Investigation of a 100 MVA electric furnace transformer with a roll-type low-voltage winding
Abstract Generated abstract
This paper analyzes the optimization of a 100 MVA, 115/0.75 kV electric furnace transformer using an external two-layer foil low-voltage winding and an internal multilayer high-voltage winding with voltage regulation by switching winding layers. Using previously derived design relations, the authors calculate winding dimensions, conductor parameters, current densities, steel and copper weights, losses, energy consumption, and estimated costs for low-voltage windings with two to six turns. The comparison indicates that the most economical design among the variants considered is a three-turn rolled low-voltage winding, with calculated costs of 60.4 thousand rubles. The paper also examines regulation behavior for a four-turn variant, showing that switching high-voltage layers changes the short-circuit voltage from 26.4 percent to 17.2 percent.
Full Text
STUDY OF A 100 MVA ELECTRIC FURNACE TRANSFORMER WITH A FOIL LOW-VOLTAGE WINDING
I. D. Kutyavin, G. A. Starovoitova
(Presented by the scientific seminar of the Department of Electric Power Stations)
The article considers questions of optimizing the dimensions and parameters of powerful furnace transformers with a foil low-voltage winding (LV) and a multilayer high-voltage winding (HV). The transformer power is 100 MVA, the voltages are 115/0.75 kV, the winding connection is \(Y/\Delta\), and regulation of the voltage of the LV winding is carried out by changing the number of winding layers within the range of 100–200%.
The LV winding is located on the outside and is made of foil copper in the form of a spiral in two layers (\(p_1 = 2\)). The width of the strip is equal to the winding height \(h\). The interturn insulation is made with insulating spacers, creating axial cooling ducts of width \(\delta_1 = 0.8\) cm. The number of layers of this winding \(m_1\) is equal to the number of turns \(N_1\).
The HV winding is located inside the LV winding and is made as a multilayer winding of a composite conductor consisting of \(p_2\) parallel elementary rectangular conductors with enamel insulation of thickness on two sides equal to \(i_e = 0.01\). On a bundle of \(p_2\) conductors an additional paper insulation is applied, with thickness on two sides equal to \(i_2 = 0.2\). The axial dimension of the copper of an elementary conductor is \(y_2\), and the radial dimension is \(z_2\). Regulation of the secondary voltage is carried out by changing the number of switched-on layers of the HV winding from the side of the leakage channel. In connection with this, the width of the leakage channel changes, and the reactive component of the short-circuit voltage (s.c.v.) \(U_p\) also changes (Table 2).
For numerical optimization of the dimensions and parameters of the transformer under consideration, the basic relations obtained in [1, 2] were used, and the optimization results are given in Table 1.
Initial data (dimensions in centimeters):
\[ a_{02}=5;\quad a_{21}=5;\quad \delta_{11}=6;\quad a_{09}=8; \]
\[ i_1=0.01;\quad i_2=0.2;\quad \delta_1=0.8;\quad \delta_2=2.4;\quad K_c=0.85;\quad K_p=0.95;\quad K_t=1.3;\quad K_m=0.9; \]
\[ K_n=0.8;\quad K_{dc}=1.093;\quad K_{d1}=1.0833;\quad \rho=2.14\cdot 10^{-6}\ \Omega\text{m}/\text{cm};\quad \gamma_m=8.9\ \frac{g}{cm^3}; \]
\[ \gamma_c=7.65\ \frac{g}{cm^3};\quad B_c=1.65\ \text{T};\quad \alpha_c=1.46;\quad p_n=0.125; \]
\[ p_a=0.06;\quad Z_{uk}=212\ \text{rub}/\text{kW};\quad Z'_{ue}=6.6\ \text{rub}/\text{kW}\cdot\text{h};\quad Z''_{ue}=7.3\ \text{rub}/\text{kW}\cdot\text{h}; \]
\[ q_1=q_2=0.3\ \text{W}/\text{cm}^2;\quad \alpha_1=\alpha_2=22.4\cdot 10^4;\quad T_{\max}=6000\ \text{h};\quad \tau=4600\ \text{h}; \]
\[ t=8760\ \text{h};\quad \beta=3.5 \]
Table 1
| Item No. | Number of layers of the LV winding | $m_1=N_1$ | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| 1 | $x_{10}=\sqrt{\dfrac{1.73}{C N_1}}$ | cm | 0.970 | 0.791 | 0.685 | 0.613 | 0.559 |
| 2 | $d=\sqrt{\dfrac{4Q_c}{N_1}}$ | cm | 125 | 100 | 90 | 80 | 71 |
| 3 | $Q_{c\Delta}=K_c\dfrac{\pi d^2}{4}$ | cm$^2$ | 10450 | 6680 | 5410 | 4270 | 3370 |
| 4 | $\Delta_1=\dfrac{454}{\sqrt{x_{10}}}$ | A/cm$^2$ | 461 | 510 | 549 | 580 | 607 |
| 5 | $h=\dfrac{I_{\mathrm{LV}}}{x_{10}\Delta_1}$ | cm | 100 | 110 | 118 | 125 | 130 |
| 6 | $\varphi_1=\varphi_2=\varphi=N_1x_{10}\Delta_1$ | A/cm | 895 | 1210 | 1505 | 1780 | 2040 |
| 7 | $z_2=V\cdot i_z$ | cm | 0.247 | 0.216 | 0.195 | 0.182 | 0.171 |
| 8 | $p_2=\sqrt{\dfrac{2V-1}{3}}$ | pcs. | 4 | 4 | 4 | 3 | 3 |
| 9 | $x_{20}=p_2z_2$ | cm | 0.988 | 0.864 | 0.780 | 0.546 | 0.513 |
| 10 | $\Delta_2=\sqrt{\dfrac{a_2}{K_{d2}x_{20}}}$ | A/cm$^2$ | 471 | 504 | 530 | 634 | 650 |
| 11 | $y_2=\dfrac{I_{\mathrm{HV}2}}{x_{20}\Delta_2}$ | cm | 1.08 | 1.15 | 1.22 | 1.45 | 1.50 |
| 12 | $m_2=\dfrac{N_2(y_2+i_2)K_y}{h}$ | pcs. | 2 | 3 | 4 | 5 | 6 |
| 13 | $G_{\mathrm{M}1}=3\gamma_{\mathrm{M}}\cdot10^{-6}\cdot\pi Q_{\mathrm{M}1}d_1$ | t | 2.77 | 3.45 | 4.38 | 5.15 | 5.90 |
| 14 | $G_{\mathrm{M}2}=3\gamma_{\mathrm{M}}\cdot10^{-6}\cdot\pi Q_{\mathrm{M}2}d_2$ | t | 2.05 | 2.60 | 3.17 | 3.00 | 3.37 |
| 15 | $G_{\mathrm{M}}=G_{\mathrm{M}1}+2G_{\mathrm{M}2}$ | t | 6.87 | 8.64 | 10.72 | 11.06 | 12.70 |
| 16 | $G_c=\gamma_c\cdot10^{-6}Q_cl_c$ | t | 109 | 71 | 59 | 43 | 35 |
| 17 | $G_{\mathrm{CY}}=4.9\gamma_c\cdot10^{-6}Q_cd$ | t | 39.2 | 27.5 | 24.0 | 20.0 | 16.5 |
| 18 | $G_a=G_c+G_{\mathrm{M}}$ | t | 115.9 | 79.7 | 69.6 | 54.0 | 47.7 |
| 19 | $G_p=G_c+\beta G_{\mathrm{M}}$ | t | 133.0 | 101.3 | 96.0 | 81.7 | 79.5 |
| 20 | $P_c=K_{dc}a_c(G_c+0.93G_{\mathrm{CY}})$ | kW | 237 | 158 | 133 | 101 | 83 |
| 21 | $P_{\mathrm{M}}=\dfrac{\rho}{\gamma_{\mathrm{M}}}(K_{d1}\Delta_1^2G_{\mathrm{M}1}+K_{d2}\Delta_2^2G_{\mathrm{M}2})$ | kW | 267 | 396 | 562 | 748 | 920 |
| 22 | $W=tP_c+K_{\mathrm{M}}\tau P_{\mathrm{M}}S_x^2$ | thousand kWh | 3090 | 3020 | 3490 | 3973 | 4525 |
| 23 | $Z_{ta}=K_t(p_n+p_a)G_p$ | thousand rubles | 32.0 | 24.4 | 23.0 | 19.7 | 19.1 |
| 24 | $Z_c=Z_{uk}p_n(P_c+K_{\mathrm{M}}P_{\mathrm{M}}S_x^2)$ | thousand rubles | 12.6 | 13.6 | 16.9 | 20.5 | 24.1 |
| 25 | $Z_p=Z'_{uz}P_ct+Z''_{uz}\tau P_{\mathrm{M}}S_x^2$ | thousand rubles | 22.7 | 22.4 | 26.6 | 30.9 | 35.7 |
| 26 | $Z_r=Z_{ta}+Z_c+Z_p$ | thousand rubles | 67.3 | 60.4 | 66.5 | 71.1 | 78.9 |
In calculating the data in the table, the number of turns of the LV winding $N_1$, equal to the number of layers of this winding $m_1$, which varied within the range 2–6, was taken as the independent variable. The radial thickness of the copper of the layer $x_{10}$ (optimal) was determined from expression (21) [1] for $y_1=h$ and $i_1=0$. From the known number of turns $N_1$ and the rated volta-
the LV winding, the core diameter \(d\) is found, which is rounded to a standardized value. After this, the refined cross-sectional area of the core steel \(Q_{\text{st}}\) is calculated. The current density \(\Delta_1\) was found from expression (1–28) [2]. Then the winding height \(h\), the specific magnetizing force \(\varphi_1=\varphi_2\) (for windings of equal height), and the number of turns of the HV winding, \(N_2\), were calculated.
To determine the dimensions of the elementary conductors of the HV winding, relation (26) [1] was used:
\[ 3 i_{\text{e}}(x_{20}+i_{\text{e}})^2=(z_2+i_{\text{e}})^2(2z_2-i_{\text{e}}), \tag{1} \]
where \(x_{20}\) is the optimum radial thickness of the copper of an HV-winding layer, consisting of \(p_2\) elementary conductors of radial thickness \(z_2\), so that
\[ x_{20}\simeq p_2z_2;\qquad 3p_2^2=\frac{2z_2-i_{\text{e}}}{i_{\text{e}}};\qquad K_{\text{d}2}=\frac{2z_2}{2z_2-i_{\text{e}}};\qquad i_{\text{e}}=0; \tag{2} \]
To determine \(x_{20}\), expression (21) [1] may be used
\[ x_{20}=\sqrt{\frac{1.73(y_2+i_2)}{c\,m_2y_2}}. \tag{3} \]
The heat balance of an HV-winding layer of height \(y_2\) per one running centimeter is
\[ 2K_{\text{p}}y_2q_2=\rho K_{\text{d}2}\Delta_2^2x_{20}y_2. \tag{4} \]
The current density in the HV winding is
\[ \Delta_2=\sqrt{\frac{\alpha_2}{K_{\text{d}2}x_{20}}};\qquad \alpha_2=\frac{2K_{\text{p}}q_2}{\rho}. \tag{5} \]
The specific magnetizing force of the HV winding, taking (5) into account, is
\[ \varphi_2=\frac{m_2x_{20}y_2\Delta_2}{y_2+i_2} = \frac{m_2y_2}{y_2+i_2}\sqrt{\frac{\alpha_2x_{20}}{K_{\text{d}2}}}. \tag{6} \]
From (3) and (6), let us determine the relation
\[ \frac{m_2y_2}{y_2+i_2} = \frac{\sqrt{3}}{cx_{20}^2} = \varphi_2\sqrt{\frac{K_{\text{d}2}}{\alpha_2x_{20}}}; \qquad x_{20}=\sqrt[3]{\frac{3\alpha_2}{K_{\text{d}2}c^2\varphi_2^2}}. \tag{7} \]
Let us square both sides of equality (7), take into account relations (2), and introduce the new variable \(V=\dfrac{z_2}{i_{\text{e}}}\). Then from (7) we obtain
\[ V^2(2V-1)^{1/4}=U= \frac{3}{c\,i_{\text{e}}\varphi_2} \sqrt{\frac{\sqrt{3}\,\alpha_2}{2i_{\text{e}}}}. \tag{8} \]
The figure plots the dependence \(U(V)\), which can be used to calculate \(z_2\) and \(p_2\). For this, \(U\) is found from the right-hand side of (8), \(V\) is determined from the curve in Fig. 1, and then the values \(z_2\), \(p_2\), \(x_{20}\), \(K_{\text{d}2}\), and \(\Delta_2\) can be calculated. The expressions for \(y_2\) and \(m_2\) are given in the table. But \(m_2\) is the number of layers of the unregulated part of the HV winding. If it is desired that the voltage of the LV winding be regulated twofold (from
from \(U_{\mathrm{H}1}\) to \(0.5\,U_{\mathrm{H}1}\)) at rated current, the number of layers of the HV winding must be \(2m_2=m_2^1\). In this case, the conductor cross section of the regulating layers may be reduced, taking into account the current flowing through them.
The radial widths of the winding sections are determined:
\[ a_1=N_1x_{10}+(N_1-1)\delta_1; \]
\[ a_2=m_2'x_{20}+(m_2'-1)\delta_2; \tag{9} \]
the mean diameters of the windings
\[ d_2=d+2a_{02}+a_2; \]
\[ d_1=d_2+a_2+2a_{21}+a_1; \tag{10} \]
the mean diameter of the leakage channel
\[ d_{12}=d+2a_{02}+2a_2+a_{21}; \tag{11} \]
the width of the core window
\[ a=2(a_{02}+a_2+a_{21}+a_1+0.5a_{11}); \tag{12} \]
the winding cross-sectional areas
\[ Q_{\mathrm{m}1}=x_{10}hN_1;\qquad Q_{\mathrm{m}2}=\frac{m_2'x_{20}y_2}{y_2+i_2}h. \tag{13} \]
Fig. 1. Calculated dependence \(u=f(v)\).
The formulas for determining the weights of copper and steel, the losses in them, and the calculated costs are given in Table 1. The most advantageous variant proved to be the one with a three-turn rolled LV winding. The calculated costs for this variant amount to 60.4 thousand rubles.
Table 2
| No. | Number of included layers of the HV winding during regulation | Symbol | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|
| 1 | Current in the HV winding | \(I_2\) | 500 | 400 | 333 | 286 | 250 |
| 2 | Number of turns in the HV winding | \(N_2\) | 354 | 442 | 530 | 620 | 708 |
| 3 | Number of rows of conductor in the regulating layers | \(p_2\) | 4 | 3 | 3 | 2 | 2 |
| 4 | Radial thickness of the copper of the layers | \(x_2\) | 0.780 | 0.585 | 0.585 | 0.390 | 0.390 |
| 5 | Mean diameter of the HV winding | \(d_2\) | 110.3 | 113.3 | 116.3 | 119.1 | 121.9 |
| 6 | Mean diameter of the LV winding | \(d_1\) | 158.9 | 158.9 | 158.9 | 158.9 | 158.9 |
| 7 | Mean diameter of the leakage channel | \(d_{21}\) | 137.2 | 140.2 | 143.2 | 146.0 | 148.8 |
| 8 | Width of the reduced leakage channel | \(a_p\) | 21.4 | 19.1 | 16.9 | 14.9 | 12.8 |
| 9 | Short-circuit voltage | \(U_p\) | 0.264 | 0.240 | 0.217 | 0.196 | 0.172 |
| 10 | Width of the leakage channel | \(a_{21}\) | 16.6 | 13.6 | 10.6 | 7.8 | 5.0 |
The variation of the current, the number of turns of the HV winding, and other quantities when regulating the voltage of the LV winding by switching the number of layers of the HV winding for the variant with a four-turn rolled winding is shown in Table 2. When the first four layers are connected into the HV circuit, the short-circuit voltage is 26.4%, and when all layers are connected it is 17.2%.
REFERENCES
-
I. D. Kutyavin. Design of transformer windings with minimal copper losses. Elektrotekhnika, 1969, No. 7.
-
I. D. Kutyavin et al. Some questions of optimizing the geometry and parameters of transformers and autotransformers. TPI, 1970.