On the Natural Mode of Compensated Lines
Abstract Generated abstract
This paper examines how the natural operating mode of long compensated power transmission lines differs from the corresponding mode in uncompensated lines. Using calculated compensated-line circuits and experimental confirmation, it analyzes voltage and current distributions, vector phase shifts, surge impedance, and reactive current components for lines with series capacitors and shunt inductors. The paper argues that compensation distorts voltage distribution and local phase shifts, while preserving equality of voltage and current magnitudes and phase displacement only at the line ends. It concludes that a compensated line in natural mode may carry reactive load current because its surge impedance can be complex, yet it still has no reactive losses and equal apparent power at both ends.
Full Text
ON THE NATURAL MODE OF COMPENSATED LINES
O. B. TOLPYGO
(Presented by the scientific seminar of the Faculty of Power Engineering)
The development of problems of transmitting electric power over long distances, carried out by Soviet scientists and engineers, has shown that the transmission of power of the order of the natural power over a distance of 1000–1200 km is entirely possible.
The natural operating mode of electric transmission, as is known, is characterized by the absence of reflected waves, which is indicated by the equality of the powers of the electric and magnetic fields of the line.
In the natural mode the power of the incident waves is completely absorbed by the load, whose resistance is equal to the surge impedance of the line.
The external indications of the natural mode are:
1) constancy of the magnitude of the effective voltage and current along the line;
2) equality of the shifts of the voltage and current vectors along the line to its wavelength;
3) absence of reactive losses and reactive load, i.e., equality of the power factor along the entire length of the line. On the basis of calculations of a whole series of compensated-line circuits, confirmed by experiment, the above-stated indications of the natural mode for compensated lines appear somewhat different.
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Speaking of the constancy of the moduli of the voltages along an uncompensated line, this cannot be asserted for a compensated line, into which concentrated resistances and conductances are inserted: capacitors for series compensation and inductive coils for shunt compensation. The presence of concentrated resistances causes a concentrated voltage drop in them, while concentrated conductances cause a reduction of the voltage at individual points of the line due to the flow through them of inductive current. As a result, the pattern of voltage distribution along the line is distorted. But in the natural mode the equality of the moduli of voltages and currents at the line ends is preserved. As an example one may cite the voltage-distribution curves along the line in the natural mode for certain circuits of compensated lines, shown in Figs. 1 and 2.
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The angles of shift of the voltage (current) vectors along individual points of a compensated line in the natural mode are not equal to the wavelengths of the sections, but depend on the point of insertion and on the type of compensating devices used. Only at the line ends is the shift of the voltage (current) vectors equal to the wavelength of the compensated lines (Figs. 3, 4). Thus, in the natural mode the voltage (current) vectors are shifted by an angle equal to the wavelength only at the line ends.
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The surge impedance of a compensated line in the general case is a complex quantity, containing, in addition to an active component—
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
..., as well as a reactive component. The natural mode is characterized by the fact that the load resistance will be equal to the surge impedance of the compensated line. Therefore the load current
\[ \dot I_2=\frac{\dot U_2}{Z_n}=\frac{\dot U_2}{Z_{ve}}=I_a+jI_r \]
is, in the general case, a complex quantity. Consequently, in the natural mode a current will flow along the line containing both an active and a reactive component. Table 1 gives the values of the active and reactive currents in the natural mode for several compensated lines.
Table No. 1
| No. | Type of line circuit | \(Z_{ve}\) | \(I_a\) | \(jI_r\) |
|---|---|---|---|---|
| 1 | series capacitor, then series inductor | \(0.88\angle -14^\circ 30'\) | 1.1 | 0.285 |
| 2 | series inductor, series capacitor, series inductor | 0.74 | 1.35 | 0 |
| 3 | series inductor, then series capacitor | \(0.88\angle 14^\circ 30'\) | 1.1 | -0.285 |
| 4 | shunt inductor at the left end, then series inductor | \(1.16\angle 14^\circ 50'\) | 0.83 | -0.22 |
| 5 | series inductor, shunt inductor in the middle, series inductor | 1.34 | 0.745 | 0 |
| 6 | series inductor, then shunt inductor at the right end | \(1.16\angle -14^\circ 50'\) | 0.83 | 0.22 |
| 7 | shunt inductor at the left end, then series inductor, series capacitor, series inductor | \(0.9\angle 11^\circ 30'\) | 1.09 | -0.225 |
| 8 | series inductor, shunt inductor, series capacitor, series inductor | \(1.0\angle -8^\circ 35'\) | 0.99 | 0.15 |
| 9 | series inductor, series capacitor, shunt inductor, series inductor | \(1.0\angle 8^\circ 35'\) | 0.99 | -0.15 |
| 10 | series inductor, series capacitor, series inductor, shunt inductor at the right end | \(0.9\angle -11^\circ 30'\) | 1.09 | 0.225 |
| 11 | series capacitor, series inductor, shunt inductor at the right end | \(1\angle -30^\circ 10'\) | 0.875 | 0.505 |
In the natural mode, when a reactive current flows due to the reactive component of the surge impedance, the equality of the voltage magnitudes at the ends of the line is not disturbed.
Reactive losses in the natural mode are absent regardless of whether the surge impedance has a reactive component or not. In fact, the power at the end of the line in the natural mode is
\[ W_2=\hat{U}_2 \dot{I}_2 . \]
The voltages and current at the beginning of the line are
\[ \dot{U}_1=A\dot{U}_2+B\dot{I}_2=U_2 e^{\lambda} \]
\[ \dot{I}_1=CU_2+D\dot{I}_2=I_2 e^{\lambda} \]
\[ W_1=\hat{U}_1\dot{I}_1=\hat{U}_2 e^{-\lambda} I_2 e^{\lambda}=U_2\dot{I}_2, \]
i.e., in the natural mode the apparent powers at the beginning and end of a compensated line are equal, and reactive losses in the line are absent, although reactive load currents do flow.
On the basis of the foregoing, the natural mode of compensated transmission lines should, in our opinion, be characterized by the following features:
1) constancy of the magnitudes of the effective voltages and currents at the ends of the compensated line;
2) equality of the shifts of the voltage (current) vectors by the electrical length of the line at its ends;
3) absence of reactive losses and equality of the power factor along the entire length of the line in the presence of a reactive load in the natural mode.