CORONA ON A LINE WITH A GROUNDED PHASE

V. K. Shcherbakov

Professor, Doctor of Technical Sciences

Critical Corona Voltage

There is reason to suppose that the critical corona voltage under the conditions of a three-phase line with one grounded phase [1, 2] will be different than under the conditions of a symmetrical three-phase line, since, other conditions being equal, the potentials of the conductors, the charges, and the configuration of the fields in these two cases are entirely different. However, despite these differences, in both cases Peek’s expression [3] for the gradient of the visible corona on the surface of smooth parallel cylinders of radius \(r\), at air density \(\delta\), may be considered valid:

\[ E_v = 30\delta \left(1 + \frac{0.301}{\sqrt{r \cdot \delta}}\right)\ \text{kV}_{\max}/\text{cm}. \tag{1} \]

The disruptive gradient corresponding to the onset of a self-sustained discharge in air between parallel conductors having a radius greater than \(0.2\) cm was determined by Peek as

\[ E_0 = 30\ \text{kV}_{\max}/\text{cm}. \]

Further, on the basis of the geometry of the electric field between two parallel cylinders, there are relations between the maximum and minimum gradients—\(E_{\max}\) and \(E_{\min}\)—on the one hand, and the potential difference \(U\) applied to the cylinders, on the other [4]:

\[ E_{\max} = \frac{U}{ r \sqrt{\frac{D-r}{D+r}} \cdot \ln \frac{D+\sqrt{D^2-r^2}}{r} }, \tag{2} \]

\[ E_{\min} = \frac{U}{ r \sqrt{\frac{D+r}{D-r}} \cdot \ln \frac{D+\sqrt{D^2-r^2}}{r} }, \tag{3} \]

where \(D\) is the distance between the cylinders.

In high-voltage power-transmission lines, the distances between the conductors \(D\) are incommensurably greater than the radius of the conductors, and therefore the difference between the maximum and minimum gradients on the surface of the conductor disappears, and formulas (2) and (3) take the form

\[ E = \frac{U}{r \ln \frac{D}{r}}. \tag{4} \]

Peek’s well-known formula for calculating the critical disruptive voltage was obtained from expression (4) with the addition of coefficients,

taking into account the surface condition—\(m_{0}\), the weather condition—\(m_{\mathrm{w}}\), and the air pressure—\(\delta\):

\[ U_{\mathrm{cr}(\phi)}=21.1\cdot m_{0}\cdot m_{\mathrm{w}}\cdot \delta\cdot r\cdot \ln \frac{D}{r}\ \text{kilovolts}. \tag{5} \]

Thus, for practical calculations it is customary to assume the field near the surface of the conductors to be uniform around the circumference, and, consequently, the charge density over the surface of the conductors to be the same. But then the gradient at the surface of the conductor can be calculated as a function of the charge of the conductor \(q\) as

\[ E=\frac{q\ (\text{coul/km})}{2\Pi r\ (\text{cm})}\cdot \frac{1}{8.84\cdot 10^{-6}}\ \text{kV/cm}, \]

and the disruptive critical gradient as

\[ E_{0}=\frac{q_{0}}{2\Pi r}\cdot \frac{1}{8.84\cdot 10^{-6}}\ \text{kV/cm}, \tag{6} \]

i.e., to the critical (disruptive) gradient there corresponds a quite definite critical charge on the conductor, and, bearing in mind the assumption of uniformity of the field around the conductor at its surface, one that does not depend on the mutual arrangement of the conductors and on the ratio of the conductor potentials.

The charges of the conductors of the line can be calculated from the conditions that:

\[ \begin{aligned} \overline{q}_{1}&=C_{11}\cdot \overline{U}_{10}+C_{12}(\overline{U}_{10}-\overline{U}_{20})+C_{13}(\overline{U}_{10}-\overline{U}_{30}),\\ \overline{q}_{2}&=C_{12}(\overline{U}_{20}-\overline{U}_{10})+C_{22}\cdot \overline{U}_{20}+C_{23}(\overline{U}_{20}-\overline{U}_{30}),\\ \overline{q}_{3}&=C_{13}(\overline{U}_{30}-\overline{U}_{10})+C_{23}(\overline{U}_{30}-\overline{U}_{20})+C_{33}\cdot \overline{U}_{30}. \end{aligned} \tag{7} \]

For a symmetrical line, provided the condition

\[ \overline{U}_{20}=a\cdot \overline{U}_{10}, \]

\[ \overline{U}_{30}=a^{2}\cdot \overline{U}_{10}, \]

is observed, the charge, let us say, of the conductor of the 2nd phase according to (7) is obtained equal to

\[ \overline{q}=C_{12}(a\cdot U_{10}-U_{10})+C_{22}\cdot a\cdot U_{10}+C_{23}(a\cdot U_{10}-a^{2}U_{10})= \]

\[ =U_{1}\left[C_{12}(a-1)+C_{22}\cdot a+C_{23}(a-a^{2})\right]. \tag{8} \]

The same on the conductor of the 2nd phase of a line with a grounded phase (phase 1—\(a\) is grounded), when the condition holds

\[ U_{10}=0, \]

\[ \overline{U}_{20}=(a-1)\cdot U_{1}, \]

\[ \overline{U}_{30}=(a^{2}-1)\cdot U_{1}, \]

the charge is equal to

\[ \overline{q}'_{2}=C_{12}[(a-1)\cdot U_{1}-0]+C_{22}(a-1)\cdot U_{1}+C_{23}(a-1)\cdot U_{1}-(a^{2}-1)\cdot U_{1}]= \]

\[ =U_{1}\left[C_{12}(a-1)+C_{22}(a-1)+C_{23}(a-a^{2})\right]. \tag{9} \]

Expressions (8) and (9) indicate a linear dependence of the charges on the phase voltage of the line and make it possible to establish a constant ratio of the charges of a line with a grounded phase and of a symmetrical line (in the present case, of the 2nd phase), at one and the same nominal voltage. This ratio will be equal to

\[ \frac{\bar q'_2}{q_2} = \frac{C_{12}(\alpha - 1) + C_{22}(\alpha - 1) + C_{23}(\alpha - \alpha^2)} {C_{12}(\alpha - 1) + C_{22}\cdot \alpha + C_{23}(\alpha - \alpha^2)} \]
\[ = \frac{\alpha(C_{12}+C_{22}) + C_{23}(\alpha-\alpha^2) - (C_{12}+C_{22})} {\alpha(C_{12}+C_{22}) + C_{23}(\alpha-\alpha^2) - C_2}. \tag{10} \]

Or, since \(\alpha-\alpha^2=-j\sqrt{3}\), then

\[ \frac{\bar q'_2}{q_2} = \frac{ -\dfrac{3}{2}(C_{12}+C_{22}) - j\dfrac{\sqrt{3}}{2}\,(C_{12}+C_{22}+2C_{23}) }{ -\dfrac{1}{2}(C_{12}+C_{22}+2C_{12}) - j\dfrac{\sqrt{3}}{2}\,(C_{12}+C_{22}+2C_{23}) } = \]
\[ = \frac{ 3(C_{12}+C_{22}) + j\sqrt{3}(C_{12}+C_{22}+2C_{23}) }{ (3C_{12}+C_{22}) + j\sqrt{3}(C_{12}+C_{22}+2C_{23}) }. \tag{11} \]

The ratio of the absolute values of the charges from (11) is obtained in the form

\[ \frac{q'_2}{q_2} = \sqrt{ \frac{ [3(C_{12}+C_{22})]^2 + 3[C_{12}+C_{22}+2C_{23}]^2 }{ [3C_{12}+C_{22}]^2 + 3[C_{12}+C_{22}+2C_{23}]^2 } }. \tag{12} \]

On the basis of exactly the same reasoning, the charge ratio for the 3rd phase has the form

\[ \frac{q'_3}{q_3} = \sqrt{ \frac{ [3(C_{13}+C_{33})]^2 + 3[C_{13}+C_{33}+2C_{23}]^2 }{ [3C_{13}+C_{33}]^2 + 3[C_{13}+C_{33}+2C_{23}]^2 } }. \tag{13} \]

Formulas (12) and (13) show that the charge ratio is determined only by the position of the conductors on the supports; therefore these ratios are better associated not with the phases of the transmission line, but with the conductors of the line.

With application to the dimensions of the Shatura 110-kV supports (Fig. 1), the author calculated the partial capacitances and the charge ratios of the conductors of two phases for cross sections AC-185 and AC-120, respectively: 1) for the ordinary (symmetrical) state of the line and 2) for the state when the upper conductor (conductors) is grounded, but without changing the position of the upper conductor on the support; the results of the calculations are given in Tables 1 and 2.

The data of Tables 1 and 2 indicate that, for a definite arrangement of the conductors on the supports, the ratio of the charges on conductors of the same name is almost independent of their cross sections. Thus, for a definite type of support (in the sense of a definite arrangement of the conductors on the supports), with unchanged distances between conductors, a constant value of the ratio of the charges of conductors of the same name can be established, independent of the conductor cross section. The data of the same tables indicate that, at identical phase (or interphase) voltages, the charges on the conductors of a line with a grounded phase are greater than the charges on conductors of the same name of a symmetrical line, and that the degree of this difference in charges is greater for single-circuit lines than for double-circuit lines.

Fig. 1

Table 1. Two circuits in operation

Table 2. One circuit in operation, without taking into account the influence of the wires of the other circuit

In a line with a grounded phase, therefore, the charges on the ungrounded wires will reach the value corresponding to the disruptive gradient at the surface of the wires at lower voltages than is the case in symmetrical power transmissions. From equations (8) and (9) it follows that equal charges on the wires in the two transmission systems under consideration correspond to different phase voltages, and the ratio of these voltages at $q'=q$ is expressed as

\[ \frac{U'}{U}= \frac{C_{12}(a-1)+C_{22}\cdot a+C_{23}(a-a^2)} {C'_{12}(a-1)+C'_{22}(a-1)+C'_{23}(a-a^2)}. \tag{14} \]

Comparing (10) and (14), we note that

\[ \frac{U'}{U}=\frac{q}{q'}. \tag{15} \]

Consequently, upon transition to operation with a grounded phase, the critical corona voltage of the wires of a given line is reduced to the same extent as the charges of the wires increase. Thus, in the example considered above (Table 1), the critical corona voltage of the 2nd wire with two circuits on the towers is reduced by a factor of 1.19, and the critical voltage of the 3rd wire by a factor of 1.24. In single-circuit lines this reduction is more significant and, according to Table 2, is expressed by a factor of 1.28 for the 2nd wire and by a factor of 1.33 for the 3rd wire.

The protective role of the upper wire requires a certain elevation of this wire above the other two wires, as is shown by way of example in Fig. 2, where, while preserving all other tower dimensions according to Fig. 1, the upper wires are additionally raised by two meters. In this case, when the upper wires of the line are grounded, the degree of reduction of the critical corona voltage is smaller, as is seen from Tables 3 and 4.

Table 3. Double-circuit line

Table 4. One circuit, without taking into account the influence of the neighboring circuit

Calculating the magnitude of the critical corona voltage for the wires of a line with one grounded phase by comparing the charges of the wires of such a line with the charges of the wires of a line with a symmetrical system of wire potentials and with the usual arrangement of the wires on the towers does not make sense. Usually the wires of lines on towers are arranged in such a way that the charges of the wires of one and the same line are unequal in magnitude, and hence, as a consequence, the critical corona voltages of the wires are unequal. [1 and to [3] indicates that, with a horizontal arrangement of the wires on the towers, the critical voltage of the outer wires is 6% higher, and that of the middle wire 4% lower, than the critical voltage of the same wires when they are arranged at the vertices of an equilateral triangle.

It is most convenient to calculate the magnitude of the critical corona voltage of a line with a grounded phase on the basis of the following considerations.

Fig. 2

As was shown earlier, the critical (disruptive) field strength on the surface of a conductor can be expressed in terms of the magnitude of the conductor charge

\[ E_{\mathrm{cr}}=\frac{q_{\mathrm{cr}}(\mathrm{coul}/\mathrm{km})}{2\pi r(\mathrm{cm})}\cdot\frac{1}{8.84\cdot10^{-6}}\ \mathrm{kV/cm}; \]

taking \(E_{\mathrm{cr}(\phi)}=21.1\ \mathrm{kV/cm}\) (according to Peek), we find the magnitude of the charge corresponding to the appearance of the critical field strength on the surface of the conductor in the form

\[ q_{\mathrm{cr}}=21.1\cdot 2\pi\cdot r\cdot 8.84\cdot10^{-6}\ \mathrm{coul/km}. \]

But since for each conductor the relation

\[ q=U_{\phi} C_{\mathrm{work}} \]

actually holds, then

\[ U_{\mathrm{cr}(\phi)}=\frac{q_{\mathrm{cr}}}{C_{\mathrm{work}}} =\frac{r}{C_{\mathrm{work}}}\cdot 21.1\cdot2\Pi\cdot8.84\cdot10^{-6}\cdot10^{-3} =1.175\frac{r}{C_{\mathrm{work}}}\cdot10^{-6}\ \mathrm{kV}. \]

Introducing into formula (16) coefficients accounting for the condition of the conductor surface (\(m_o\)) and the weather condition (\(m_p\)), as well as the factor \(\delta\)—the air density, we finally obtain an expression for calculating the critical corona voltage in the form

\[ U_{\mathrm{cr}(\mathrm{ph})}=m_o\cdot m_p\cdot\delta\cdot1.17\cdot\frac{r}{C_{\mathrm{work}}}\cdot10^{-6}\ \mathrm{kV}. \tag{17} \]

Here \(r\) is in cm and \(C_p\) is in farads per kilometer.

Formula (17) is equally valid both for lines with a grounded phase and for symmetrical lines.

The working capacitance \(C_p\) from (7), let us say, for the second conductor, is obtained as

\[ C_{p(2)}=-\frac{q_2}{U_{\phi}}; \]

but from (9) it follows that

\[ q_2=U_{\phi}\cdot \sqrt{\frac{9}{4}(C_{12}+C_{22})^2+ \frac{3}{4}(C_{12}+C_{22}+C_{23})^2}, \]

therefore,

\[ C_{p(2)}=\frac{1}{2} \sqrt{9(C_{12}+C_{22})^2+3(C_{12}+C_{22}+2C_{23})^2}; \tag{18} \]

and in exactly the same way for the 3rd conductor—

\[ C_{p(3)}=\frac{1}{2} \sqrt{9(C_{13}+C_{33})^2+3(C_{13}+C_{33}+2C_{23})^2}. \tag{19} \]

Table 5 gives the values of the working capacitances and critical voltages for AC-120 and AC-185 conductors and the towers according to Fig. 2. The data of Table 5 show that corona on 110 kV lines with a grounded phase presents no practical difficulties for double-circuit lines, since AC-120 conductors (radius—0.765 cm) do not corona, and it appears possible to use AC-95 conductors (radius—0.695). As regards single-circuit 110 kV lines, however, it is necessary to note the difficulties involved in using conductors of radii smaller than 0.765 cm, since AC-120 conductors (radius—0.765 cm) in good weather may already produce weak corona at the beginning of the line.

Table 5

It is of interest to note that in all practically possible cases the grounded conductor will not develop corona. Indeed, according to (7), the value of the operating capacitance of the grounded conductor will be equal to

\[ C_{\mathrm{op}(1)} \simeq 3 C_{12}, \]

and this means (see Tables 3, 4, and 5) that the relation will always hold

\[ 2 C_{\mathrm{op}(1)} < C_{\mathrm{op}(2)}, \]

i.e., the critical voltage of the grounded conductor will always be approximately twice as high as the critical voltage of the two other conductors (see formula 16). True, the cross-section of the grounded conductor may be taken as less than half the cross-sections of the two other conductors, but even in this case the radius of the grounded conductor will be smaller than the radius of the two other conductors only by a factor of \(\sqrt{2}\), and, consequently, the critical corona voltage of this conductor will always be higher than the operating voltage of the electric transmission line.

Also deserving attention is the question of the influence, on the magnitude of the critical corona voltage, of a neighboring circuit when that circuit is disconnected and grounded. In the preceding calculations it was assumed that there was a complete absence of influence of the neighboring disconnected circuit, when single-circuit lines or the operation of only one circuit were under consideration. However, as calculation examples show, the influence of a neighboring disconnected and grounded circuit nevertheless exists in lines with a grounded phase. In order to clarify this question, the author calculated the partial capacitances of a line with AC-120 conductors arranged on the support according to Fig. 2, when one circuit is in operation and the other is disconnected and grounded; Table 6 gives the values of the operating capacitances

critical corona voltage for three characteristic states of this line.

Table 6

As was to be expected, the presence of a neighboring circuit, located on the same towers and in a disconnected and grounded state, increases the operating capacitance of the conductors and, consequently, lowers the critical corona voltage of the operating circuit, if this operating condition of a double-circuit line is compared with operation of a single-circuit line.

220 kV lines are usually built with AC-400 or ACU-400 conductors. For the tower shown in Fig. 3 the operating capacitances are:

\[ C_{p(2)} = C_{p(3)} = 0.01070 \cdot 10^{-6}\ \text{F/km}, \]

\[ C_{p(1)} = 0.00303 \cdot 10^{-6}\ \text{F/km}; \]

the critical corona voltage, for \(m_p = 1\) and \(\delta = 1\), is obtained as follows:

a) for the lower conductors

\[ U_{\text{cr}(ph-ph)} = \sqrt{3}\cdot 0.85 \cdot 1.17 \frac{1.39}{0.0107} = 224\ \text{kV}, \]

b) for the upper conductor

\[ U_{\text{cr}(ph-ph)} = \sqrt{3}\cdot 0.85 \cdot 1.17 \frac{1.075}{0.00303} = 610\ \text{kV}. \]

Fig. 3

The critical voltage could have been raised somewhat in this case by using ACU-400 conductors \(\left(U_{\text{cr}(ph-ph)} = 233\ \text{kV}\right)\); the second circuit, located at a short distance, could also have produced the desired effect.

Corona losses

Power loss due to corona in lines is usually calculated by Peek’s formula:

\[ P_k = \frac{241}{\delta}(f+25) \sqrt{\frac{r}{D}} \left(U - U_{\text{cr}}\right)^2 \cdot 10^{-5}\ \text{kW/km}, \tag{20} \]

where
\(\delta\) — the density of air;
\(f\) — the frequency of the alternating current;
\((U-U_{\mathrm{cr}})\) — the difference between the operating and critical voltage of the line; in calculating losses for one conductor, the phase values of \(U\) and \(U_{\mathrm{cr}}\) are substituted.

However, this formula, being experimental, does not indicate the mechanism by which corona losses are formed and therefore, when analyzing this formula, it is not possible to notice how the losses change in passing to lines with a grounded phase.

Holm gives a formula for calculating corona losses of a more complicated form than Peek’s formula, but Holm’s formula was obtained as the result of logical reasoning taking into account modern physical conceptions of phenomena in air under corona [3]. According to Holm, the energy loss to corona per half-period is equal to

\[ W = U_{\mathrm{av}} \cdot Q, \tag{21} \]

where \(U_{\mathrm{av}}\) is a certain constant average value of the voltage between the conductors at which corona losses occur;
\(Q\) is the quantity of electricity that must be supplied from the energy source to create the space charge in the space surrounding the corona and to neutralize part of the space charge from the preceding half-period.

On the basis of relation (21), an expression is obtained for the power lost to corona in the form

\[ P_{\mathrm{k}} = W \cdot 2f = 2f \cdot U_{\mathrm{av}} \cdot Q. \tag{22} \]

In the subsequent arguments, according to Holm, the quantity of electricity \(Q\) is calculated as a complex capacitive charge, the magnitude of which is determined by the voltage between the conductors and the geometrical dimensions of the line.

But for one and the same \(U_{\mathrm{av}}\), and on the conductors of one and the same line, the quantity of electricity \(Q\) in formulas (21) and (22) will be different depending on the operating regime of the line: in the case of operation of a line with a grounded phase, as was shown above, the magnitude \(Q\) is greater than in the case of operation of the same line under the usual (symmetrical) regime, in the ratio

\[ \frac{C_{\mathrm{op.}\,(\mathrm{ground.})}}{C_{\mathrm{op.}\,(\mathrm{symm.})}} = \beta. \]

The corona power in a line with a grounded phase can therefore be calculated as

\[ P'_{\mathrm{k}} = 2f \cdot U_{\mathrm{av}} \cdot (\beta \cdot Q). \tag{23} \]

Taking the ratio of (23) and (22), we find that

\[ \frac{P'_{\mathrm{k}}}{P_{\mathrm{k}}} = \frac{2f \cdot U_{\mathrm{av}}(\beta \cdot Q)} {2f \cdot U_{\mathrm{av}} \cdot Q} = \beta, \]

i.e., the power loss to corona in a line with a grounded phase is \(\beta\) times greater than in a symmetrical line.

Consequently, in order to determine the corona power in the ungrounded conductors of a line with a grounded phase, it is necessary to find this power by the known formulas for the symmetrical operating regime of the line, and then to increase this result in the ratio

\[ \frac{C_{\mathrm{p}\,(\mathrm{ground.})}}{C_{\mathrm{p}\,(\mathrm{symm.})}}: \]

\[ \Delta P_{\mathrm{ground.}} = \beta \cdot \Delta P_{\mathrm{symm.}} \]

Experimental verification

In order to verify the propositions for determining corona losses in lines with a grounded phase, a series of experiments was carried out with measurement of the power losses due to corona in an experimental line with a grounded phase. The measurements were made on a 3-phase line 30 meters long, with iron conductors 3.2 mm in diameter and arranged at the vertices of an equilateral triangle. The line was supplied from a three-phase transformer of experimental type, with a power of 30 kW and a voltage of 40 kV. The line was mounted in the open air; the transformer was indoors; the voltage was brought out through porcelain bushings mounted in wooden panels.

Fig. 4

The active power was measured with a milliwattmeter specially made for this purpose, operating on the ferrodynamic principle, with a measuring range up to 200 milliwatts; the voltage measuring range was 100 V, the resistance of the voltage coil 3000 Ω; the full current measuring range 5–6 mA, the resistance of the current coil 120 Ω. The error of the measurements was not more than 0.5% of the scale limit.

When measuring losses in the symmetrical mode, the wattmeter was connected according to the Peek–Hendricks circuit, i.e. the current winding was connected at the ends of the phase windings before the neutral, which was grounded, and the voltage winding—to the greater part of the turns of the phase high-voltage windings near the neutral (Fig. 4).

Fig. 5

Fig. 6

Grounding one of the phases required insulating the neutral from ground for the phase voltage and therefore, in this mode of transformer operation, the wattmeter could not be connected according to the Peek–Hendricks circuit. Only the winding of the grounded phase at the grounding point was accessible, and therefore the instrument was connected at this point of the winding, as shown in Fig. 5. Assuming equality of the losses in the ungrounded conductors,

which is close to reality, since, according to the above considerations (see Table 6), the operating capacitances and critical voltages of ungrounded conductors differ little even when the distances of these conductors to the ground and to the grounded conductor are sharply unequal; one may conclude that the magnitudes of the active components of the currents of the ungrounded conductors are equal and, consequently, that it is possible to measure, in the grounded phase of the transformer, one third of the total active power lost by the transmission line to corona. In Fig. 6 a vector diagram of the voltages and currents of the transmission line is shown in accordance with the proposition that the active currents in the phases are equal; in doing so it is quite permissible to regard the symmetry of the phase voltages as undisturbed, since in our experiments the load on the transformer was very small and, correspondingly, the voltage drop in the transformer windings was very small. The capacitive current in the grounded phase leads the phase voltage by \(90^\circ\) and therefore will not be registered by the wattmeter. Thus, if the measured power of the grounded phase is denoted by \(P_1\), then the total power of the transmission line expended on corona is determined as

Fig. 7

Fig. 8

\[ P_k = 3P_1 . \tag{26} \]

On the basis of the experiments carried out, the curves of Fig. 7 and Fig. 8 were constructed. In the upper part of Figs. 7 and 8 the critical corona voltage was found; it was assumed that, during the experiments, the proportionality between the voltage applied to the line and the current of active conductivity remained unchanged, excluding the conductivity current due to corona. The curves shown and many other measurements and calculations confirmed this assumption within the range of voltages used in the experiments. At the bottom of Figs. 7 and 8 the degree of agreement between the measured and calculated values is shown.

...the corona-loss curves as a function of voltage. One can see good agreement between the results of calculation by Peek’s formula and the experimental data; moreover, the losses calculated by Peek’s formula were in each case multiplied by the coefficients

\[ \beta=\frac{C_{\text{op. (ground.)}}}{C_{\text{op. (sym.)}}}. \]

Since the line conductors had a radius of less than \(0.2\) cm, on Peek’s recommendation the losses were calculated by the formula

\[ P_k=241(f+25)\sqrt{\frac{r+\frac{6}{D}+0.04}{D}}\,(U-U_{\text{cr}})^2\cdot 10^{-5}\ \text{kW/km}. \tag{27} \]

For small conductor radii and not very large values of the difference \((U-U_{\text{cr}})\)—according to Peek himself—the losses do not follow formula (27) quite rigorously; nevertheless, the measurement results, and especially according to Fig. 8, gave good agreement of the calculations with experiment. The coefficient \(\beta\) for the experimental line was found to be equal to 1.21.

General Conclusions

1. In a line with a grounded phase, the critical corona voltage is lower, other conditions being equal, than in a symmetrical line; the critical voltage is reduced by the same factor by which the operating capacitance of the conductor increases.

2. Practically, the reduction in the value of the critical voltage in double-circuit lines on common supports is expressed as \(15\text{--}25\%\), and conductors of grade AS-95 already give insignificant losses in 110 kV lines; in single-circuit lines the degree of reduction in the value of the critical voltage is greater and is expressed as \(24\div32\%\), and therefore the use of conductors of grade AS-120 is associated with the allowance of corona losses, though very small in magnitude.

3. The calculation of the critical corona voltage may be carried out by formula (V, 17):

\[ U_{\text{cr}(\phi)}=1.17\,\frac{r}{C_{\text{op}}}\cdot m_o\cdot m_p\cdot \delta\cdot 10^{-6}\ \text{kV}, \]

with the operating capacitance in this case determined from formulas (18) and (19).

1. Under practical operating conditions of power-transmission lines, the grounded conductor of the line does not corona, even if the cross-section of this conductor is half the cross-section of the other two conductors.

2. The power loss to corona may be found as

\[ \Delta P_{\text{ground.}}=\beta\cdot \Delta P_{\text{sym.}}, \]

where \(\Delta P_{\text{sym.}}\) is determined for the symmetrical regime by known formulas (for example, by Peek’s formula).

Literature

1. V. K. Shcherbakov.—Lines using the conductor of one of the phases also as protection against lightning strokes. Proceedings of the Tomsk Industrial Institute, vol. 59, 1941.

2. V. K. Shcherbakov.—Electrical calculation of asymmetrical power transmissions (with a grounded phase). Proceedings of the Tomsk Polytechnic Institute, vol. 63.

3. F. W. Peek.—Dielectric Phenomena in High-Voltage Engineering. Translated from English. 1934.

4. A. A. Smurov.—High-Voltage Electrical Engineering and Power Transmission. 1931.

5. High-Voltage Engineering, edited by Prof. L. I. Sirotinsky, issue 1, 1940.