12}\right). \tag{41} \]

For simplicity, let us denote the left-hand side of the resulting equation by \(R\), and bear in mind that

\[ b_2=(x_2+i_2)m'_2 n_2+n'_k\delta'_2+n_{k2}\delta_2, \tag{42} \]

then, taking into account (5), (37), (40), we derive the design formula for \(\Delta_1\):

\[ \Delta_1=\frac{-V-\sqrt{V^2-4UP}}{2U},\quad \frac{a}{cM^2}. \tag{43} \]

Here

\[ V=m_1\alpha_1\left[ \frac{I_{B1}KBhq_c}{i_1+\delta_1\kappa'} \left(\delta_2+3\delta_{12} -\frac{R}{d+2a_{02}+2b_2+\delta_{12}}\right) -i_1Sk_p \right]; \tag{44} \]

\[ U=Sk_p\left[ \frac{R}{d+2b_2+2a_{02}+\delta_{12}} -(b_2+3\delta_{12}) \right]; \tag{45} \]

\[ P=\frac{I_{B1}\alpha_1m_1\left(Sk_p+m_1\alpha_1 i_1 KBhq_c\right)} {i_1+\delta_1\kappa'}. \tag{46} \]

Thus, the eight variables are now represented as a functional dependence on the winding height. Using the relations given in [2] for calculating the weights \(Q_{\text{st}}, Q_{\text{y}}, Q_{m1}\), and \(Q_{m2}\) through the variables, the formula for the calculated costs (1) will likewise be a function only of the winding height \(h\).

The optimal value of \(h\) corresponds to the minimum value of the calculated costs. Because of its complexity and the presence of a large number of auxiliary equations, minimization of the calculated-cost formula (1) is most expediently performed by a numerical method with the aid of a digital computer. Having determined the optimal value of \(h\) and carried out the calculation according to the formulas given above, we find the optimal values for the remaining variables, the necessary dimensions, weights, etc.

Conclusions

1. The method developed in [2] for techno-economic studies of oil-filled transformers of frames 1 and 2 can also be used for the study of mine dry-type transformers. In this case the system of “physical constraints” must be supplemented by an equation for the heat-flux density from the surface of the magnetic core.

2. Because of the great complexity of the thermal calculation, the corresponding calculated quantities \(\jmath_1, \jmath_2, \sigma_0\) are introduced into the system of “physical constraints” instead of the actual densities of the heat fluxes from the surfaces of the windings and the magnetic core.

3. The set of initial data in the studies must include specific values of \(\jmath_1, \sigma_2, \sigma_0\). In order to determine the optimal combinations of \(\jmath_1, \sigma_2, \sigma_0\) for each transformer, an auxiliary study must be carried out.

4. As for oil-filled transformers, all design formulas for costs, weights, dimensions, losses, etc., are expressed as functions of a single independent variable, which it is expedient to take as the winding height \(h\).

5. In the studies, all the most important technical and operational requirements are automatically satisfied, since they are incorporated into the mathematical apparatus as “physical constraints.”

The results of techno-economic studies of mine dry-type transformers by the present method will be presented in subsequent publications.

Symbols

\(\kappa_{\mathrm{po}}\) — coefficient accounting for the covering of part of the cooling surface of the magnetic core by insulating parts.

\(D_{\mathrm{avg}}\) — average diameter of the HV winding, cm.

\(r_{\mathrm{v}}\) — resistance of one parallel conductor of the HV winding, cm.

\(m_{2}^{1}\) — number of parallel conductors of the LV winding layer in the radial direction.

\(q_{w2}, r_{\mathrm{c}}\) — cross section and resistance of a turn of the LV winding, \(\text{cm}^{2}\), \(\Omega \cdot \text{cm}\).

\(p\) — number of cooling surfaces of the LV winding layer.

\(n_{k2}, n_{k}'\) — numbers of ducts and interlayer spacers of the LV winding.

\(n_{0}\) — number of cooling ducts in the core.

\(\delta_{0-2}\) — calculated dimension of the cooling duct between the core and the LV winding, cm.

References

1. V. A. Khorunzhii, E. N. Zikeev. Electrical equipment for mines—to series production. Bulletin of the Electrical Industry, No. 3, 1963.

2. I. D. Kutyavin, V. A. Zorin. Technical and economic substantiation of the selection of optimal dimensions of low-power power transformers. Reliability and Economy of Power Supply at Petrochemical Plants, issue 1, Omsk, 1968.

3. Yu. I. Gerasimov, G. B. Fridman. Mine explosion-proof transformer substations. Gosenergoizdat, 1963.

4. G. B. Fridman. Thermal calculation of mine transformers. Bulletin of the Electrical Industry, No. 3, 1963.