Protection Against Sustained Earth Faults in Compensated Networks*

Postgraduate student R. A. Vainshtein

This article describes a method for creating protection against sustained earth faults in compensated networks by artificially superimposing on the network a control current with a frequency different from the network frequency. The frequency of the control current is taken to be 25 Hz. The block diagram of the protection is shown in Fig. 1. To superimpose on the network a current with a frequency of 25 Hz, an additional coil is connected in series with the arc-suppression coil; it is made on two identical closed cores, each of which has two windings. The windings are connected in opposition with respect to windings П. A pulsating current with a frequency of 25 Hz flows through windings П; therefore the cores of the additional coil are periodically saturated, and its inductance also changes at a frequency of 25 Hz. An electromagnetic parametric frequency divider is used as the source of the current with a frequency of 25 Hz. Frequency-selective operating elements, fed from a zero-sequence current filter, are installed on each line of the protected network. The principal element of the operating element is an electromagnetic parametric amplifier. Its circuit is shown in Fig. 2. A polarized relay is connected to the output of the parametric amplifier through a rectifier. Winding \(W_y\), which is intended for periodic variation of the inductance of the coil of the amplifier oscillatory circuit, is fed from a zero-sequence voltage filter. Windings \(W_k\) and \(W_c\) are connected in opposition with respect to winding \(W_y\). Windings \(W_c\) are connected to the secondary winding of a zero-sequence current transformer.

Below, by means of simple mathematical derivations, the fundamental possibility of implementing protection with the described elements is shown. Let us consider the device for superimposing on the network a current with a frequency of 25 Hz. Suppose that the inductance of the additional coil varies according to the law:

\[ L = L_0(1 + m \sin \Omega t) \tag{1} \]

where \(L_0\) is the mean value of the inductance;
\(m\) is the depth of modulation of the inductance:

\[ \Omega = 2\pi f, \qquad \text{and } f = 25\ \text{Hz}. \]

* The work was carried out under the supervision of Professor I. D. Kutyavin.

When a ground fault occurs, for the circuit consisting of a section of the damaged line and the arc-suppression-coil circuit, neglecting the zero-sequence resistance of the line and the transient resistance at the fault point, the following equation of electrical equilibrium may be written:

Fig. 1.

\[ L_k \cdot \frac{di}{dt} + \frac{d(li)}{dt} = U_{\phi m} \cdot \sin (2 \Omega t + \varphi) \tag{2} \]

where \(U_{\phi m}\) is the amplitude of the phase voltage of the network;
\(L_k\) is the inductance of the arc-suppression coil.

We solve equation (2) by the method of harmonic balance. If, in a circuit in which a sinusoidal voltage with angular frequency \(2\Omega\) acts, there is an inductance varying according to law (1), then the current contains harmonics with frequencies:

\[ 2\Omega; \qquad 2\Omega - \Omega; \qquad 2\Omega + \Omega \]

In accordance with this, let us represent the solution of the equation in the form:

\[ i = I_1 \cdot \sin(\Omega t + \varphi_1) + I_2 \cdot \sin(2\Omega t + \varphi_2) + I_3 \cdot \sin(3\Omega t + \varphi_3). \tag{3} \]

Let us resolve all sinusoidal quantities into sine and cosine components:

\[ \begin{aligned} U_{\phi m}\sin(2\Omega t+\psi) &= U_{\phi 1}\cdot \sin 2\Omega t + U_{\phi 2}\cdot \cos 2\Omega t,\\ I_1\cdot \sin(\Omega t+\varphi_1) &= I_{11}\cdot \sin \Omega t + I_{12}\cdot \cos \Omega t,\\ I_2\cdot \sin(2\Omega t+\varphi_2) &= I_{21}\cdot \sin 2\Omega t + I_{22}\cdot \cos 2\Omega t \end{aligned} \tag{4} \]

and so on.

Fig. 2.

Substitute (4) and (2), and compose equations expressing the equality of the sine components with frequency \(\Omega\), the cosine components with frequency \(\Omega\), and so on.

Denoting

\[ a=\Omega (L_k+L_0);\qquad b=\Omega m L_0, \tag{5} \]

we obtain a system of six equations with six unknowns

\[ \begin{aligned} - a I_{12}-\frac{1}{2} b I_{21}&=0,\\ a I_{11}-\frac{1}{2} b I_{23}&=0,\\ -2a I_{22}+b I_{11}-b I_{31}&=U_{\phi 1},\\ 2a I_{21}+b I_{12}-b I_{32}&=U_{\phi 2},\\ -3a I_{32}+\frac{3}{2} b I_{21}&=0,\\ 3a I_{31}+\frac{3}{2} b I_{23}&=0. \end{aligned} \tag{6} \]

Solving system (6) and using the known relations

\[ \tg \varphi_1=\frac{I_{12}}{I_{11}};\qquad \tg \varphi_2=\frac{I_{22}}{I_{21}},\qquad \tg \varphi_3=\frac{I_{32}}{I_{31}}, \tag{7} \]

we obtain

\[ I_1=\frac{m L_0 U_{\phi m}}{\Omega\left[m^2 L_0^2-2(L_k+L_0)^2\right]}; \tag{8} \]

\[ I_2=\frac{(L_k+L_0)U_{\phi m}}{\Omega\left[m^2 L_0^2-2(L_k+L_0)^2\right]} \tag{9} \]

\[ I_3=\frac{m L_0 U_{\phi m}}{\Omega\left[m^2 L_0^2-2(L_k+L_0)^2\right]} \tag{10} \]

\[ \varphi_1=\vartheta,\qquad \varphi_2=\psi-90^\circ,\qquad \varphi_3=\vartheta. \tag{11} \]

Let us determine the relation between \(L_k\) and \(L_0\). Denoting \(\dfrac{I_1}{I_2}=p\) and dividing (8) by (9), we obtain

\[ L_0=\frac{p\cdot L_k}{m-p} \tag{12} \]

Taking into account that the current with frequency 25 cps required for operation of the protection is many times smaller than the current of the arc-suppression coil, and assuming that the cores of the additional coil are strongly saturated by current pulses with frequency 25 Hz, one may take \(m\approx 1\), and \(p\ll 1\), that is,

\[ L_0\approx p L_k \tag{13} \]

Next let us consider the response element of the protection with a parametric amplifier. Suppose that the inductance of the parametric amplifier varies according to the law

\[ l=L_0(1+m\sin 2\Omega t) \tag{14} \]

where \(m\ll 1\), and all loss sources are concentrated in one conductance, \(q\). The circuit is connected to a current source. The current of this source varies according to the law

\[ i=I_m\cdot \sin(n\Omega t+\varphi); \tag{15} \]

where \(n=1, 2, 3\ldots\), which corresponds to frequencies of 25, 50, 75 Hz, etc. The adopted assumptions correspond to the equivalent circuit of the amplifier shown in Fig. 3.

Fig. 3.

It is known that the amplification of oscillations in a parametric amplifier occurs due to the energy introduced into the oscillatory circuit by a periodic change in inductance or capacitance. Let us determine the average value of the power over the period \(T=\dfrac{\pi}{\Omega}\) that is released in the variable part of the inductance when a current of angular frequency \(n\Omega\) flows through it.

\[ P=\frac{1}{T}\int\limits_{0}^{T}\frac{d\left(l_n\cdot i_L\right)}{dt}\cdot i_L\cdot dt \tag{16} \]

where

\[ l_n=mL_0\cdot \sin 2\Omega t, \]

\[ i_L=I_{Lm}\cdot \sin\left(n\Omega t+\varphi+\theta-90^\circ\right) \tag{17} \]

is the current flowing through the windings \(W_k\),

\[ I_{Lm}= \frac{I_m} {\sqrt{\left(n\Omega C-\dfrac{1}{n\Omega L_0}\right)^2+g^2}\cdot n\Omega L_0}. \tag{18} \]

\[ \theta=\operatorname{arctg}\frac{n\Omega C-\dfrac{1}{n\Omega L_0}}{q}. \tag{19} \]

Carrying out the integration, we obtain

\[ \begin{aligned} P={}&\frac{(n-2)mL_0}{4(2n-2)T} I_{\mathrm{Lm}}^{2} \{ \sin 2[(2n-2)\pi+\alpha]-\sin 2\alpha \}\\ &-\frac{(n+2)mL_0}{4(2n+2)T} I_{\mathrm{Lm}}^{2} \{ \sin 2[(2n+2)\pi+\alpha]-\sin 2\alpha \}\\ &-\frac{mL_0}{4(n-1)T} I_{\mathrm{Lm}}^{2} \{ \sin 2[(n-1)\pi+\alpha]-\sin 2\alpha \}\\ &-\frac{mL_0}{4(n+1)T} I_{\mathrm{Lm}}^{2} \{ \sin 2[(n+1)\pi+\alpha]-\sin 2\alpha \}, \end{aligned} \tag{20} \]

where

\[ \alpha=\varphi+\theta-90^\circ . \tag{21} \]

Let us examine expression (20) for various \(n\). For all values of \(n\), except \(n=1\), \(P=0\); for \(n=1\) an indeterminacy of the form \(0/0\) is obtained, resolving which we get

\[ P=-\frac{3}{4}\,mL_0\cdot I_{\mathrm{Lm}}^{2}\cos 2\alpha . \tag{22} \]

The minus sign means that energy is being supplied to the circuit. The best conditions for supplying energy to the circuit occur when

\[ \alpha=k\pi , \tag{23} \]

where \(k=0,1,2,3\), etc.

This condition must be satisfied for a ground fault on any phase. Let us write the law of variation of the inductance for faults on phases A, B, and C*).

\[ \begin{aligned} l_A&=L_0[1+m\sin(2\Omega t+\beta_A)],\\ l_B&=L_0[1+m\sin(2\Omega t+\beta_B)],\\ l_C&=L_0[1+m\sin(2\Omega t+\beta_C)]. \end{aligned} \tag{24} \]

Taking (11), (21), (23), and (24) into account, we obtain

\[ \begin{aligned} \beta_A&=k\pi-\theta+90^\circ,\\ \beta_B&=k\pi+120^\circ-\theta+90^\circ,\\ \beta_C&=k\pi+240^\circ-\theta+90^\circ . \end{aligned} \tag{25} \]

All the results obtained cannot be regarded as design relations, since the analysis in no way takes into account the characteristic of the ferromagnetic material from which the cores of the auxiliary coil and the parametric amplifier are made. However, these results make it possible to draw the following important fundamental conclusions:

1. With sufficiently high sensitivity of the responding elements, the average inductance of the auxiliary coil is many times smaller than the inductance of the arc-suppression coil, as follows from (13). Therefore

* The initial voltage angles of phases A, B, C are taken as \(0^\circ\), \(240^\circ\), \(120^\circ\), respectively.

switching on the additional coil will not substantially affect the compensation setting.

1. The parametric amplifier according to the circuit of Fig. 2, by its operating principle, amplifies only current with a frequency of 25 cps, as follows from (20).

2. The phase of the current with a frequency of 25 cps changes with the change in the phase of the current flowing through the coil in accordance with (11). To achieve the same sensitivity of the responding element when any phase is short-circuited to the zero-sequence voltage filter feeding the parametric amplifier, according to (25) a voltage with the reverse phase sequence should be applied.

Some laboratory-test data are given below. Fig. 4 shows the dependence of the voltage on the oscillatory circuit on the primary current of the zero-sequence transformer

Fig. 4.

one of the prototypes of a parametric amplifier at frequencies of 25 cps and 50 cps. A standard transformer of type T. Z. was used as the zero-sequence transformer.

Fig. 5 shows the circuit on which the interaction of all the protection elements was checked. The polarized relay, connected at the output of the parametric amplifier, operates reliably if, for example, the acting value of the first harmonic of the current is 2 a, and the component at a frequency of 25 cps in it is 0.2—0.25 a. In the absence of a component at a frequency of 25 cps, the relay operates at a current of 10 a. If the phase of the zero-sequence voltage is set so that maximum sensitivity is achieved for a fault on phase A, then the same sensitivity for faults on the other phases is ensured by interchanging phases B and C at the voltage transformer, which is consistent with (25).