ARC EXTINCTION IN THE PRESENCE OF A CAPACITOR IN THE ARC-SUPPRESSION COIL CIRCUIT

P. T. ANOKHIN

(Presented by the scientific seminar of the Department of Electric Power Stations)

Protection against ground faults with one-hundred-percent coverage of the stator winding of a unit generator using an auxiliary DC source has its own particular features. The connection diagram of the device for protection RZ is shown in Fig. 1, a. The arc-suppression coil—

Fig. 1. Equivalent circuit for studying the character of the transient process during a ground fault in the generator circuit: a—calculation circuit; b—zero-sequence equivalent circuit during a ground fault; c—zero-sequence equivalent circuit after arc interruption.

the ASCG and the neutral point of the high-voltage winding of the voltage transformer TN, which is to be grounded according to the conditions of its operation, are con—

grounded, respectively, through capacitors \(C_1\) and \(C_2\). Grounding can also be carried out through a single capacitor, as shown in [1].

The operating principle of the protection is very simple. The main units of the ground-fault protection are a direct-current source and a measuring element.

In normal operation the capacitors \(C_1\), \(C_2\) and the capacitances of the generator and transformer to ground, \(C_g\) and \(C_t\), are at the potential of the direct-current source. When a ground fault occurs at any point of the generator-voltage circuit, a direct current flows in the measuring element, depending on the magnitude of the transient resistance at the fault point, \(R_p\). The use of polarized relays makes it possible to implement protection sensitive to faults through transient resistances of the order of \(300\ \text{k}\Omega\) and more at source voltage \(U_p \simeq 300\ \text{V}\). The magnitude of the direct current can be limited by the internal resistance of the ground-fault protection circuit to the required value, for example \(40\text{--}50\ \mu\text{A}\).

Grounding the generator neutral through an arc-suppression coil solves two basic problems. First, the current at the fault point in the steady-state regime is substantially reduced; second, the rate of recovery of the voltage \(U_d\) across the arc gap after arc interruption is reduced.

The tuning of an arc-suppression coil is usually carried out with slight overcompensation; therefore, the inclusion of capacitor \(C_1\) leads to some increase in the current at the fault point, \(J_z\). The current \(J_z\) can be limited both by increasing the capacitance of capacitor \(C_1\) and by changing the tuning of the arc-suppression coil. However, more importantly, the series connection of the capacitance \(C_1\) and the inductance of the coil \(L_k\) is a prerequisite for a qualitative change in the character of the transient process during a ground fault in the generator circuit. In this connection it is necessary to evaluate the conditions for arc extinction at the fault point, taking into account the process of recovery of the voltage \(U_d\) across the arc gap after arc interruption.

A detailed analysis of the transient process during a ground fault is given in [2]. We shall touch on this question only insofar as the presence of capacitor \(C_1\) in the coil circuit affects the transient process.

The free component of the capacitive current \(J_c''(t)\) has the character of rapidly damped oscillations of mean frequency \(\omega_c'\), which exceeds the fundamental frequency \(\omega\) several times [2].

The free component of the arc-suppression-coil current \(J_k''(t)\) decays much more slowly than \(J_c''(t)\). The coil has practically no effect on the current \(J_c''(t)\) [4]; therefore we shall consider these two processes separately and confine ourselves to determining the current \(J_k''(t)\).

Figure 1,b gives the zero-sequence equivalent circuit for determining the current \(J_k''(t)\). In setting it up the following assumptions were adopted:

1) the zero-sequence resistance of the generator is not taken into account, since it is very small in comparison with the resistance of the arc-suppression coil;

2) the three-phase system is assumed to be symmetrical; but in normal operation the neutral potential is equal to the voltage of the direct-current source \(U_p\);

3) the feed from the direct-current source is neglected.

The analysis we carried out shows that, in the case of a metallic ground fault \((R_p = 0)\), the transient process in the coil circuit

practically always has the character of oscillations of low frequency \(\omega_{\mathrm{н}}'\), which is several times less than the fundamental frequency \(\omega\).

The total current at the point of fault is determined by the expression

\[ I_{3}(t)=I_{3}'(t)+I_{3}''(t), \tag{1} \]

where \(I_{3}'(t)\) and \(I_{3}''(t)\) are, respectively, the forced and free components of the earth-fault current,

\[ I_{3}''(t)=I_{\mathrm{k}}''(t)+I_{\mathrm{c}}''(t). \tag{2} \]

The current \(I_{3}(t)\) depends on the voltage of the damaged phase at the instant of fault, i.e., on the initial phase \(\psi\).

We give approximate expressions for the components of the current \(I_{3}(t)\) for the most characteristic cases, namely: \(\psi=\dfrac{\pi}{2}\) and \(\psi=0\). The calculations are carried out in relative units. As base quantities we take the amplitudes of the capacitive current \(I_{mc}\) and of the generator phase voltage \(U_{m\phi}\).

We introduce the following abbreviated notation:

\[ K_{1}=\frac{1}{3\omega^{2}L_{\mathrm{k}}C_{\mathrm{r}}} \]
— the degree of tuning of the oscillatory circuit;

\[ V_{1}=1-K_{1} \]
— the degree of detuning of the same circuit;

\[ d_{\mathrm{k}}=\frac{R_{\mathrm{k}}}{\omega L_{\mathrm{k}}} \]
— a coefficient characterizing the losses in the copper and in the steel core of the coil;

\[ K_{\mathrm{c}}=\frac{3C_{\mathrm{r}}}{C_{1}};\qquad K_{\mathrm{p}}=\frac{U_{\mathrm{p}}}{U_{m\phi}};\qquad \omega_{\mathrm{н}}'=\sqrt{\omega_{\mathrm{н0}}^{2}-\delta^{2}}; \]

\[ \omega_{\mathrm{н0}}=\frac{1}{\sqrt{L_{\mathrm{k}}C_{1}}};\qquad \delta=\frac{R_{\mathrm{k}}}{2L_{\mathrm{k}}}=\frac{d_{\mathrm{k}}}{2}\,\omega; \]

\[ \varphi_{2}=\operatorname{arctg}\frac{1-K_{1}(1+K_{\mathrm{c}})}{d_{\mathrm{k}}}. \]

For \(\psi=\dfrac{\pi}{2}\) we have

\[ I_{3}'(t)= \frac{\sqrt{d_{\mathrm{k}}^{2}+\left[1-K_{1}(1+K_{\mathrm{c}})\right]^{2}}} {1-K_{\mathrm{c}}K_{1}}\cdot \cos(\omega t+\varphi_{2}); \tag{3} \]

\[ I_{\mathrm{k}}''(t)=- \frac{K_{1}\left[K_{1}K_{\mathrm{c}}+K_{\mathrm{p}}(1-K_{1}K_{\mathrm{c}})\right]} {(1-K_{1}K_{\mathrm{c}})\sqrt{K_{1}K_{\mathrm{c}}}} \sin \omega_{\mathrm{н}}'t\cdot e^{-\delta t}; \tag{4} \]

\[ I_{\mathrm{c}}''(t)= \frac{2}{3}\frac{\omega_{\mathrm{c}}'}{\omega}\cdot \sin \omega_{\mathrm{c}}'t\cdot e^{-\delta_{\mathrm{c}}t}. \tag{5} \]

For \(\psi=\dfrac{\pi}{2}\) we have

\[ I_{3}'(t)= \frac{\sqrt{d_{\mathrm{k}}^{2}+\left[1-K_{1}(1+K_{\mathrm{c}})\right]^{2}}} {1-K_{1}K_{\mathrm{c}}}\cdot \sin(\omega t+\varphi_{2}); \tag{6} \]

\[ I_{\mathrm{k}}''(t)= \frac{K_{1}}{1-K_{1}K_{\mathrm{c}}} \left[ \cos \omega_{\mathrm{н}}'t - \frac{d_{\mathrm{k}}+2K_{\mathrm{p}}(1-K_{1}K_{\mathrm{c}})} {2\sqrt{K_{1}K_{\mathrm{c}}}}\cdot \sin \omega_{\mathrm{н}}'t \right]\cdot e^{-\delta t}; \tag{7} \]

\[ I_{\mathrm{c}}''(t)= -\frac{2}{3}\cos \omega_{\mathrm{c}}'t\cdot e^{-\delta_{\mathrm{c}}t}. \tag{8} \]

Figure 2 shows curves constructed from expressions (3)—(8). The following values of the coefficients were adopted:

\[ K_1 = 1.0;\qquad K_c = 0.0278;\qquad K_n = 0.03;\qquad d_k = 0.02. \]

Let us consider the influence of the capacitor \(C_1\) on arc extinction according to the existing theories on this question.

The greatest contribution to the development of the theory of arcing ground faults was made by Petersen. Among other authors who took a further step in its development are Peters and Slepian. As a result, two basic theories were advanced, differing in their assumptions concerning the instant of arc extinction [2].

According to the first theory (Petersen), the arc is extinguished when the medium-frequency oscillation current \(J_c''(t)\) passes through zero. For this, the current \(J_c''(t)\) must be the determining component in the total ground-fault current.

The second theory (Peters and Slepian) assumes the possibility of arc extinction only when the power-frequency current passes through zero, i.e., after the current \(J_c'(t)\) has decayed.

We believe that, in the case where an arc-suppression coil is grounded through a capacitor, according to the second theory the arc can be extinguished when the low-frequency oscillation current passes through zero.

Let us give a comparative estimate of the possibility of arc extinction in connection with the change in the transient process at the point of the ground fault. Most investigators agree that both mechanisms of arc extinction are possible [3]. We shall follow their example, assuming that a fault is probable at any initial phase \(\Psi\). For this we shall use the curves given in Fig. 2, where the approximate dependence of the free component of the capacitive current \(J_c''(t)\) is also shown.

For \(\Psi = \dfrac{\pi}{2}\) (Fig. 2, \(a\)) we have a current component \(J_c(t)\) of considerable magnitude, which is several times greater than the capacitive current of the steady-state condition, taken by us as 1. At the same time the amplitude of the current \(J_k''(t)\) is several times less than 1. According to the first theory, the arc can be extinguished within a quarter-period of the power frequency, i.e., at \(t < 0.005\) sec. According to the second theory, it will be extinguished only after a half-period of the low-frequency oscillations, i.e., at \(t \approx 0.06\) sec. It should be noted that before \(C_1\) was switched in, at \(\Psi = \dfrac{\pi}{2}\), in the transient condition the free component was represented only by the current \(J_c''(t)\); consequently, according to the second theory the arc could be extinguished at \(t \approx 0.01\) sec.

For \(\Psi = 0\) (Fig. 2, \(b\)) the amplitude of the current \(J_c''(t)\) does not exceed \(2/3\), while the amplitude of the current \(J_k''(t)\) is approximately equal to 1. During the existence of the medium-frequency oscillations the total current does not pass through zero; therefore, the probability of arc interruption according to the first theory is absent. The arc can be extinguished only when the low-frequency oscillation current \(J_k''(t)\) passes through zero, i.e., at \(t \approx 0.025\) sec. It is characteristic that before the capacitor \(C_1\) was switched in, at \(\Psi=0\), it could be extinguished only after the aperiodic component of the current of the arc-suppression coil had decayed (curve \(e^{-d_k\omega t}\)). Obviously, for other values of \(d_k\) as well, the curve \(J_k''(t)\) will pass through zero considerably earlier than the curve \(e^{-d_k\omega t}\) decays.

Consequently, according to the first theory, after switching in the capacitor \(C_1\), the conditions for arc extinction practically did not change. According to the second theory, the instant of time of the probable interruption of the arc depends on the initial—

Fig. 2. Components of the earth-fault current for different values of the initial phase:
a — for \(\psi=\dfrac{\pi}{2}\); b — for \(\psi=0\).

of the initial phase \(\Psi\). For \(\Psi\) close to \(\pm \dfrac{\pi}{2}\), when the free component of the current of the arc-suppression coil is small, after switching on capacitor \(C_1\) the arcing time increases. However, for \(\Psi\) close to zero or \(2\pi\), when the free current of the coil is approximately equal to 1, after switching on capacitor \(C_1\) the arcing time until the first current zero crossing is sharply reduced; therefore the conditions for arc extinction become more favorable than before switching on \(C_1\).

In determining the recovery voltage \(U_{\text{d}}\) across the arc gap, we shall use the method set forth in [4]. The simplified equivalent circuit is shown in Fig. 1,c.

Capacitor \(C_1\) and the arc-suppression coil have practically no effect on the equalization of the charges of the capacitances of the healthy and faulted phases; therefore we do not consider the intermediate-frequency oscillation circuit.

After the damping of the intermediate-frequency oscillations, the capacitances of all three phases begin to discharge through the inductance \(L_{\text{k}}\) (Fig. 1,c). This process is damped and proceeds with the angular frequency of the free oscillations

\[ \omega'=\sqrt{\omega_0^2-\alpha^2}, \tag{9} \]

where

\[ \omega_0=\frac{1}{\sqrt{L_{\text{k}}C_{\text{e}}}}, \tag{10} \]

\[ C_{\text{e}}=\frac{C_1\cdot 3C_{\text{g}}}{C_1+3C_{\text{g}}} =\frac{3C_{\text{g}}}{1+K_{\text{c}}}, \tag{11} \]

from which it is seen that changing the capacitance of the oscillatory circuit by a factor of \(\dfrac{1}{1+K_{\text{c}}}\) will lead to a change in \(\omega'\) approximately in the ratio \(\sqrt{1+K_{\text{c}}}\). Thus, for example, even with a very significant coefficient \(K_{\text{c}}=0.04\), \(\omega'\) will increase by no more than a factor of 1.02.

It is known that after interruption of the arc, the voltage component of the free oscillations and the phase voltage are in phase opposition. Since \(\omega'\) is close to \(\omega\), the voltage across the arc gap \(U_{\text{d}}\) rises slowly and has a beat character.

Switching on capacitor \(C_1\) has led to an insignificant increase in \(\omega'\); therefore the rate of recovery of the voltage across the arc gap will practically not change.

Conclusions

From the standpoint of existing theories concerning arc extinction, switching a capacitor into the circuit of the arc-suppression coil is permissible. Moreover, in faults accompanied by a free component of the coil current of considerable magnitude, the conditions for arc extinction in the presence of the capacitor will be more favorable than before it is switched on.

REFERENCES

1. M. N. Raik. Protection against ground faults of generator-transformer units. Energetika za rubezhom, relay protection series, issue 3. Moscow–Leningrad, Gosenergoizdat, 1960.

2. R. Wilheim, M. Waters. Grounding of neutrals in high-voltage systems. Moscow–Leningrad, Gosenergoizdat, 1959.

3. A. I. Dolginov. High-voltage engineering in electric power systems. Moscow, “Energiya,” 1968.

4. P. V. Borisoglebskii, L. F. Dykhovskaya, et al., under the general editorship of D. V. Razevig. High-voltage engineering. Moscow–Leningrad, “Energiya,” 1964.