On the Modeling of Power Transformers on Direct-Current Computing Devices
Abstract Generated abstract
The paper addresses the difficulty of representing power transformers with variable transformation ratios on direct-current computing devices, especially when supply-side voltage deviations and on-load tap changers create many possible operating conditions. It proposes selecting transformer ratios by least-squares approximation over a specified range of primary voltages, using reciprocal voltage functions to simplify otherwise cumbersome equations. The method is derived for two-winding transformers and extended to three-winding transformers with two variable ratios, yielding algebraic or nonlinear systems solvable by standard numerical methods such as Newton’s method. A numerical example for a two-winding transformer shows how the calculated coefficient is converted to the nearest standard tap setting.
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On the Modeling of Power Transformers on Direct-Current Computing Devices
R. I. Borisov, V. V. Litvak
(Presented by the Department of Electrical Networks and Systems)
For three-winding transformers equipped with OLTC devices, when there are voltage deviations on the side of the supply source it is possible to establish 45 values of transformation ratios, which greatly complicates the solution of the problem of modeling transformers on direct-current computing devices. Approximation with respect to transformation ratios for given voltage deviations reduces the number of calculated operating conditions and facilitates the solution of the problem.
The determination of the transformation ratios of power transformers that do not have OLTC devices is performed by selection according to the calculated voltage values in maximum and minimum operating conditions. Approximation to the desired voltage is carried out for fixed values of the voltages on the side of the supply source.
It is proposed that the choice of transformation ratios be made taking into account possible changes in the voltage on the side of the supply source, which will ensure smaller voltage deviations at consumers and, consequently, better operating conditions for them.
Establishing the range and scale of values of the transformation ratios of power transformers and autotransformers is also required in their design.
Let us consider the problem of determining the transformation ratio of a power two-winding transformer at a given load and with changes in voltage on the side of the supply source. For the equivalent circuit in Fig. 1, the following expression can be written for the voltage at the load terminals as a function of the transformation ratio \(K\) and the circuit parameters:
\[ U_{\mathrm{н}} = \frac{U_1 z_{\mathrm{н}} K^2} {z_{s_1} + (z_{s_2}+z_{\mathrm{н}})K^2} \cdot \frac{1}{K} = \frac{U_1} {\dfrac{z_{s_1}}{z_{\mathrm{н}}K} + \dfrac{z_{\mathrm{н}}+z_{s_2}}{z_{\mathrm{н}}}K} = \]
\[ = \frac{U_1} {\dfrac{\alpha_1}{K}+\beta_1 K + j\left(\dfrac{\alpha_2}{K}+\beta_2 K\right)} , \]
where
\[ \frac{z_{s_1}}{z_{\mathrm{н}}}=\alpha_1+j\alpha_2; \qquad 1+\frac{z_{s_2}}{z_{\mathrm{н}}}=\beta_1+j\beta_2. \]
The leakage reactances of the windings may be regarded as independent of the position of the tap changer. Assuming that the voltage on the primary side of the transformer can vary within the limits \(U_1'' \div U_1'\) for each calculated loading condition, we determine the value of the transformation coefficient “\(K\)” that provides the smallest deviations from the desired voltage on the secondary side. The nominal network voltage is chosen as the desired voltage. Approximation with respect to the transformation coefficient is carried out by the least-squares formula
\[ \frac{\partial}{\partial K}\cdot \int_{U_1''}^{U_1'} \left[ U_{2\text{ж}}- \frac{U_1}{ \dfrac{a_1+j a_2}{K}+(\beta_1+j\beta_2)K } \right]^2 dU_1=0. \tag{1} \]
Transforming equation (1) with respect to the modulus or the real part of the complex \(U_1(K)\) proves to be very cumbersome. Therefore it is proposed to approximate \(U_{2\text{ж}}\) by inverse functions, which gives
Fig. 1. Equivalent circuit of a two-winding transformer with variable transformation coefficient.
a difference in the approximations of \(0.2\%\) when the functions differ by \(10\%\). Such convergence of the process makes it possible to regard both methods of approximation, with the same variables, as satisfying the corresponding requirements for calculation accuracy.
\[ \frac{1}{U_1(K)} = \frac{ \dfrac{a_1+j a_2}{K}+(\beta_1+j\beta_2)K }{U_1}; \]
\[ \frac{1}{\left|U_1(K)\right|} = \frac{ \left[ \left(\frac{a_1}{K}+\beta_1K\right)^2+ \left(\frac{a_2}{K}+\beta_2K\right)^2 \right]^{\frac12} }{U_1} = \frac{\gamma^{\frac12}(K)}{U_1}. \]
Then
\[ \frac{\partial}{\partial K} \int_{U_1''}^{U_1'} \left[ \frac{1}{U_{2\text{ж}}} - \frac{\gamma(K)}{U_1} \right]^2 dU_1 = -2\frac{1}{U_{2\text{ж}}} \ln U_1\bigg|_{U_1''}^{U_1'} \frac{\partial}{\partial K}\gamma^{\frac12}(K) - \frac{1}{U_1}\bigg|_{U_1''}^{U_1'} \frac{\partial}{\partial K}\gamma K = 0. \]
Let us denote
\[ -\frac{2}{U_{2\text{ж}}} \ln U_1\bigg|_{U_1''}^{U_1'} = -A; \qquad -\frac{1}{U_1}\bigg|_{U_1''}^{U_1'} = B. \]
Consequently,
\[ -\frac{1}{2}A\gamma^{-\frac12}(K)\cdot\gamma'(K) + B\gamma'(K) = \gamma'(K) \left[ B-\frac{1}{2}A\gamma^{-\frac12}(K) \right] = 0. \]
which gives a biquadratic equation with respect to \(K\)
\[ (\beta_1^2+\beta_2^2)K^4+(2\alpha_1\beta_1+2\alpha_2\beta_2)K^3-\frac{A^2}{4B^2}K^2+(\alpha_1^2+\alpha_2^2)=0. \tag{2} \]
For three-winding transformers, the approximation should be carried out with respect to two variable transformation ratios: between the high- and medium-voltage windings \(K\), and the high- and low-
Fig. 2. Equivalent circuit of a three-winding transformer with GPN.
voltage windings \(N\) (Fig. 2). The voltages across the loads \(z_{\mathrm{н}2}\) and \(z_{\mathrm{н}3}\), reduced to the corresponding stage, are:
\[ \dot U_2= \frac{U_1 z_{\mathrm{н}3}} {\dfrac{z_{s_1}}{K}+(z_{s_2}+z_{\mathrm{н}2})K+ \dfrac{K z_{s_1}(z_{s_2}+z_{\mathrm{н}2})}{N^2(z_{s_3}+z_{\mathrm{н}3})}} = U_2(KN), \]
\[ \dot U_3= \frac{U_1 z_{\mathrm{н}3}} {\dfrac{z_{s_1}}{N}+(z_{s_3}+z_{\mathrm{н}3})N+ \dfrac{N z_{s_1}(z_{s_3}+z_{\mathrm{н}2})}{K^2(z_{s_2}+z_{\mathrm{н}2})}} = U_3(KN). \]
The approximation equations for the moduli of the reciprocal functions \(\dfrac{1}{|U_2(KN)|}\) and \(\dfrac{1}{|U_3(KN)|}\) have the following form:
\[ \frac{\partial}{\partial K} \int_{U_1''}^{U_1'} \left[ \frac{1}{U_{2\mathrm{ж}}} - \frac{1}{|U_2(KN)|} \right]^2 \,dU_1=0, \tag{3} \]
\[ \frac{\partial}{\partial N} \int_{U_1''}^{U_1'} \left[ \frac{1}{U_{3\mathrm{ж}}} - \frac{1}{|U_3(KN)|} \right]^2 \,dU_1=0. \tag{4} \]
Let us reduce these equations to a form convenient for solution:
\[ \dot U_2(KN)= \frac{U_1} {\dfrac{a_1-j\beta_1}{K}+ \left(a_2+j\beta_2+\dfrac{a_3+j\beta_3}{N^2}\right)K} = \]
\[ = \frac{U_1} {\dfrac{a_1}{K}+a_2K+\dfrac{a_3K}{N^2} +j\left(\dfrac{\beta_1}{K}+\beta_2K+\dfrac{\beta_3K}{N^2}\right)} = \frac{U_1}{\alpha(KN)+j\beta(KN)}; \]
\[ \frac{1}{|U_2(KN)|}=\frac{[a^2(KN)+\beta^2(KN)]^{1/2}}{U_1} =\frac{\varphi(KN)^{1/2}}{U_1}; \]
\[ \frac{1}{|U_3(KN)|}=\frac{[a^2(KN)+b^2(KN)]^{1/2}}{U_1} =\frac{\psi(KN)^{1/2}}{U_1}. \]
Equations (3) and (4) are written as follows:
\[ \frac{\partial}{\partial K} \int_{U_1''}^{U_1'} \left[ \frac{1}{U_{2\mathrm{zh}}} - \frac{\varphi(KN)^{1/2}}{U_1} \right]^2 dU_1 = -2\frac{1}{U_{2\mathrm{zh}}}\ln U_1\bigg|_{U_1''}^{U_1'} \frac{\partial}{\partial K}\varphi(KN)^{1/2} - \]
\[ -\frac{1}{U_1} \int_{U_1''}^{U_1'} \frac{\partial}{\partial K}\varphi(KN) = -A\frac{\partial}{\partial K}\varphi(KN)^{1/2} + B\frac{\partial}{\partial K}\varphi(KN) = \]
\[ = -\frac{1}{2}A\varphi(KN)^{-1/2}\varphi'(KN) + B\varphi'(KN) = \varphi'(KN)\times \]
\[ \times \left[ B-\frac{1}{2}A\varphi(KN)^{-1/2} \right] =0. \]
\[ \frac{\partial}{\partial N} \int_{U_1''}^{U_1'} \left[ \frac{1}{U_{3\mathrm{zh}}} - \frac{\psi(KN)^{1/2}}{U_1} \right]^2 dU_1 = \psi'(KN) \left[ B-\frac{1}{2}C\psi(KN)^{-1/2} \right] =0. \]
\[ C=-2\frac{1}{U_{3\mathrm{zh}}}\ln U_1\bigg|_{U_1''}^{U_1'} . \]
This gives a system of nonlinear equations for the sought transformation coefficients
\[ \varphi(KN)-\frac{A^2}{4B^2}=0. \]
\[ \psi(KN)-\frac{C^2}{4B^2}=0. \]
This system can be solved by Newton’s method, taking the nominal values of the transformation coefficients as the initial approximations.
Example. Determine the value of the transformation coefficient \(K\) of a two-winding power transformer for voltage deviations on the primary side within the range \(1.05\)—\(0.9\) and full load of a TDT-10/110 transformer.
\[ U_{\mathrm{n}_1}=110\ \mathrm{kV},\qquad U_{\mathrm{n}_2}=38.5\ \mathrm{kV},\qquad S_{\mathrm{n}}=10\ \mathrm{MVA},\qquad \cos\varphi_{\mathrm{n}}=0.9, \]
\[ e_k=10.5\%,\qquad \Delta P_{\mathrm{k.z.}}=97.5\ \mathrm{kW}. \]
\[ z_{\mathrm{n}}=\frac{35^2}{10}(0.9+j0.435)=110+j53.25; \]
\[ x_{S_1}=\frac{0.05\cdot 110^2}{10}=61\ \Omega,\qquad R_1=\frac{48.75\cdot 110^2}{10^5}=5.92\ \Omega, \]
\[ x_{S_2}=\frac{0.05\cdot 38.5^2}{10}=7.42\ \Omega,\qquad R_2=\frac{48.75\cdot 38.5^2}{10^5}=0.72\ \Omega, \]
\[ \frac{z_{S_1}}{z_{\mathrm{n}}} = a_1+ja_2 = \frac{5.91+j61}{110+j53.25} = 0.255+j0.42; \]
\[ 1+\frac{z_{s_2}}{z_{\mathrm{н}}}=\beta_1+j\beta_2 =1+\frac{0.72+j7.42}{110+j53.25} =1.03+j0.051, \]
\[ -A=-2\frac{1}{U_{э\mathrm{к}}}\ln U_1\bigg|_{U_1''}^{U_1'} =-2\frac{1}{35}\cdot\frac{1}{0.433}\cdot0.068=-0.009; \]
\[ B=-\frac{1}{U_1}\bigg|_{0.9\cdot110}^{1.05\cdot110} =-\frac{1}{1.05\cdot110}+\frac{1}{0.9\cdot110}=0.0015; \]
\[ \frac{A^2}{4B^2}=\frac{0.009^2}{4\cdot0.0015^2}=9; \]
\[ \alpha_1^2+\alpha_2^2=0.255^2+0.42^2=0.241; \]
\[ \beta_1^2+\beta_2^2=1.03^2+0.03^2=1.0609; \]
\[ 2\alpha_1\beta_1+2\alpha_2\beta_2 =2\cdot0.255\cdot1.03+2\cdot0.42\cdot0.03=0.55. \]
Solving equation (2) with respect to \(K\),
\[ 1.069K^4-8.45K^2+0.241=0;\qquad K=2.8. \]
We choose the nearest standard value \(K=2.79\), which is established with taps of 107.25 on the 110 kV side and the nominal tap on the secondary side.