Control of Operating Parameters of a Long-Distance AC Power Transmission Line with an Intermediate System by Regulating Interconnecting Transformers
Abstract Generated abstract
This paper develops a method for estimating how on-load tap changes of coupling transformers affect the steady-state parameters of a long-distance AC transmission line connected to an intermediate power system. Using generalized static characteristics of adjacent system sections, the authors linearize reactive-power, voltage, active-power, and transformation-ratio relationships and derive equations for calculating parameter increments, including variants for inverse regulation problems and for cases with fixed-voltage control or transformer reactance dependence on tap position. A numerical example for a line with an intermediate system shows that changing the autotransformer tap can significantly alter line voltage and reactive-power flows, suggesting that the intermediate system can provide useful regulation of transmission-line operating conditions under suitable assumptions.
Full Text
CONTROL OF OPERATING PARAMETERS OF A LONG-DISTANCE AC TRANSMISSION LINE WITH AN INTERMEDIATE SYSTEM BY REGULATING COUPLING TRANSFORMERS
R. I. BORISOV, V. I. GOTMAN
(Presented by the Department of Electrical Systems and Networks)
Methods for calculating changes in the steady-state operating conditions of interconnected power systems using the static response characteristics of parts of power systems are of particular interest, since they possess a certain clarity and make it possible to solve certain local problems of controlling the operating conditions of individual parts by means of information obtained only within their limits.
The generalized static characteristics of parts of a power system with respect to node \(D\) (Fig. 1) represent the dependences:
\[ P_i = P(U_D), \]
\[ Q_i = Q(U_D). \tag{1} \]
The operating parameters of point \(D\) can be determined by the method of intersection of the static characteristics of parts \(A\), \(B\), \(C\) of the system adjoining this point [3, 4]. The condition for the existence of an operating state is the fulfillment at the coupling point \(D\) of the boundary conditions:
\[ U_D^A = U_D^B = U_D^C,\quad \Delta S_D = 0,\quad f_D^A = f_D^B = f_D^C. \tag{2} \]
The slope of the static characteristics is determined: for parts \(A\) and \(B\)
\[ \frac{1}{\gamma_1} = \frac{d\left(Q_{DB} - Q_{AD}\right)}{dU_D} \geq 0 \tag{3} \]
and for \(C\)
\[ \frac{1}{\gamma_2} = \frac{dQ_{CD}}{dU_D} \leq 0. \tag{4} \]
When the operating condition changes in one of the parts of the interconnection, for example \(C\) (Fig. 1), the characteristic of this part will shift parallel to itself. If unchanged operating conditions are maintained in part \(A—B\) of the system, point \(D\) will move along the static characteristic of this part. The change in operating conditions in each part can be determined by knowing the increment of one of the parameters and the slope coefficients of the characteristics \(\gamma_1\) and \(\gamma_2\).
\[ \Delta Q = Q_2 - Q_1 = \frac{\Delta U_1}{\gamma_1} = \frac{\Delta U_2}{\gamma_2}. \tag{5} \]
Relation (5) also reflects the conditions for voltage regulation by an ideal transformer with an on-load tap changer. The change in the coefficien-
of transformation leads to a voltage increment at its terminals with different signs \((\gamma_1 > 0,\ \gamma_2 < 0)\). The voltage increments, referred to one voltage level, are in an inversely proportional dependence on the slope coefficient of the corresponding static characteristics of the parts of the system adjacent to the transformer.
The influence of intermediate connections on the operating conditions of a long-distance transmission line depends on the method of connection and, consequently, on the equipping of the coupling autotransformers with on-load tap-changing devices and on their operation. The choice
Fig. 1. On determining the steady-state operating condition of a long-distance transmission line with an intermediate system by the method of intersection of static characteristics
of the correct regulation law may prove essential for obtaining the desired change in the calculated operating parameter.
The reactive powers flowing through the transformer are functions of the voltages at its terminals, the active powers, and the transformation ratio:
\[ Q_i = Q(U_1, U_2, U_3, P_1, P_2, P_3, K)\quad i = 1, 2, 3. \tag{6} \]
If only a longitudinal-regulation device is present, the expressions for the active powers are independent functions
\[ P_i = P(\Pi) = \mathrm{const}. \tag{7} \]
Taking into account the smooth character of the functions \(Q_i\), under the assumption of continuity of the argument \(K\), the increments of power in the windings are represented in the form of a Taylor series retaining the first terms of the expansion:
\[ \begin{aligned} \Delta Q_1 &= \sum \frac{\partial Q_1}{\partial U_i}\Delta U_i + \sum \frac{\partial Q_1}{\partial P_i}\Delta P_i + \frac{\partial Q_1}{\partial K}\Delta K,\\ \Delta Q_2 &= \sum \frac{\partial Q_2}{\partial U_i}\Delta U_i + \sum \frac{\partial Q_2}{\partial P_i}\Delta P_i + \frac{\partial Q_2}{\partial K}\Delta K,\\ \Delta Q_3 &= \sum \frac{\partial Q_3}{\partial U_i}\Delta U_i + \sum \frac{\partial Q_3}{\partial P_i}\Delta P_i + \frac{\partial Q_3}{\partial K}\Delta K, \end{aligned} \tag{8} \]
\[ i=1,2,3. \]
In these equations, with unchanged conditions in systems I, II, III (Fig. 2, b), the voltage changes \(\Delta U_i\) are functions of the change in the transformation ratio. On the other hand, variations of the values of the transformation ratio are, for each part of the system, an external disturbance; the changes in the values \(\Delta Q_i\) and \(\Delta P_i\) will occur in accordance with the static characteristics of these parts and, consequently, can be determined from them.
Suppose that in each of the parts of the system adjacent to the transformer there is at least one load node. Then the static characteristics of these parts will be complex functions of the form (see Appendix):
\[ Q_i = Q\,[U_i, Q_{\mathrm{н}j}(U_i), P_i(U_i)], \tag{9} \]
\[ P_i = P(U_i), \qquad i=1,2,3, \]
where \(i\) is the number of the part of the system relative to the transformer terminals;
\(j\) is the number of the load nodes of the parts of the system.
Then the increments of reactive powers from (9) may be written as
\[ \Delta Q_i = \left( \frac{\partial Q_i}{\partial U_i} + \frac{\partial Q_i}{\partial Q_{\mathrm{н}j}}\frac{dQ_{\mathrm{н}j}}{dU_i} + \frac{\partial Q_i}{\partial P_i}\frac{dP_i}{dU_i} \right)\Delta U_i, \tag{10} \]
\[ i=j=1,2,3. \]
Introduce the notation
\[ \frac{\partial Q_i}{\partial U_i}=\beta_i,\qquad \frac{\partial Q_i}{\partial Q_{\mathrm{н}j}}=\eta_i,\qquad \frac{dQ_{\mathrm{н}j}}{dU_i}=\Psi_i, \tag{11} \]
\[ \frac{\partial Q_i}{\partial P_i}=\varepsilon_i,\qquad \frac{dP_i}{dU_i}=a_i. \]
We rewrite equation (10), taking into account the adopted notation (11), with respect to \(\Delta U_i\):
\[ \Delta U_i = \frac{1}{\beta_i+\eta_i\Psi_i+\varepsilon_i a_i}\,\Delta Q_i = \gamma_i \Delta Q_i . \tag{12} \]
From the equations of the static characteristics of active power,
\[ \Delta P_i = a_i \Delta U_i . \tag{13} \]
Substituting relations (12), (13) into equations (8), we obtain a system of equations with respect to the increments \(\Delta Q_1, \Delta Q_2, \Delta Q_3\):
\[ \left[ \frac{\partial Q_1}{\partial U_1}\gamma_1 + \frac{\partial Q_1}{\partial P_1}\gamma_1 a_1 -1 \right]\Delta Q_1 + \left[ \frac{\partial Q_1}{\partial U_2}\gamma_2 + \frac{\partial Q_1}{\partial P_2}\gamma_2 a_2 \right]\Delta Q_2 + \]
\[ +\left[\frac{\partial Q_1}{\partial U_3}\gamma_3+\frac{\partial Q_1}{\partial P_3}\gamma_3\alpha_3\right]\Delta Q_3 = -\frac{\partial Q_1}{\partial K}\Delta K; \]
\[ \left[\frac{\partial Q_2}{\partial U_1}\gamma_1+\frac{\partial Q_2}{\partial P_1}\gamma_1\alpha_1\right]\Delta Q_1 +\left[\frac{\partial Q_2}{\partial U_2}\gamma_2+\frac{\partial Q_2}{\partial P_2}\gamma_2\alpha_2-1\right]\Delta Q_2+ \tag{14} \]
\[ +\left[\frac{\partial Q_2}{\partial U_3}\gamma_3+\frac{\partial Q_2}{\partial P_3}\gamma_3\alpha_3\right]\Delta Q_3 = -\frac{\partial Q_2}{\partial K}\Delta K; \]
\[ \left[\frac{\partial Q_3}{\partial U_1}\gamma_1+\frac{\partial Q_3}{\partial P_1}\gamma_1\alpha_1\right]\Delta Q_1 +\left[\frac{\partial Q_3}{\partial U_2}\gamma_2+\frac{\partial Q_3}{\partial P_2}\gamma_2\alpha_2\right]\Delta Q_2+ \]
\[ +\left[\frac{\partial Q_3}{\partial U_3}\gamma_3+\frac{\partial Q_3}{\partial P_3}\gamma_3\alpha_3-1\right]\Delta Q_3 = -\frac{\partial Q_3}{\partial K}\Delta K. \]
The solution of these equations has the form
\[ \Delta Q_i=\frac{D_i}{D}=\frac{D_i'}{D}\Delta K. \tag{15} \]
The values of the variables in equation (14) should be calculated from the parameters of the initial operating condition (before the transformation ratios are changed). The magnitude of the change in the transformation ratio is equal to \(\Delta K=K-K_0\), where \(K_0\) is the initial value and \(K\) the new value.
From the known increments of the reactive-power flows one can determine the changes in the voltages at the transformer terminals and in the active-power flows
\[ \Delta U_i=\gamma_i\Delta Q_i, \tag{12} \]
\[ \Delta P_i=\alpha_i\Delta U_i. \tag{13} \]
The change in voltage at the load nodes is determined as follows:
\[ \Delta U_j=\Delta U_i\frac{dU_j}{dU_i}. \tag{16} \]
The new values of the parameters \(\Pi\) are determined from the initial values \(\Pi_0\) and the computed increments \(\Delta \Pi\):
\[ \Pi=\Pi_0+\Delta \Pi. \tag{17} \]
It is possible to solve the inverse problem—determining the value \(\Delta k\) that provides a specified change in voltage or reactive power on one side of the transformer.
In this case, the increments of the remaining operating parameters assume forced values according to the static characteristics of the sections of the power system adjacent to the transformer sides.
If the voltage on one side of the transformer \(U_i\) is regulated according to an astatical law, the corresponding coefficient \(\gamma_i\) becomes equal to zero, and expressions (14) are substantially simplified.
If it is necessary to take into account the dependence of the transformer reactances (TM) on the transformation ratio, this is done when calculating the partial derivative \(dQ_i/dK\); in this case \(Q_i\) is regarded as a composite function
\[ Q_i=Q\left[K,\ x_1(K),\ x_2(K),\ x_3(K)\right], \]
\[ i=1,2,3, \tag{18} \]
\[ \frac{\partial Q_i}{\partial K} = \left.\frac{\partial Q_i}{\partial K}\right|_{x_j=\mathrm{const}} +\sum_{j=1}^{3}\frac{\partial Q_i}{\partial x_j}\frac{dx_j}{dK}. \]
In the case when, in the course of regulation, the sign of the reactive-power flow changes, the calculated values \(\Delta Q_p\), obtained from (15), can be refined by introducing a correction for the transformer magnetizing power \(Q_\mu\). Then, for example, when changing the direction
Fig. 2. Diagram of a long-distance transmission line with an intermediate system
of the flow from \(1—2\) to \(2—1\) (Fig. 1), the actual increments of reactive powers \(\Delta Q_\phi\) can be determined from the formulas:
\[ \begin{aligned} \Delta Q_{1\phi} &= \Delta Q_{1p} - Q_\mu,\\ \Delta Q_{2\phi} &= \Delta Q_{2p} + Q_\mu . \end{aligned} \tag{19} \]
Appendix
Equations of the static characteristics
- Long-distance transmission line with respect to buses \(U_1\), with unchanged values \(U_{10}\) and \(U_{20}\),
\[ \left. \begin{aligned} Q_1 &= Q(U_1)=\\ &= \frac{U_1^2 \sin(\lambda_1+\lambda_2)} {\sin \lambda_1 \cdot \sin \lambda_2} - \left[ \sqrt{\left(\frac{U_{10}U_1}{\sin\lambda_1}\right)^2-P_{10}^{\,2}} + \sqrt{\left(\frac{U_{20}U_1}{\sin\lambda_2}\right)^2-P_{20}^{\,2}} \right],\\ P_1 &= P(U_1)=\mathrm{const}. \end{aligned} \right\} \tag{П-1} \]
- Of the intermediate system with respect to \(U'_2\)
\[ \begin{gathered} Q_2=\left[-\frac{U_2^{\prime 2}}{x_{\mathrm{r}2}}+ \sqrt{\left(\frac{U'_2U'_{\mathrm{r}}}{x_{\mathrm{r}2}}\right)^2-P_2^{\prime 2}}-Q_{\mathrm{n}2}(U'_2)\right] -\frac{Q_2^{\prime 2}+P_2^{\prime 2}}{U_2^{\prime 2}}\,x_{\ell 1},\\ P_2=P'_2-P_{\mathrm{n}2}(U'_2) \end{gathered} \tag{P-2} \]
The equation relating the voltages \(U_2\) and \(U'_2\)
\[ U_2=\sqrt{\left(U'_2-\frac{Q'_2}{U'_2}x_{\ell 1}\right)^2+ \left(\frac{P'_2-P_{\mathrm{n}2}(U'_2)}{U'_2}x_{\ell 1}\right)^2}. \tag{P-3} \]
Linearized equations of the static load characteristics
\[ \begin{gathered} P_{\mathrm{n}2}=P_{\mathrm{nom}}\left[1+(U'_2-U'_{2\mathrm{nom}})\frac{dP}{dU}\right],\\ Q_{\mathrm{n}2}=Q_{\mathrm{nom}}\left[1+(U'_2-U'_{2\mathrm{nom}})\frac{dQ}{dU}\right]. \end{gathered} \tag{P-4} \]
Equations of the reactive powers of the three-winding autotransformer according to its equivalent circuit (Fig. 3, b).
\[ Q_1=-\frac{U_1^2}{x_1K_{12}^2}+ \sqrt{\left(\frac{U_1U_2}{x_1K_{12}}\right)^2-P_1^2}, \tag{P-5} \]
\[ Q_2=\frac{U_2^2}{x_{22}}- \sqrt{\left[\frac{U_2U_3}{x_3}+ \sqrt{\left(\frac{U_1U_2}{x_1K_{12}}\right)^2-P_1^2}\right]^2+(P_1^2-P_2^2)}, \tag{P-6} \]
\[ Q_3=\frac{U_3^{\prime 2}}{x_3}-\frac{U_2U_3}{x_3}, \tag{P-7} \]
where
\[ x_{22}=x_{22}K_{12}^2=\frac{x_{13}x_{33}}{x_{13}+x_{33}}. \]
Example. For the interconnection shown in Fig. 2, a, calculate the increments and new values of the operating-mode parameters when the transformation ratio of the coupling autotransformer is changed, in order to determine the regulating capabilities of the intermediate system.
Data for the system elements:
\[ \mathrm{T}_1\text{--AODTsT }6\times135/500;\ 500;\ 242;\ 11;\ U_{\mathrm{k\,vs}}=10\%;\ U_{\mathrm{k\,vn}}=33.8\%; \]
\[ U_{\mathrm{k\,sn}}=22.5\%. \]
Transformation ratios in the initial operating mode:
\[ K_{12}=\frac{500}{242}=2.066;\quad K_{13}=\frac{50}{11}=45.45;\quad K_{23}=\frac{242}{11}=22. \]
\[ \mathrm{T}_2\text{--TDTsPG }2\times630/220;\ 242;\ 15.75;\ U_{\mathrm{k}}=12.8\%;\ K_{\mathrm{t}2}=15.36. \]
\[ S_{\mathrm{load}}=720-j348\ \text{(MVA)}. \]
Sections of the EHV line: \(l_{10}=400\ \text{km}\) \((\lambda_1=24^\circ)\); \(l_{20}=600\ \text{km}\) \((\lambda_2=36^\circ)\),
\[ l_1=50\ \text{km}\ 2\times\mathrm{ASO}\text{-}300. \]
Assume: \(S_{\mathrm{b}}=900\ \text{MVA};\ U_{\mathrm{b}}=500\ \text{kV}.\)
a)
\[ U_1\ (500\ \mathrm{kV}) \]
\[ U_3\ (11\ \mathrm{kV}) \]
\[ U_2\ (220\ \mathrm{kV}) \]
\[ K_{12}=\frac{U_{1\mathrm{nom}}}{U_{2\mathrm{n}}}, \qquad K_{23}=\frac{U_{2\mathrm{n}}}{U_{3\mathrm{n}}} \]
\[ K_{13}=\frac{U_{1\mathrm{nom}}}{U_{3\mathrm{n}}}=K_{12}K_{23} \]
b)
\[ U_1,\quad x_{13},\quad P_1,Q_1 \]
\[ U_2,\quad x_{23},\quad P_2,Q_2 \]
\[ U_3,\quad x_{53},\quad Q_3 \]
Fig. 3. Calculation scheme for deriving the operating-mode equations of a regulated transformer.
The transformation ratios of the coupling autotransformer are subject to the relation
\[ K_{13}\cdot K'_{32}\cdot K'_{21}=1. \]
Let
\[ K_{12},\ K_{13}\ — \mathrm{var};\quad K_{23}=\mathrm{const}. \]
Take \(K_{12}\) as the independent coefficient; then
\[ K_{13}=K_{23}\cdot K_{12}. \]
Numerical values of the parameters of the system elements and of the initial operating condition, required for calculating the partial derivatives and coefficients.
Parameters of the system elements:
\[ x_{\mathrm{t}2}=\frac{12.8\%\cdot 242^2\cdot 900}{100\%\cdot 2\cdot 630\cdot 500^2}\cdot 2.066^2=0.0812; \]
\[ x_{\ell 1}=\frac{0.4\cdot 50\cdot 900}{2\cdot 500^2}=0.1366; \]
\[ x_{13}=x_1K_{12}^2 =\frac{10\%\cdot 242^2\cdot 900}{100\%\cdot 6\cdot 135\cdot 500^2}\cdot K_{12}^2 =0.0231\cdot K_{12}^2; \]
\[ x_{23}=0; \]
\[ x_{33}=0.0532\cdot K_{12}^2;\quad \sin\lambda_1=0.4067;\quad \sin\lambda_2=0.588; \]
\[ \sin(\lambda_1+\lambda_2)=0.866. \]
Values of the parameters of the initial operating condition:
\[ U_{10}=U_{30}=1;\quad U_1=0.95;\quad U'_2=0.975;\quad U_{\mathrm{r}}=1.02; \]
\[ P_{10}=1;\quad P_{20}=1.3; \]
\[ P_1=0.3;\quad P_{\mathrm{load}}=0.9;\quad Q_{\mathrm{load}}=0.435;\quad \frac{dP}{dU}=2;\quad \frac{dQ}{dU}=4; \]
\[ U_{23}=U'_2K_{12}=\frac{0.9711}{2.066}\cdot K_{12}=0.47\cdot K_{12};\quad U_{13}=U_1=0.95; \]
\[ U_{33}=U_3\cdot K_{12}=0.491\cdot K_{12}. \]
Compute the values of the partial derivatives with respect to the independent variables from the autotransformer equations (p-5, p-6, p-7)
\[ \frac{\partial Q_1}{\partial U_1} = -\frac{2U_1}{x_1K_{12}^2} + \left(\frac{U_1U'_2}{x_1K_{12}^2}\right) \left[ \left(\frac{U_1U_2}{x_1K_{12}}\right)^2 -P_1^2 \right]^{-1/2} =-9.397, \]
\[ \frac{\partial Q_1}{\partial U'_2}=19.885,\quad \frac{\partial Q_1}{\partial P_1}=-0.0321,\quad \frac{\partial Q_1}{\partial K_{12}}=4.321, \]
\[ \frac{\partial Q_2}{\partial U'_1}=-9.389,\quad \frac{\partial Q_2}{\partial U'_2}=29.171,\quad \frac{\partial Q_2}{\partial P_2}=0.022,\quad \frac{\partial Q_2}{\partial K_{12}}=4.52, \]
\[ \frac{\partial Q_3}{\partial U'_2}=-9.232,\quad \frac{\partial Q_3}{\partial K_{12}}=0. \]
The values of the coefficients \(x_i,\ \gamma_i\) are determined from the equations of the static characteristics of the parts of the system.
Long-distance transmission line. From equation (p-1), \(\beta_1 = 1.298\);
\[ \gamma_1=\frac{1}{\beta_1}=0.77;\qquad \alpha_1=0. \]
Intermediate system. From equation (p-3),
\[ \frac{dU_2}{dU'_2}=2.845. \]
From equation (p-2), taking (p-3) and (p-4) into account,
\[ \alpha_2=\frac{dP_2}{dU_2}=\frac{dP_2}{dU'_2}\cdot\frac{dU'_2}{dU_2} =-P_{\text{nom}}\frac{dP}{dU}\frac{dU'_2}{dU_2}=-0.668; \]
the other coefficients are calculated similarly:
\[ \gamma_2=-0.222;\qquad \alpha_3=0;\qquad \gamma_3=0\quad (U_3=\text{const}). \]
We shall determine the derivative with respect to the transformation ratio by taking, as the independent variable, the number of taps of the regulated transformer:
\[ K'_{12}=\frac{U_{1\text{n}}+\Delta U\cdot N}{U_{2\text{n}}},\qquad \frac{dK}{dN}=\frac{\Delta U}{U_{2\text{n}}}=\frac{10}{242}=0.0413,\quad \Delta U=2\%U_{\text{n}1}=10\text{ kV}. \]
The increments of the parameters and their new values for \(N=+6\) are
\[ \Delta Q_1=\frac{D'_1}{D}\cdot\frac{dK}{dN}N =\frac{-12.32}{-28.1}\cdot0.0413\cdot6=0.1087, \]
\[ \Delta U_1=\gamma_1\Delta Q_1=0.0837,\qquad Q_1=0.1979+0.1087=0.3066; \]
\[ U_1=0.95+0.0837=1.0338. \]
The results of the calculations are given in the table.
Table
| Operating condition | \(N\) | \(U_1\), kV | \(U_2\), kV | \(U'_2\), kV | \(Q_1\), Mvar | \(Q_2\), Mvar | \(Q_3\), Mvar | \(P_1=P_2\), MW |
|---|---|---|---|---|---|---|---|---|
| Initial condition | 0 | 475 | 235 | 236 | 158,4 | 19,5 | 157 | 240 |
| Final condition (after changing the tap) | +6 | 516,8 | 232,8 | 235,2 | 245,4 | 51,4 | 222,0 | 247,7 |
| Increments | +6 | 41,8 | −2,2 | −0,8 | 87,0 | 31,9 | 65,0 | 4,7 |
Thus, under certain conditions the IS has fairly good regulating capabilities and makes it possible to change the operating parameters of a long-distance transmission line in the desired direction.
References
- A. F. Bermant. A Short Course in Mathematical Analysis. Nauka, Moscow, 1965.
- B. P. Demidovich, I. A. Maron. Fundamentals of Computational Mathematics. Nauka, Moscow, 1963.
- I. M. Markovich. Operating Conditions of Electric Power Systems. Gosenergoizdat, Moscow–Leningrad, 1963.
- Kh. F. Fazylov. Theory and Methods of Calculating Electric Power Systems. Academy of Sciences of the Uzbek SSR, Tashkent, 1953.
- L. M. Zisman. Calculation of the operating condition of a regulated transformer taking into account the static characteristics of power systems. Elektrichestvo, No. 12, 1965.
- Yu. I. Boryu, N. M. Grabovskaya. Calculation of short-circuit impedances of autotransformers with regulating windings. Elektrichestvo, No. 6, 1963.