Influence of the Regulating Effects of Intermediate Power Systems on the Static Stability of Long-Distance Power Transmission

Yu. V. Khrushchev, V. I. Gotman

(Presented by the scientific seminar of the departments of electrical stations and electrical networks and systems)

Under the adopted plan for the electrification of the country, it is proposed to transmit large quantities of electric power to the European part from the eastern regions of the country over direct-current transmission lines with a voltage of 1.5 million V and alternating-current lines of 750 kV. In this case the alternating-current lines will inevitably play the role of intersystem ties between neighboring electric-power systems. In view of this, operating questions of the joint work of long-distance power transmissions and intermediate power systems acquire great importance. Moreover, it is proposed to take advantage of the favorable influence of intermediate power systems on the operating conditions of long-distance transmission lines and to expand the possibilities for transmitting energy by high-voltage alternating current.

The possibility and expediency of joint operation of long-distance power transmissions and intermediate power systems require comprehensive investigations. In the present article only one question of this problem is considered—the influence of various laws of variation of the exchange power between an intermediate power system and a long-distance power transmission on the aperiodic stability of a remote generating station. In order to simplify the calculations, the intermediate power system is assumed to be balanced in active and reactive power at the rated voltage on the medium-voltage buses of the tie autotransformer \(T_2\) (Fig. 1). In this case, obviously, the influence of voltage-regulating effects at the point of connection can be completely excluded only with smooth and inertia-free regulation. With the aid of the on-load tap changer of a real autotransformer, as is known, only stepwise voltage regulation can be carried out. Therefore, in each particular established operating condition the influence of the on-load tap changer of the tie autotransformer will not be manifested, and, in estimating the aperiodic stability of the long-distance transmission according to the practical criterion \(\dfrac{\partial P}{\partial \delta}\), the real regulating effects of the intermediate power system should be taken into account. If the latter are such that the derivative \(\dfrac{\partial P}{\partial \delta}\) increases its value, then, obviously, one may speak of a favorable influence of the regulating…

...ing effects of the intermediate power system on the aperiodic stability of a long-distance transmission line.

With a simplified allowance for the automatic voltage regulation (AVR) of the generators of the remote power station \(G_1\), and in the absence of regulating effects of the balanced intermediate power system, for the circuit (Fig. 1) the following equation of small oscillations will be valid [1]:

\[ T_j p^2 \Delta \delta_1 + S_{Ex\delta}\Delta \delta_1 = 0, \tag{1} \]

where \(T_j\) is the inertia constant;
\(S_{Ex\delta}\) is the synchronizing torque, determined with the e.m.f. of the generator constant, applied behind a certain reactance \(x\).

Taking into account the characteristics of the intermediate power system, the equation of small oscillations of the rotors can be written in a somewhat different form [1, 2]:

\[ \left(T_j p^2 + \frac{\partial P_{1A}}{\partial \delta_1}\right)\Delta \delta_1 + \frac{\partial P_{1A}}{\partial \delta_A}\Delta \delta_A + \frac{\partial P_{1A}}{\partial U_A}\Delta U_A . \tag{2} \]

The latter equation should be supplemented by the equations of active- and reactive-power balance at the junction point:

\[ \begin{aligned} \Delta P_{A1} + \Delta P_{A0} + \Delta P_C &= 0,\\ \Delta Q_{A1} + \Delta Q_{A0} + \Delta Q_C &= 0. \end{aligned} \tag{3} \]

Expressing the increments of powers \(\Delta P\) and \(\Delta Q\) in the last equations through deviations of the parameters \(\Delta U_A\), \(\Delta \delta_A\), \(\Delta \delta_1\), we obtain the balance equations in the following form:

\[ \left. \begin{aligned} \frac{\partial P_{A1}}{\partial \delta_1}\Delta \delta_1 + \left(\sum \frac{\partial P}{\partial \delta_A} + K_{P\delta}\right)\Delta \delta_A + \left(\sum \frac{\partial P}{\partial U_A} + K_{PU}\right)\Delta U_A &= 0,\\ \frac{\partial Q_{A1}}{\partial \delta_1}\Delta \delta_1 + \left(\sum \frac{\partial Q}{\partial \delta_A} + K_{Q\delta}\right)\Delta \delta_A + \left(\sum \frac{\partial Q}{\partial U_A} + K_{QU}\right)\Delta U_A &= 0, \end{aligned} \right\} \tag{4} \]

where

\[ \sum \frac{\partial P}{\partial \delta_A} = \frac{\partial P_{A1}}{\partial \delta_A} + \frac{\partial P_{A0}}{\partial \delta_A}; \]

\[ \sum \frac{\partial P}{\partial U_A} = \frac{\partial P_{A1}}{\partial U_A} + \frac{\partial P_{A0}}{\partial U_A}; \]

\[ \sum \frac{\partial Q}{\partial \delta_A} = \frac{\partial Q_{A1}}{\partial \delta_A} + \frac{\partial Q_{A0}}{\partial \delta_A}; \]

Fig. 1. Transmission scheme.

\[ \sum \frac{\partial Q}{\partial U_A} = \frac{\partial Q_{A1}}{\partial U_A} + \frac{\partial Q_{A0}}{\partial U_A}. \]

By \(K_{P\delta}, K_{PU}, K_{Q\delta}, K_{QU}\) are denoted the regulating effects of the active and reactive power of the intermediate power system with respect to the angle and magnitude of the voltage vector \(\overline{U}_A\).

Eliminating the variables \(\Delta \delta_A\) and \(\Delta U_A\) from the system of equations (2) and (4), we obtain the equation of rotor motion in the following form:

\[ a_0 T_j p^2 \Delta \delta_1 + a_2 \Delta \delta_1 = 0, \tag{5} \]

where

\[ a_0 = \left( \sum \frac{\partial P}{\partial \delta_A} + K_{P\delta} \right) \left( \sum \frac{\partial Q}{\partial U_A} + K_{QU} \right) - \left( \sum \frac{\partial Q}{\partial \delta_A} + K_{Q\delta} \right) \left( \sum \frac{\partial P}{\partial U_A} + K_{PU} \right); \]

\[ a_2 = \left| \begin{array}{ccc} \dfrac{\partial P_{1A}}{\partial \delta_1} & \dfrac{\partial P_{1A}}{\partial \delta_A} & \dfrac{\partial P_{1A}}{\partial U_A} \\[1.2em] \dfrac{\partial P_{A1}}{\partial \delta_1} & \displaystyle \sum \frac{\partial P}{\partial \delta_A} + K_{P\delta} & \displaystyle \sum \frac{\partial P}{\partial U_A} + K_{PU} \\[1.2em] \dfrac{\partial Q_{A1}}{\partial \delta_1} & \displaystyle \sum \frac{\partial Q}{\partial \delta_A} + K_{Q\delta} & \displaystyle \sum \frac{\partial Q}{\partial U_A} + K_{QU} \end{array} \right|. \]

Comparing equations (1) and (5), we find that the synchronizing torque of the line, taking into account the regulating effects of the intermediate power system, is equal to the ratio of the coefficients \(a_2\) and \(a_0\).

\[ S_{Ex\delta} = \frac{a_2}{a_1} \tag{6} \]

Let us determine how each of the regulating effects individually influences the synchronizing torque \(S_{Ex\delta}\). As an example, consider a long-distance lossless power transmission line of length \(l = 1200,\ 2400\), and \(3600\) km. The voltages at the ends are assumed constant and equal to the rated value; the intermediate power system is connected at the midpoint of the line (Fig. 1).

The results of calculations for the natural operating condition are presented in Figs. 2, 3, 4. In these figures, regions of favorable action of the regulating effects are distinguished (solid lines), as are regions of unfavorable action (dashed lines). It is characteristic that the dependences \(S_{U1\delta} = f_1(K_{QU})\) and \(S_{U1\delta} = f_2(K_{P\delta})\) (Figs. 2, 3) have the form of hyperbolic curves with two branches; moreover, for lines shorter than 3000 km, one of the branches is entirely located in the region of negative values of the regulating effects \(K_{P\delta}\) and \(K_{QU}\) and has a favorable effect on the synchronizing torque \(S_{U1\delta}\). The second branches of these curves increase the synchronizing torque only for positive values of the regulating effects. The regions of unfavorable influence shrink as the line length approaches 3000 km and lie within the following limits:

\[ 0 > K_{P\delta} > \frac{ \displaystyle \sum \frac{\partial P}{\partial U_A} \cdot \sum \frac{\partial Q}{\partial \delta_A} - \sum \frac{\partial P}{\partial \delta_A} \cdot \sum \frac{\partial Q}{\partial U_A} }{ \displaystyle \sum \frac{\partial Q}{\partial U_A} }; \]

Fig. 2. Dependences \(S_{U_1\delta}=f_1(K_{QU})\) for \(P_{*1}=P_{*2}=1,\ \lambda_1=\lambda_2\):
1—\(l=1200\) km; 2—\(l=2400\) km; 3—\(l=3600\) km.

\[ 0<K_{QU}> \frac{ \sum \frac{\partial P}{\partial U_A}\cdot \sum \frac{\partial Q}{\partial \delta_A} - \sum \frac{\partial P}{\partial \delta_A}\cdot \sum \frac{\partial Q}{\partial U_A} }{ \sum \frac{\partial P}{\partial \delta_A} }. \]

The dependences \(S_{\Pi 1\delta}=f_3(K_{PU})\) and \(S_{U1\delta}=f_4(K_{Q\delta})\) coincide under the adopted conditions and are expressed by straight lines (Fig. 4).

For lines with \(l<3000\) km, the negative values of the regulating effects \(K_{PU}\) and \(K_{Q\delta}\) exert a favorable influence. The influence of these effects is intensified as the line length approaches \(l=3000\) km.

For lines of length \(l>3000\) km, all the regulating effects exert, in a certain sense, the opposite influence compared with lines of length \(l<3000\) km.

Thus, for a line with \(l=3600\) km, the regulating effects \(K_{P\delta}\) and \(K_{QS}\) exert a negative influence for practically all their values (Fig. 3), with the exception of a small segment located on the positive semiaxis. The regulating effects \(K_{PU}\) and \(K_{Q\delta}\) increase the synchronizing torque when their values are positive. In this case it may prove expedient to connect the lines to the load buses of intermediate power systems, which, as is known, possess positive regulating effects of active power with respect to voltage.

Fig. 3. Dependences \(S_{U_1\delta}=f_2(K_{p\delta})\) for \(P_{*1}=P_{*2}=1;\ \lambda_1=\lambda_2\):
1—\(l=1200\) km; 2—\(l=2400\) km; 3—\(l=3600\) km.

Fig. 4. Dependences \(S_{u1\delta}=f_3(K_{pu})\) and \(S_{u1\delta}=f_4(K_{q\delta})\) for \(P_{*1}=P_{*2}=1\); \(\lambda_1=\lambda_2\):
1 — \(l=1200\) km; 2 — \(l=2400\) km; 3 — \(l=3600\) km.

References

1. V. A. Venikov. Transient electromechanical processes in electrical systems. Energiya, 1964.

2. L. V. Tsukernik. Analysis of the coefficient matrix of the equations of disturbed motion of a complex power system and determination of the order of the characteristic equation. Collection of papers of the Institute of Electrotechnics, issue 16. Publishing House of the Academy of Sciences of the Ukrainian SSR, 1959.

3. V. A. Venikov, L. V. Zhukov. Regulation of the operating conditions of electric power systems and long-distance power transmissions and improvement of their stability by means of controllable static IRM. Elektrichestvo, No. 6, 1967.