On the Calculation of High-Power Power Transformers with a Roll-Type Low-Voltage Winding
Abstract Generated abstract
This article develops a calculation procedure for optimizing high-power three-phase two-winding transformers with a rolled low-voltage winding and a multilayer high-voltage winding. The design variable is the number of low-voltage winding layers, and the method combines heat-balance equations, leakage-field eddy-current loss constraints, optimal conductor dimensions, and techno-economic cost estimates. A worked calculation for a 250 MVA transformer compares several layer counts, including resulting winding dimensions, material masses, losses, annual energy costs, and total calculated costs. The example indicates that broad variation of the low-voltage layer number is impractical because of nontechnological transformer height and core diameter, and that the minimum calculated cost in the considered case occurs at five low-voltage winding layers.
Full Text
ON THE CALCULATION OF HIGH-POWER POWER TRANSFORMERS WITH A ROLLED LOW-VOLTAGE WINDING
I. D. Kutyavin, S. A. Stepanov
The article considers the problem of optimizing the dimensions and parameters of high-power three-phase two-winding transformers for a minimum of calculated costs, under the condition of a minimum of additional losses in the copper from eddy currents caused by the axial component of the leakage field. The problem is solved as applied to transformers of size V with a rolled low-voltage (LV) winding and a multilayer high-voltage (HV) winding. The LV winding is made of rolled copper or aluminum, with the strip width equal to the winding height \(h\). The turn insulation of the LV winding may be made by applying a polyimide layer to the side surfaces of the strip, or by means of a continuous interlayer between turns made of polymer film.
The number of layers of the LV winding \(m_1\) is taken as the optimized variable quantity.
The basic relations necessary for solving the problem are as follows.
Nominal currents in the windings
\[ I_{\mathrm{n}1}=z_1 h \Delta_1,\qquad I_{\mathrm{n}2}=n_2 z_2 y_2 \Delta_2, \tag{1} \]
where \(z_1, z_2\) are the radial copper thicknesses of the elementary conductors of the windings;
\(\Delta_1, \Delta_2\) are the current densities in the windings;
\(n_2\) is the number of elementary parallel conductors in the HV winding;
\(y_2\) is the axial height of the elementary conductor of the HV winding.
Magnetomotive forces per unit height of the windings
\[ \varphi_1=p_1 z_1 m_1 \Delta_1,\qquad \varphi_2=p_2 z_2 m_2 \Delta_2 \frac{y_2}{y_2+i_2}, \tag{2} \]
where \(p_1\) is the number of turns in a layer of the LV winding;
\(p_2\) is the number of rows of elementary conductor in the HV winding;
\(m_2\) is the number of layers of the HV winding;
\(i_2\) is the thickness of insulation on the two sides of the conductor of the HV winding.
\(\varphi_1=\varphi_2\) for windings of equal height.
Numbers of turns of the windings
\[ N_1=p_1 m_1,\qquad N_2=\frac{h m_2}{n_2(y_2+i_2)}. \tag{3} \]
The equations of the heat balance of the winding layer per one centimeter of the mean turn:
\[ 2 k_{\mathrm{p}1} q_1 h = \rho k_{\partial 1}\Delta_1^2 z_1 p_1 h; \qquad 2 k_{\mathrm{p}2} q_2 y_2 = \rho k_{\partial 2}\Delta_2^2 z_2 y_2 p_2, \tag{4} \]
where \(k_{\mathrm{p}1}, k_{\mathrm{p}2}\) are the coefficients of coverage of the winding surfaces by insulating structures;
\(q_1, q_2\) are the specific heat outputs of the winding surfaces;
\(\rho\) is the calculated specific resistivity of the winding material;
\(k_{\partial 1}, k_{\partial 2}\) are coefficients taking into account the additional losses in the copper of the windings from eddy currents.
From equations (4) one can find the current densities in the windings:
\[ \Delta_1=\sqrt{\frac{a_1}{k_{\partial 1}p_1 z_1}};\quad a_1=\frac{2k_{\mathrm{p}1}q_1}{\rho};\quad \Delta_2=\sqrt{\frac{a_2}{k_{\partial 2}p_2 z_2}};\quad a_2=\frac{2k_{\mathrm{p}2}q_2}{\rho}. \tag{5} \]
The optimal copper thicknesses of the winding layers [2]:
\[ x_{10}=p_1z_1=\sqrt{\frac{1.73}{c_1m_1}};\quad x_{20}=p_2z_2=\sqrt{\frac{1.73(y_2+i_2)}{c_1m_2y_2}};\quad c_1=\frac{\pi f \mu_0}{\rho}, \tag{6} \]
where \(f\) is the current frequency, \(\mu_0=4\pi\cdot10^{-9}\ \mathrm{H/cm}\).
In [1] the optimal dependence was obtained
\[ i_{\mathrm{e}}=\frac{(z_2+i_{\mathrm{e}})^2(2z_2-i_{\mathrm{e}})} {3(x_2+i_{\mathrm{e}})^2}, \tag{7} \]
which gives the relation between the variables \(p_2\) and \(z_2\). After some transformations we obtain
\[ 3p_2^2=3\frac{(x_2+i_{\mathrm{e}})^2}{(z_2+i_{\mathrm{e}})^2} =\frac{2z_2-i_{\mathrm{e}}}{i_{\mathrm{e}}};\qquad k_{\partial 2}=1+\frac{1}{3p_2^2}=\frac{2z_2}{2z_2-i_{\mathrm{e}}}, \tag{8} \]
where \(i_{\mathrm{e}}\) is the insulation on two sides of the elementary conductor of the HV winding.
From expressions (2) and (6), taking into account (5) and (8), we determine the ratio
\[ \frac{m_2y_2}{y_2+i_2} =\varphi \sqrt[4]{\frac{12i_{\mathrm{e}}}{a_2^2(2z_2-i_{\mathrm{e}})^2}} =\frac{3\sqrt{3}\cdot i_{\mathrm{e}}}{cz_2^2(2z_2-i_{\mathrm{e}})}. \tag{9} \]
After some simplification of (9) and the substitution \(v_2=\dfrac{z_2}{i_{\mathrm{e}}}\), we obtain
\[ \sqrt[4]{v_2^4(v_2-1)^{0.5}}=u_2 =\sqrt[4]{\frac{2za_2}{2\sqrt{3}\varphi^2c_1^2i_{\mathrm{e}}^3}}. \tag{10} \]
Expression (10) may be replaced by the design curve \(u_2=f(v_2)\), shown in Fig. 1. Then, determining \(u_2\) from the right-hand side of (10), from the design curve we find \(v_2\) and calculate \(z_2=v_2 i_{\mathrm{e}}\). In order to straighten the design curve, we extract the fourth root from the right- and left-hand sides of (10).
As an example, a calculation is given for a three-phase power transformer with a rating of \(250\ \mathrm{MVA}\), with voltages of \(15,\ 75/115\ \mathrm{kV}\), with a rolled LV winding and a multilayer HV winding.
Initial data for the calculation (dimensions in centimeters):
\(i_1=i_3=0.01\) — insulation thickness of the elementary conductors on two sides; \(i_2=0.2\) — additional insulation for the \(p_2\) conductors of the HV winding;
Fig. 1. Calculated dependence \(U=f(V)\).
Fig. 2. Element of transformer windings.
\(\delta_1=0.8\) and \(\delta_2=2.4\) — radial dimensions of the axial cooling ducts of the windings;
\(a_{12}=5,\ a_{01}=3,\ a_{0я}=8,\ a_{22}=6\) — main-insulation distances;
\(U_p=10.5\%\) — reactive component of the short-circuit voltage;
\(k_c=0.85\) — coefficient of filling the core cross-section with steel;
\(B_c=1.65\ \mathrm{T}\) — calculated induction in the core limb;
\(k_p=0.95\) — Rogowski coefficient;
| Calculation expression | 4 | 5 | 6 | 7 |
|---|---|---|---|---|
| $x_1(\mathrm{cm})$ | 0,695 | 0,621 | 0,567 | 0,526 |
| $\Delta_1\left(\dfrac{A}{\mathrm{cm}^2}\right)$ | 459 | 484 | 506 | 526 |
| $z_2(\mathrm{cm})$ | 0,193 | 0,181 | 0,169 | 0,161 |
| $x_2(\mathrm{cm})$ | 0,589 | 0,543 | 0,507 | 0,483 |
| $\Delta_2\left(\dfrac{A}{\mathrm{cm}^2}\right)$ | 492 | 506 | 525 | 540 |
| $y_2(\mathrm{cm})$ | 4,4 | 4,59 | 4,8 | 4,84 |
| $m_2$ | 5 | 6 | 7 | 8 |
| $m_1x_1+(m_1-1)\hat{\delta}_1=a_1(\mathrm{cm})$ | 5,18 | 6,3 | 7,4 | 8,48 |
| $m_2x_2+(m_2-1)\hat{\delta}_2=a_2(\mathrm{cm})$ | 12,5 | 15,3 | 18 | 20,7 |
| $\dfrac{\varphi a'_p}{2p}\left(1+\sqrt{1+\dfrac{4pa\partial}{\varphi a'_p}}\right)=d(\mathrm{cm})$ | 80 | 100 | 125 | 150 |
| $k_c\dfrac{\pi d^2}{4}=Q_c(\mathrm{dm}^2)$ | 42,6 | 66,7 | 104 | 150 |
| $\dfrac{U_{\mathrm{n}1}\cdot10^4}{222B_cQ_c}=N_1(\text{turns})$ | 100 | 65 | 42 | 28 |
| $\dfrac{U_{\mathrm{n}2}\cdot10^4}{222\sqrt{3}\,B_cQ_c}=N_2(\text{turns})$ | 476 | 274 | 177 | 121 |
| $\dfrac{I_{\mathrm{n}1}N_1}{\sqrt{3}\varphi_1}=h(\mathrm{cm})$ | 470 | 232 | 128 | 83,3 |
| $\dfrac{N_1}{m_1}=p_1$ | 25 | 13 | 7 | 4 |
| $\dfrac{x_1}{p_1}=z_1(\mathrm{cm})$ | 0,028 | 0,048 | 0,081 | 0,132 |
| $p_1m_1z_1h=Q_{\mathrm{M}1}(\mathrm{cm}^2)$ | 1310 | 717 | 436 | 306 |
| $h\dfrac{m_2x_2y_2}{y_2+i_2}=Q_{\mathrm{M}2}(\mathrm{cm}^2)$ | 1300 | 717 | 441 | 309 |
| $3\cdot\gamma_{\mathrm{M}}\pi d_2Q_{\mathrm{M}2}10^{-3}=G_{\mathrm{M}2}(\mathrm{t})$ | 10 | 6,76 | 5,05 | 4,22 |
| $3\cdot\gamma_{\mathrm{M}}\pi d_1Q_{\mathrm{M}1}10^{-3}=G_{\mathrm{M}1}(\mathrm{t})$ | 12,8 | 8,7 | 6,25 | 5,25 |
| $G_{\mathrm{M}2}+G_{\mathrm{M}1}=G_{\mathrm{M}}(\mathrm{t})$ | 22,8 | 15,5 | 11,3 | 9,47 |
| $\gamma_cQ_cl_c10^{-3}=G_c(\mathrm{t})$ | 72 | 83,5 | 119 | 177 |
| $G_{\mathrm{M}}+G_c=G_a(\mathrm{t})$ | 94,8 | 99 | 130 | 187 |
| $c+\beta G_{\mathrm{M}}=G_{\mathrm{p}}(\mathrm{t})$ | 152 | 138 | 158 | 212 |
| $4{,}9\,\gamma_cQ_cd\,10^{-3}=G_{\mathrm{cy}}(\mathrm{t})$ | 13 | 25,5 | 49,6 | 86 |
| $k_{\partial c}a_c(G_c+0{,}93\,G_{\mathrm{cy}})=P_c(\mathrm{kW})$ | 135 | 171 | 264 | 410 |
| $\dfrac{\rho}{\gamma_{\mathrm{M}}}\left(k_{\partial1}\Delta_1^2G_{\mathrm{M}1}+k_{\partial2}\Delta_2^2G_{\mathrm{M}2}\right)=P_{\mathrm{M}}(\mathrm{kW})$ | 1320 | 960 | 756 | 682 |
| $t\cdot p_c+\tau\cdot p_{\mathrm{M}}S_*^2=W\cdot10^3(\mathrm{kW}\cdot\mathrm{h})$ | 6,47 | 5,35 | 5,33 | 6,3 |
| $З_{\mathrm{ta}}+З_c+З_{\mathrm{p}}=З(\text{thousand rubles})$ | 104 | 87,9 | 91,2 | 113 |
\[
\gamma_{\mathrm{m}}=8.9\ \frac{\mathrm{g}}{\mathrm{cm}^{3}},\quad
\gamma_{\mathrm{s}}=7.8\ \frac{\mathrm{g}}{\mathrm{cm}^{3}}
\]
— specific weights of copper and steel;
\[ \beta=3.5,\quad k_{\mathrm{t}}=1.3\frac{\mathrm{rub}}{\mathrm{kg}},\quad p_{\mathrm{sh}}=0.125,\quad p_{\mathrm{a}}=0.06,\quad Z_{\mathrm{sh}}=212\frac{\mathrm{rub}}{\mathrm{kW}}, \]
\[
k_{\mathrm{m}}=0.9,\quad
Z'_{\mathrm{uz}}=Z''_{\mathrm{uz}}=6\frac{\mathrm{rub}\cdot 10^{-3}}{\mathrm{kW}\cdot \mathrm{h}},\quad
S_{*}^{2}=0.8
\]
— techno-economic coefficients;
\[
k_{\mathrm{ds}}=1.093
\]
— coefficient of additional losses in the steel;
\[
\alpha_{\mathrm{s}}=1.46\frac{\mathrm{W}}{\mathrm{kg}}
\]
— specific losses in the steel;
\[
t=8760\ \mathrm{h},\quad \tau=500\ \mathrm{h}
\]
— number of hours of transformer use and number of hours of losses per year;
\(p_{2}=3\) — number of rows of the elementary conductor in the HV winding;
\(k_{1}=1,\ k_{\mathrm{d}2}=1.037\) — coefficients of additional losses.
In the calculation presented, the discreteness of the dimensions of the conductor material was not taken into account.
Conclusions
-
Variation of the number of layers of the LV winding \(m_{1}\) over a wide range is not advisable. At \(8<m_{1}<5\), the transformer height and the core diameter reach nontechnological dimensions.
-
The minimum calculated costs correspond to \(m_{1}=5\).
Literature
-
I. D. Kutyavin. Design of transformer windings with minimum copper losses. Elektrotekhnika, 1969, No. 7.
-
Dietrich W., Auslegung von Transformatorenwicklungen mit kleinsten wirkwiderstand, Elektrotechn., 1965, No. 6.