ON A METHOD FOR CALCULATING CONTROLLABLE REACTIVE POWER IN ELECTRIC POWER TRANSMISSION

V. K. Shcherbakov

In most manuals on the calculation of electrical networks and power-transmission lines (A. A. Glazunov, A. Ya. Ryabkov, M. D. Kamenskii, and others), it is customary to set forth the theory for calculating controllable reactive power in power-transmission lines and the operating conditions of compensators in the following order: first, on the basis of the vector diagram of the voltages of a power-transmission line replaced by lumped constants, the required magnitude of the controllable reactive current in the power-transmission line is determined graphically (Fig. 1); then the use of the so-called circle diagram of the power-transmission line is shown for finding the reactive power in the power-transmission line corresponding to the specified voltages at the ends of the line and to the transmitted active power (Fig. 2); and, finally, formulas are given for the analytical determination of the compensator power.

Fig. 1

In all manuals, primary attention is given to the first method (“vector diagram”); it is presented in the greatest detail, while the method of the “circle diagram” is given as a supplement useful in certain practical cases.

Fig. 2

Both methods for calculating controllable reactive power in power-transmission lines are regarded as independent methods, and nowhere is it indicated that the “vector diagram” method is a special case of the general “circle diagram” method.

It is shown below that the “vector diagram” method indirectly repeats the “circle diagram” method, and that it is meaningful to combine these two methods into a single one and, on the whole, to simplify the process of calculating controllable reactive power in power-transmission lines.

1. An electric power transmission line without intermediate power tapping can in general form be represented by Fig. 2. The voltages at the ends of the transmission line or of a part of it are related by the equation

\[ \overline{U}_{1\phi}=\dot{A}U_{2\phi}+\dot{B}I_2, \tag{1} \]

to which the vector diagram of Fig. 3 corresponds. The current at the end of the transmission line, \(\overline{I}_2\), depends on the magnitude and phase of the voltages at the ends of the transmission line:

\[ \overline{I}_2=\frac{\overline{U}_{1\phi}-\dot{A}U_{2\phi}}{\dot{B}}; \]

for \(U_1=\mathrm{const}\) and \(U_2=\mathrm{const}\), the current \(\overline{I}_2\) is a function only of the angle \(\Theta_{12}\), as follows from equation (1) and from the vector diagram of Fig. 3.

Fig. 3

The power at the end of the transmission line is determined as

\[ \overline{S}_2=P_2-jQ_2=\sqrt{3}\cdot \overline{I}_2\cdot U_2 =\frac{\overline{U}_1U_2-\dot{A}U_2^2}{\dot{B}} . \tag{2} \]

If \(\overline{U}_1=U_1\angle\Theta^\circ\), \(\dot{A}=A\angle\alpha\), \(\dot{B}=B\angle\psi\), then the expression for \(\overline{S}_2\) is better represented in the following form:

\[ P_2-jQ_2=-\frac{A}{B}U_2^2\angle \alpha-\psi +\frac{U_1U_2}{B}\angle \psi-\Theta . \tag{3} \]

A graphical representation of equation (3) is given in Fig. 4; moreover, as was to be expected, the construction of the vector diagram of Fig. 3 is repeated here, only on a different scale and in different units of measurement. The sides of the polygon of Fig. 3 (voltages), multiplied by

\[ \frac{\sqrt{3}\,U_2}{\dot{B}}, \]

give powers; as a whole, the polygon of Fig. 4 is rotated relative to the polygon of Fig. 3 through the angle \(-\psi\).

Thus, by constructing the vector diagram of the transmission-line voltages, we thereby obtain, on another scale, the basis of the circle diagram of transmission-line powers. The latter makes possible a direct and more general analysis of the operating conditions of the transmission line and includes everything that the voltage diagram could give indirectly, although in the process of analysis it too is a circle diagram.

2. In the simplest case, when an electric transmission consisting only of a line represented by a П-shaped equivalent circuit (Fig. 5) is being studied, the construction of the circular power diagram is simplified still further. Corresponding to the equivalent circuit of Fig. 5,

\[ \overline{U}_{1\phi}=U_{2\phi}+\dot I^{\,\prime\prime}_2 \dot Z, \tag{4} \]

which is represented graphically in Fig. 6.

Fig. 4                    Fig. 5

Multiplying then the left- and right-hand sides of expression (4) by

\[ \frac{U_2\sqrt{3}}{Z}, \]

we obtain:

\[ P_2-jQ'_2=-\frac{U_2^2}{\dot Z}+\frac{\overline{U}_1U_2}{\dot Z} =-\frac{U_2^2}{Z}\angle-\psi+\frac{U_1U_2}{Z}\angle-\psi+\Theta_{12}, \]

which is represented graphically in Fig. 7.

Fig. 6

The circular power diagram of Fig. 7 makes it possible to obtain the value of the reactive power \(Q_2=Q'_2+Q''_c\), accompanying the transmitted active power \(P_2\) for given \(U_1\) and \(U_2\). For other values of \(U_1\) and \(U_2\), the corresponding circle is easily plotted and the reactive power of the electric transmission \(Q_2\), corresponding to any value of the active power of the electric transmission—\(P_2\), is determined.

What could be more visual and simpler?

If it is desired to avoid redrawing the vector diagram \((\Delta — OO'a)\), the vector diagram of Fig. 6 may be converted directly into a circular power diagram. To do this, it is necessary to add the power coordinates to the voltage vector diagram at the point \(O'\), as shown in Fig. 8.

Fig. 7

In the same way, the power coordinates are added to the voltage vector diagram of Fig. 3. Here the coordinate of active power and the vector \(-\dot{A}U_2\) form the angle \(\alpha-\psi\).

1. The following concepts should be distinguished: 1) regulating power and 2) compensator power.

The regulating power in a power transmission is obtained as the difference between the required (accompanying) reactive power in the power transmission—\(Q_2\), for given \(P_2\), \(U_2\), and \(U_1\), and the reactive power required by the load, i.e. \(Q_{\text{reg}} = Q_2 - Q_n\).

Note. Here and below, reactive powers of one direction are compared.

If \(Q_2 = Q_n\), no regulating intervention of reactive power is required. In the case when \(Q_2 > Q_n\), an additional consumer of reactive power is required; and, conversely, when \(Q_2 < Q_n\), a reactive-power generator is required at the end of the power transmission.

Fig. 8

In the general case, the receiving end of a power transmission is a system in which the reactive-power balance depends on the combined operation of many reactive-power generators and on the demand of the system load for reactive power. Here the power of the compensators required by the receiving-end system is determined by the reactive-power balance—

of the receiving end of the power transmission line. And only when the power transmission line feeds an end substation with a specified consumption of active \((P_n)\) and reactive \((Q_n)\) power from the substation, is the compensator power determined entirely by the magnitude of the regulating power, i.e., by the difference between \(Q_2\) and \(Q_n\).

Conclusions

1. The direct use of the vector diagram of voltages in a power transmission line for analyzing the behavior of reactive power in the transmission line is inadvisable.

2. The power-circle diagram should be used in all cases when it becomes necessary to analyze power and voltage conditions in power transmission lines.

3. In textbooks on electric networks, the vector diagram of voltages and the power-circle diagram of power transmission lines should be organically linked, and the latter should be made just as familiar to students and engineers as the vector diagram of voltages.

4. The concept of “regulating” reactive power in power transmission lines should be introduced, and it should be considered that determining the magnitude of the “regulating” power is the first stage in calculations; determining the compensator power is the second stage of the calculation.

5. It is necessary to accustom the student and the engineer to the understanding that determining the compensator power requires taking into account the operation of at least all generators and consumers of reactive power at the receiving end, and not merely the operation of the one given power transmission line.