Autotransformers with Multilayer HV Windings
Abstract Generated abstract
This paper adapts a previously developed optimization method for two-winding power transformers to the design of autotransformers with multilayer concentric high-voltage windings. It derives relations for the rated powers, layer numbers, turn ratios, conductor cross sections, and short-circuit voltage conditions of step-up unit and network autotransformers, including the role of compensating low-voltage windings. Numerical calculations for 200 MVA, 750 kV designs compare alternative layer counts and associated dimensions, weights, and costs. The results indicate practical optimum configurations for both unit and network autotransformers and suggest that the simplified method can be used for preliminary study and calculation of such designs.
Full Text
AUTOTRANSFORMERS WITH MULTILAYER HV WINDINGS
I. D. Kutyavin, G. V. Del, L. I. Del
This paper shows the possibility of using the general study of two-winding transformers developed at the Department of Electric Power Stations [2] for autotransformers.
Autotransformers may be regarded as two-winding transformers with different rated winding powers, but in power-supply circuits they often replace three-winding transformers, being more economical in comparison with them. Such a replacement can be made in all cases when the medium-voltage (MV) and high-voltage (HV) windings can have the same connection scheme and the same neutral state. Therefore, only the combination of a voltage of 35 kV with higher voltages, and with the presence of a low-voltage (LV) winding, necessitates the use of three-winding transformers.
For step-up block autotransformers, the rated power of the primary winding (LV) is equal to the rated (through) power \(S_{\text{н}}\). The secondary autotransformer winding (with a winding-power ratio of \(100/100/100\%\)) must have the total rated power [1]
\[ S_{p2}=S_{\text{н}}(2-N), \tag{1} \]
where \(N\) is the transformation coefficient of the autotransformer windings, equal to the ratio of the medium voltage to the high voltage.
The MV winding (the common part of the autotransformer winding) has a rated power equal to the rated power \(S_{\text{н}}\), while the additional part of this winding has the rated power
\[ S_{pд}=S_{\text{н}}(1-N). \tag{2} \]
The ratio of the rated powers of the additional \(S_{pд}\) and common \(S_{pо}\) parts of the autotransformer winding is
\[ \frac{S_{pд}}{S_{pо}}=1-N. \tag{3} \]
For network autotransformers, the primary winding is one of the autotransformer windings, either MV or HV. The total rated power of the autotransformer winding is
\[ S_{p2}=2S_{\text{н}}(1-N). \tag{4} \]
The rated powers of the common and series windings are equal to each other
\[ S_{\mathrm{po}}=S_{\mathrm{pd}}=S_{\mathrm{n}}(1-N). \tag{5} \]
Network autotransformers also have an LV winding (compensating winding), connected in delta and intended to compensate the third-harmonic components of the phase magnetic fluxes and voltages [1]. Under the conditions of thermal stability during single-phase short circuits in the MV and HV networks, the rated power of this winding must be not less than one third of the nominal power. The compensating winding may be used to supply consumers at 6–35 kV. In that case its power may be increased up to the nominal value.
If the power of the compensating winding is denoted by \(S_{\mathrm{k}}=\sigma S_{\mathrm{n}}\), where \(\sigma=1/3\div 1\), then expressions (4) and (5) are valid for \(\sigma \leq (1-N)\). For \(\sigma=1\), expression (1) will be valid for a network autotransformer.
For an autotransformer, the ideal winding is a multilayer concentric winding with an MV tap at the transition from one layer to another.
For step-up unit autotransformers, the ratio of the number of layers of the series \(m_{\mathrm{d}}\) and common \(m_{\mathrm{o}}\) windings is equal to (for equal layer heights of the winding)
\[ \frac{m_{\mathrm{d}}}{m_{\mathrm{o}}}=1-N. \tag{6} \]
The total number of layers in the winding must be an integer and proportional to
\[ m=m_{\mathrm{o}}+m_{\mathrm{d}}=2-N. \tag{7} \]
Therefore it must be a multiple of
\[ m=\frac{2}{N}-1. \tag{8} \]
The number of winding layers of network autotransformers must be even and divided equally between the common and series windings.
Under these conditions, the number of turns in a layer of the common and series windings is in the ratio
\[ w_{\mathrm{o}}=Nw_{\mathrm{d}}. \tag{9} \]
If the current densities are equal, the conductor cross sections of the windings will have the inverse dependence
\[ q_{\mathrm{md}}=Nq_{\mathrm{mo}}. \tag{10} \]
Equality of the SC voltages of the LV and MV windings of a unit autotransformer corresponds to (1.18) [2], since the winding powers are equal. Equality of the SC voltages of the LV and HV windings has the expression
\[ q_{\mathrm{mn}}\Delta_{\mathrm{n}}=\Delta_{\mathrm{d}}\left(Nq_{\mathrm{mo}}+q_{\mathrm{md}}\right). \tag{11} \]
For network autotransformers with the rated power of the compensating winding
\[ S_{\mathrm{pk}}=\sigma S_{\mathrm{n}}. \]
Equality of the SC voltages of the windings is
\[ \frac{q_{\mathrm{mn}}\Delta_{\mathrm{n}}}{\sigma} = \frac{q_{\mathrm{mo}}\Delta_{\mathrm{o}}}{1-N} = \frac{q_{\mathrm{md}}\Delta_{\mathrm{d}}}{1-N} = (\mathrm{SC})_{\mathrm{nom}}. \tag{12} \]
When formula (1.19) [2] is used for the short-circuit voltage, the nominal SC from (3.49) [2] is substituted into it.
The remaining initial relations for autotransformers are the same as for two-winding transformers [2].
Tables 1 and 2 give the results of calculating the optimal dimensions and parameters for block (Table 1) and network (Table 2) autotransformers. Initial data: \(S_c = 200\) MVA;
Table 1
| \(m_2\) | Adopted | 7 | 8 | 9 | 10 | 12 |
|---|---|---|---|---|---|---|
| \(m_{02}=\dfrac{m_2}{1-N}\) | 4.2 | 4.8 | 5.4 | 6.0 | 7.2 | |
| \(x_2\) | from (1.52) [2], cm | 0.73 | 0.68 | 0.64 | 0.61 | 0.56 |
| \(b_2\) | from (1.14) | 21.0 | 16.4 | 29.8 | 33.1 | 39.7 |
| \(b_{02}\) | from (1.14) | 12.7 | 14.7 | 16.7 | 18.7 | 22.6 |
| \(k_{r2}\) | from (1.54), rel. units | 1.022 | 1.024 | 1.025 | 1.026 | 1.027 |
| \(\psi_{02}\) | from (1.84), A/cm | 1290 | 1430 | 1560 | 1690 | 1920 |
| \(b_1\) | from joint solution, cm | 4.8 | 5.4 | 6.0 | 6.6 | 7.2 |
| \(x_1\) | (2.38) and (2.39), cm | 0.31 | 0.28 | 0.27 | 0.26 | 0.24 |
| \(k_{r1}\) | from (1.41), rel. units | 1.063 | 1.065 | 1.072 | 1.080 | 1.087 |
| \(\Delta_1\) | from (1.43), A/cm\(^2\) | 452 | 445 | 440 | 436 | 432 |
| \(\Delta_2\) | from (1.28), A/cm\(^2\) | 506 | 524 | 540 | 553 | 575 |
| \(d\) | from (2.21), cm | 87 | 101 | 117 | 133 | 165 |
| \(\varphi_1\) | from (1.74), A/cm | 1240 | 1350 | 1490 | 1620 | 1820 |
| \(Q_c\) | from (2.24), t | 161 | 166 | 176 | 192 | 252 |
| \(h\) | from (1.79), cm | 850 | 574 | 387 | 277 | 158 |
| \(Q_{m1}\) | from (13), t | 19.3 | 16.6 | 14.1 | 12.3 | 9.8 |
| \(Q_{m2}\) | from (14) | 42.5 | 35.0 | 28.0 | 23.6 | 17.9 |
| \(Q_{m02}\) | from (14) | 23.6 | 19.1 | 15.6 | 13.1 | 10.0 |
| \(Q_m=Q_{m1}+Q_{m2}\) | 61.8 | 51.6 | 42.1 | 35.9 | 27.7 | |
| \(Q_a=Q_c+Q_m\) | 223 | 218 | 218 | 228 | 280 | |
| \(Z_c\) | from (3.8), thousand rubles | 65 | 67 | 71 | 78 | 102 |
| \(Z_m\) | from (15) | 157 | 134 | 111 | 97 | 80 |
| \(Z\) | \(=Z_c+Z_m\) | 222 | 201 | 182 | 175 | 182 |
Table 2
| \(m_2\) | Adopted | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|
| \(m_{02}=0.5m_2\) | 3 | 4 | 5 | 6 | |
| \(x_2\) | from (1.52) [2], cm | 0.87 | 0.75 | 0.67 | 0.62 |
| \(k_{r2}\) | from (1.54), rel. units | 1.024 | 1.027 | 1.029 | 1.031 |
| \(b_2\) | from (1.14), cm | 20.2 | 27.0 | 33.7 | 40.4 |
| \(b_{02}\) | from (1.14), cm | 8.6 | 12.0 | 15.4 | 18.7 |
| \(\psi_{02}\) | from (1.54), A/cm | 1010 | 1260 | 1480 | 1700 |
| \(b_1\) | from joint solution | 2.6 | 3.2 | 4.0 | 4.8 |
| \(x_1\) | (2.38) and (2.39), cm | 0.47 | 0.40 | 0.35 | 0.30 |
| \(k_{r1}\) | from (1.41), rel. units | 1.050 | 1.057 | 1.065 | 1.071 |
| \(\Delta_2\) | from (1.43), A/cm\(^2\) | 486 | 469 | 453 | 446 |
| \(\Delta_1\) | from (1.28) | 465 | 499 | 529 | 548 |
| \(d\) | from (2.21) | 77 | 107 | 142 | 176 |
| \(\varphi_1\) | from (1.74), A/cm | 727 | 900 | 1080 | 1215 |
| \(Q_c\) | from (2.24), t | 132 | 141 | 184 | 256 |
| \(h\) | from (1.79), cm | 920 | 387 | 180 | 105 |
| \(Q_{m1}\) | from (13), t | 10.5 | 7.10 | 5.34 | 4.44 |
| \(Q_{m2}\) | from (14) | 42.7 | 26.0 | 17.4 | 13.14 |
| \(Q_{m02}\) | from (14) | 18.6 | 11.8 | 7.8 | 6.15 |
| \(Q_m=Q_{m1}+Q_{m2}\) | 53.2 | 33.1 | 22.7 | 17.6 | |
| \(Q_a=Q_c+Q_m\) | 185 | 174 | 207 | 274 | |
| \(Z_c\) | from (3.8), thousand rubles | 53 | 57 | 74 | 103 |
| \(Z_m\) | from (16) | 114 | 77 | 60 | 45 |
| \(Z\) | \(=Z_c+Z_m\) | 167 | 134 | 134 | 148 |
\[ U_{\text{LV}} = 20\ \text{kV}; \qquad U_{\text{HV}} = 750\ \text{kV}; \qquad N = 1/3; \qquad B = 1.70\ \text{T}; \qquad S_{\text{rk}} = 0.5S_n; \]
\[ K_c = 0.253\cdot 10^{-4}; \qquad y_1 = 1.45; \qquad y_2 = 2.0; \qquad \hat{\delta}_1 = 0.8; \qquad \hat{\delta}_2 = 3.0; \qquad i_1 = 0.05; \]
\[
i_2 = 0.4;\quad a_1 = 25 \cdot 10^4;\quad a_2 = 16 \cdot 10^4;\quad U_{kns} = 0.10;\quad \delta_{01} = 5;\quad \delta_{12} = 5;
\]
\[
l_{w1} = 12;\quad l_{w2} = 28;\quad n = 3;\quad k_q = 1;\quad C = 0.92.
\]
The LV winding is helical or continuous disk; the autotransformer winding is multilayer concentric.
In calculating the data in Tables 1 and 2, it was assumed that, as the height of a layer of the autotransformer winding decreases (with a transposed cross section), the radial dimension of the copper of layer \(x_2\) increases so that the cross-sectional area of the copper of the layer remains constant.
The weight of copper of the disk winding, in accordance with (2.32) of [2], was determined from the formula
\[ Q_m = n\pi\gamma_m 10^{-3}h\,\frac{bxyd_m}{(x+i)(y+\delta)} \tag{13} \]
and that of the multilayer winding,
\[ Q_m = n\pi\gamma_m 10^{-3}h\,\frac{mxyd_m}{y+i}. \tag{14} \]
The calculated costs associated with the copper of the windings of a unit autotransformer, in accordance with [2], were determined from the formula
\[ Z_m = \beta A Q_m + E\left(k_{r1}\Delta_1^2 Q_{m1} + k_{r2}\Delta_2^2 Q_{m02}\right). \tag{15} \]
Here, as the design operating condition for determining copper losses, the loading of the MV winding to rated power was selected.
For a network autotransformer, at \(\sigma < 1 - N\), the design operating condition will be loading of the autotransformer windings to the design power. Then the calculated costs associated with the copper of the windings of network autotransformers at \(\sigma = 0.5\) can be determined from the formula
\[ Z_m = \beta A Q_m + k_2 E \Delta_2^2\left[(1-N)^2 Q_{m02} + (Q_{m2} - Q_{m02})\right]. \tag{16} \]
If the design power of the compensating winding is equal to the rated power \((\sigma = 1)\), then the calculated costs associated with the copper of the windings of network autotransformers are determined from (15). In this case, subscript one refers to the compensating winding and subscript two to the feeding autotransformer winding (or conversely).
As can be seen from Table 1, the optimum number of layers of the secondary winding of a unit autotransformer of the given power is \(m_2 = 10\); \(m_{02} = 5\); \(d = 133\); \(h = 277\). For a network autotransformer (Table 2), at \(\sigma = 0.5\), the optimum number of layers of the autotransformer winding is \(m_2 = 9\); \(d = 125\); \(h = 250\). If it is necessary to bring out the MV terminal from the transition of one layer to another, one may take \(m_2 = 10\); \(d = 142\); \(h = 180\).
Despite the adopted idealization of the problem, the relations given above and the numerical calculation example demonstrate the possibility of using the method under consideration for the study and calculation of autotransformers.
References
- G. N. Petrov, Electrical Machines. Part I. GEI, Moscow, 1956.
- I. D. Kutyavin, G. V. Del, L. I. Del, Some Problems of Optimization of Power Transformers. Design Manual. Rotaprint, TPI, 1970.