I. D. Kutyavin.

Protection of Generators.

The present article is the result of the author’s work on the problem of increasing the sensitivity of generator protection against ground faults. The first part of the article is devoted to the question of increasing the sensitivity of normal generator ground-fault protection of the AEG type, while the second part is devoted to investigations carried out by the author in the field of the arrangement of ground-fault protection and differential protection of the Ferranti type.

Protection against Ground Faults.

Frame faults are the most frequent type of damage to generators. As practice shows, damage to the stator winding almost always begins with a frame fault and only subsequently develops into a short circuit of one type or another. Therefore protection against ground faults is the most responsible element of the entire generator-protection system.

At present, for the protection of generators operating directly on the station busbars, there exists a large number of types of ground-fault protection. Among them the most advanced is the differential wattmetric protection of the AEG type, reacting to zero-sequence power, whose sensitivity is also far from being in exact conformity with the requirements imposed on protection against ground faults.

As is known, the principal indicator of the quality of protection is its good sensitivity at a small maximum ground-fault current, the lower limit of which is determined by the magnitude of the unbalance current in the secondary circuit of the protection, and the upper limit by the permissible damage to the generator in the event of a frame fault.

From the standpoint of permissible damage to the stator iron, a maximum ground-fault current (Sheet 8) of up to 40 A may be allowed, and for 4 sec., 80 A—for 2 sec., and (120–150) A—for 0.4 sec.

However, it must be borne in mind that a frame fault may cause an interturn or interphase short circuit and lead to severe damage to the generator. Therefore, for the protection of generators against internal damage, it is absolutely necessary to avoid all relay time delays, or to reduce them to a minimum, and to reduce the maximum ground-fault current to minimum values; in doing so it is necessary to take into account the magnitude of the capacitive and watt currents of grounding, fed from the busbar side.

I. Ground-fault protection of the AEG type.

1. Arrangement of the protection.

Figure 1 shows the circuit of normal AEG ground-fault protection.

The current winding of wattmetric relay 1 of the cosine type is connected differentially to three external current transformers 2, connected in ...

the Holmgreen circuit with transformation ratio \(n_{\mathrm{tv}}\), and to the zero-sequence current transformer 3 with transformation ratio \(n_{\mathrm{to}}\), where

\[ n_{\mathrm{to}}=(1.2 \div 1.5)\,n_{\mathrm{tv}}. \]

The voltage to the voltage coil of the relay is supplied from the zero-sequence voltage transformer 4.

In normal operation and during external interphase short circuits, only the unbalance current \(J_{\mathrm{nb}}\) of the external current transformers flows through the current coil of the wattmetric relay; the voltage at the relay terminals is \(U_p=0\), and the protection does not operate.

In earth faults outside the protection zone, the power at the relay due to the component \(J_{\mathrm{v}}\)—the current of the external transformers—acts on the relay in the restraining direction, while in faults inside the protection zone it acts in the tripping direction (provided there is current flowing from other generators).

The power at the relay due to the component \(J_{\mathrm{n}}\)—the current of the zero-sequence transformer—always acts in the deflection direction.

In an earth fault outside the protection zone, the difference current in the protection circuit, due to \(n_{\mathrm{to}}>n_{\mathrm{tv}}\), goes to compensate the unbalance current of the external transformers.

Fig. 1.

2. Magnitude and phase of the unbalance current of the Holmgreen circuit.

The unbalance current of the Holmgreen circuit is caused by the nonidentity of the magnetic characteristics of the transformers composing the circuit, and is equal to the geometric sum of the secondary currents:

\[ J_{\mathrm{nb}}=J_{a2}+J_{b2}+J_{c2}\ldots\ldots\ldots\ldots\ldots (1) \]

On the basis of the vector diagram of the current transformer (Fig. 2), equation (1) can be expressed through the primary currents referred to the number of turns of the secondary winding:

\[ J_{\mathrm{nb}}=(J_{a0}-J_{a1})+(J_{b0}-J_{b1})+(J_{c0}-J_{c1})\ldots (2) \]

where \(J_0=J_r+J_a\) is the sum of the reactive and active components of the magnetizing current, hereinafter called the no-load current of the transformer.

But the sum of the primary currents under the normal operating condition of the generator is zero. Then equation (2) takes the form:

\[ J_{\mathrm{nb}}=+(J_{a0}+J_{b0}+J_{c0})\ldots\ldots\ldots\ldots (3) \]

If the primary currents are symmetrical, then, on the basis of the vector diagram of the current transformer (Fig. 2), equation (3) can be represented in a more visual form, namely:

\[ J_{\mathrm{nb}}=\left[J_{a0}e^{-j\psi_a}+a^2J_{b0}e^{-j\psi_b}+aJ_{c0}e^{-j\psi_c}\right].(4) \]

where

\[ \psi=\alpha+\varphi_2+\delta. \]

and

\[ a=e^{j\frac{2\pi}{3}}. \]

Fig. 2.

\(\varphi_2\) is the phase shift of the current and electromotive force in the secondary circuit of the transformer, and \(\delta\) is the angular error.

Consequently, with the same transformation ratio of the circuit transformers, the magnitude and phase of the unbalance current will be determined by the magnitude and phase of the no-load currents of the transformers.

3. Sensitivity of the AEG-type ground-fault protection

In the case of a metallic ground fault in the protected zone, the power at the terminals of the wattmeter relay, when one generator is operating, is

\[ P_p = (J_n - J_{nbn})\, U_p \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (5) \]

where \(J_{nbn}\) is the unbalance current of the Holmgreen circuit at the nominal load of the transformers. In order for the relay to operate, the sensitivity condition must be satisfied:

\[ P_p \geq P_{pt} \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (6) \]

where \(P_{pt}\) is the relay pickup power.

In the case of a ground fault outside the protected zone, the spurious power at the relay terminals is

\[ P_{pl} = [J_{nbn} - (J_v - J_n)]\, U_p \ldots\ldots\ldots\ldots\ldots\ldots (7) \]

In this case, in order that the relay not operate, the maximum spurious power must be less than the pickup power:

\[ P_{pt} = K_n P_{pl\ \max}, \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots (8) \]

where \(K_n = 1.25\) is the security factor.

The simultaneous solution of equations (5) and (7), subject to conditions (6) and (8), gives the basic relations for the protection parameters (sheet 4): the relay pickup power

\[ P_{pt} = 0.005 \sqrt{K_n}\, J_{nbn}\, U_{p\ \max}\, \alpha_0; \ldots\ldots\ldots\ldots\ldots (9) \]

the maximum ground-fault current

\[ J_{zp} = (1 + \sqrt{K_n})\, 100\, \frac{J_{nbn}\, n_{tv}}{\alpha_0} \ldots\ldots\ldots\ldots\ldots (10) \]

and the ratio of transformation coefficients

\[ n_{ta} = 1.36\, n_{tv}, \quad \text{for } K_n = 1.25 \ldots\ldots\ldots\ldots\ldots (11) \]

where \(\alpha_0\) is the percentage of unprotected turns, and \(U_{p\ \max}\) is the voltage at the relay terminals during a ground fault at the generator terminals.

To assess the quality of normal AEG-type protection, Fig. 3 gives the dependence of the maximum ground-fault current on the rated generator current, expressed by equation (10). In constructing the plot it was assumed that the secondary rated current of the external transformers is \(5\) A, the percentage of protected turns is \(80\%\), and \(K_n = 1.25\).

It is evident from Fig. 3 that the maximum ground-fault current for large generators, with \(80\%\) of the turns protected, goes far beyond the permissible current, which makes it necessary to reduce the percentage of protected turns.

Fig. 3.

In order to lower the maximum ground-fault current \(AEG\), transformers with a secondary rated current of up to \(20a\) are manufactured for this protection; that is, the transformation ratio of the external transformers \(n_{\mathrm{tv}}\) is reduced. With a reduction in the transformation ratio, the unbalance current of the Holmgreen circuit must increase. It is therefore of interest to determine the dependence of the unbalance current on the factor by which the secondary rated current of the transformers is increased.

For the maximum magnetic induction \(B_{\max}\), within the range \(0\text{--}7000\) gauss, the dependence \(aw_{\mathrm{ud}} = f(B_{\max})\) (according to data from the Elektroapparat plant [l. 7]) may, with sufficient accuracy for approximate calculations, be expressed by the equation

\[ aw_{\mathrm{ud}} = K B_{\max}^{0.6} \ldots\ldots\ldots\ldots\ldots\ldots (12) \]

where \(K = 0.0023\text{--}0.0044\), depending on the grade of iron.

On the basis of equation (12), one may also write the dependence of the no-load current on the secondary electromotive force of the transformer:

\[ J_0 \simeq \frac{K_1 E_2^{0.5}}{w_2^{1.6}} \ldots\ldots\ldots\ldots\ldots\ldots (13) \]

But from equation (3) the unbalance current is proportional to the no-load current of the transformers,

\[ J_{\mathrm{nb}} \simeq K_1 J_0 \simeq \frac{K_2 E_2^{0.6}}{W_2^{1.6}} \ldots\ldots\ldots\ldots\ldots\ldots (14) \]

When the secondary rated current is increased \(m\)-fold, the number of turns of the secondary winding and the length of its conductor decrease \(m\)-fold, while the conductor cross section must be increased \(m\)-fold; then the internal resistance of the secondary winding will decrease \(m^2\)-fold. At the same time, the magnitude of the secondary electromotive force will also decrease \(m\)-fold; consequently, the magnitude of the unbalance current according to equation (14)

\[ J'_{\mathrm{nb}} \simeq \left(\frac{E_2}{m}\right)^{0.6} \frac{K_2}{\left(\frac{W_2}{m}\right)^{1.6}} = K_2 \frac{E_2^{0.6}}{W_2^{1.6}}\, m \ldots\ldots\ldots (15) \]

will increase \(m\)-fold.

At the same time, the magnitude of the maximum ground-fault current, according to equation (10), will remain constant (Fig. 3), since according to equation (15)

\[ \frac{n_{\mathrm{tv}}}{m}\cdot J_{\mathrm{nb}}\cdot m = n_{\mathrm{tv}} J_{\mathrm{nb}} = \mathrm{const}. \]

But with an increase in the secondary rated current of the transformers, the grounding current in the secondary circuit of the protection increases, and this makes it possible to obtain sufficient sensitivity of the protection with coarser relays.

A considerable reduction of \(J_{\mathrm{zm}}\) can be obtained by reducing the unbalance current of the Holmgreen circuit through the matching of current transformers. The matching must be carried out by the plant manufacturing the transformers and must guarantee a definite value of the unbalance current at the rated load of the transformers.

Figure 4 gives a circuit that may be recommended for carrying out the matching of the transformers, and Fig. 5 gives the vector diagram for this circuit.

With the connection shown in the diagram (Fig. 4), the voltages at the terminals of the primary windings \(U'_1\) and \(U''_1\) and the secondary electromotive forces \(E'_2\) and \(E''_2\) of the transformers may differ in magnitude and phase because of the unequal components \(J_m\) and \(J_w\) of the no-load currents of the transformers. The ammeter in the secondary circuit of the transformers will indicate the unbalance current of the circuit, caused by the difference of the secondary emfs \(\Delta E_2\). The wattmeter in the circuit is needed only to determine the direction of energy in the secondary circuit; moreover, to increase the sensitivity, its voltage coil may be connected to the terminals of the resistance of the primary circuit.

Fig. 4.

When selecting according to the circuit (Fig. 4) by comparing several transformers with one transformer taken as the reference, it is necessary to plot the characteristics \(J_{\text{unb}}=\pm f(J_0)\) (the sign is then determined by the wattmeter), and then, from the characteristics, to select sets of transformers. In Fig. 6 the characteristics are given for four current transformers of type \(TK\)-2 25/5a.

The tests carried out by the author on six current transformers of type \(TK\)-2 25/5a showed that

1) the unbalance current in the circuit (Fig. 4) is inversely proportional to the resistance of the instruments; therefore the ammeter and wattmeter should be taken with measuring ranges of \(0\)—\(1a\), not lower, and

2) unbalance currents that are equal in magnitude and identical in sign, at the same primary current, coincide in phase; therefore coincidence of the characteristics \(J_{\text{unb}}=f(J_0)\) indicates the identity of the transformers being compared.

Fig. 5.

Fig. 6.

Of the six current transformers mentioned, two combinations were selected which, when connected according to the circuit, gave the following unbalance currents: the best combination, \(4ma\), and the worst, \(15ma\). In this case the mismatch of the characteristics for the best combination was expressed by the ratio \(1:1,\ 4:1,\!7\), and for the worst \(1:1,\!4:5,\!2\). The unbalance current was measured with a wattmeter, with the fine winding supplied by an asynchronous voltage.

In the periodical literature ([6]) there are proposals to reduce the unbalance current of the Holmgreen circuit by adjusting the resistances conne—

connected in series with the secondary windings of the transformers. When the resistance of the transformer secondary winding changes, the secondary electromotive force will change, and, as a consequence, the no-load current will also change. But on the basis of equation (4), the unbalance current depends not only on the magnitude, but also on the phase of the transformer no-load current; therefore the proposed method is unlikely to prove especially effective.

Of far greater interest is reducing the unbalance current by lowering the internal resistance of the transformers by increasing the cross-section of the secondary winding; in this case the dependence of the unbalance current on the internal resistance of the transformer can be determined according to equation (14).

\[ J_{\mathrm{unb}} \cong \frac{K_{2} E_{2}^{0.6}}{W_{2}^{1.5}} \cong K_{2}\frac{(J_{2} Z_{2})^{0.6}}{W_{2}^{1.6}} \cong K_{3} Z_{2}^{0.6} \ldots \ldots \ldots (16) \]

where \(Z_{2} \cong R_{2}\), since the inductance of the secondary winding is small.

4. Increasing the sensitivity of AEG-type protection by grounding the generator neutral through iron resistors.

A number of courses on relays and relay protection mention the extremely great advantage of grounding the generator neutral through iron resistors made in the form of lamps. As is known, iron has a large temperature coefficient of resistance; consequently, the resistance of the iron of the rheostat, as the point of the ground fault moves away from the generator neutral, will increase, which makes it possible, at a permissible maximum ground-fault current, to ensure sufficient sensitivity of the protection.

Let us determine the basic parameters of the iron resistor on the assumption that the maximum ground-fault current is sufficiently large; therefore the time delay of the protection must be no more than 0.5--1 sec.

Under this assumption, the heating process of the iron resistor may be considered adiabatic. Then the energy-balance equation is written in the form: \(J_{z}^{2} R\,dt = g c\,dT\),

where \(R\) is the resistance of the iron,

\(J_{z}\) is the ground-fault current flowing through the iron resistance,

\(g\) is the weight of the iron, and \(C\) is the mean heat capacity of the iron.

Substitute in the preceding equation

\[ R = R_{n}(1+\beta\tau); \]

\[ J_{z} = \frac{x U_{\phi}}{R_{n}(1+\beta\tau)} \ldots \ldots \ldots (17) \]

and separate the variables; then we obtain:

\[ \frac{X^{2} U_{\phi}^{2}}{g c R_{n}}\,dt = (1+\beta\tau)\,d\tau, \]

where \(R_{n}\) is the initial resistance of the iron,

\(X\) is the number of turns of the generator phase short-circuited to ground, in fractions of unity,

\(U_{\phi}\) is the phase voltage of the generator, and

\(\beta\) is the temperature coefficient of resistance of iron.

We now integrate the preceding equation over the limits \(0\)--\(t\) and \(0\)--\(\tau\); then we obtain:

\[ \frac{X^{2} U_{\phi}^{2}}{g c R_{n}}\,t = \frac{1}{2\beta}\left[(1+\beta\tau)^{2}-1\right] \ldots \ldots \ldots (18). \]

From equation (18) one can determine the weight and dimensions of the iron resistance, if one sets \(t=t_m\)—the design time delay of the relay, \(\tau=\tau_m\)—the design temperature of heating of the iron, and \(X=1\):

\[ \text{weight of iron } g= \frac{2\beta U_{\phi}^{2}t_m}{cR_n\left[(1+\beta\tau_m)^2-1\right]} \ \text{kg} \tag{19} \]

\[ \text{cross-section }\quad S=\frac{U_{\phi}}{R_n} \sqrt{\frac{2\beta\rho\,t_m\,10^3}{c\gamma\left[(1+\beta\tau_m)^2-1\right]}} \ \text{mm}^2 \]

\[ \text{and length }\quad l=U_{\phi} \sqrt{\frac{2\beta t_m\,10^3}{c\gamma\rho\left[(1+\beta\tau_m)^2-1\right]}} \ \text{m} \tag{20} \]

where \(\rho\), \(\Omega\text{ mm}^2/\text{m}\), is the specific resistance of iron at normal temperature, and

\[ \gamma \]

is the specific weight of iron.

The value of the initial resistance of the iron may be determined from the equation

\[ R_n=\frac{U_{\phi}}{J_{z m}(1+\beta\tau_m)} \tag{21} \]

To clarify the character of the change in the ground-fault current as a function of time and of the number of short-circuited turns, we determine the value of \((1+\beta\tau)^2\) from equation (18), substituting \(g\) from (19):

\[ (1+\beta\tau)^2 = 1+X^2\frac{t}{t_m}\left[(1+\beta\tau_m)^2-1\right] \tag{22} \]

Let us now substitute the obtained value for \((1+\beta\tau)^2\) and the value of \(R_n\) from (21) into equation (17); then we obtain:

\[ J_z= \frac{XJ_{z m}(1+\beta\tau_m)} {\sqrt{1+X^2\dfrac{t}{t_m}\left[(1-\beta\tau_m)^2-1\right]}} \tag{23} \]

Fig. 7 gives the attenuation curves of the ground-fault current with time for the function

\[ J_z=f\left(\frac{t}{t_m}\right) \]

according to equation (23), for values \(X=0.1;\ 0.2;\ 0.5\) and \(1\), with \(\tau_m=600^\circ\text{C}\) and \(\beta=0.009\ 1/^\circ\text{C}\) (on average for technically pure iron within the temperature range \(0\text{–}600^\circ\text{C}\)).

From Fig. 7 it is evident that, when choosing the magnitude of the permissible ground-fault current and the relay time delay, it is necessary to take account of the magnitude of the transient current, since the initial ground-fault current may exceed the steady-state current several times.

Fig. 8 shows the character of the change of the ground-fault current as a function of the number of short-circuited turns, in percent, for the following conditions: curve 1—according to equation (23) as a function of

\[ J_z=f(X\%)\quad\text{with }t=t_m;\ \tau_m=600^\circ\text{C} \]

and \(\beta=0.009\ 1/^\circ\text{C}\),

Fig. 7.

curve 2—for the iron-hydrogen Nernst rheostat \((U = 15\ \mathrm{V}\) and \(J = 0.5\ \mathrm{A})\) under the established heating-temperature regime,

curve 3—for an iron wire of diameter \(0.9\ \mathrm{mm}\), with open heating to brown tempering, and

curve 4—for a constant resistance.

On the basis of Fig. 8 it may be said that the character of the change in ground-fault current as a function of the number of short-circuited turns for iron resistances made in the form of a Nernst rheostat, owing to the insignificant influence of cooling on the temperature regime, almost completely coincides with theoretical curve 1 and obeys an equation of the form:

Fig. 8.

\[ J_3 = \frac{A J_{3m} X}{\sqrt{1 + X^2(A^2 - 1)}} . \tag{24} \]

where \(A\) is the factor by which the resistance of the iron increases when the generator terminal is grounded. For the iron-hydrogen Nernst rheostat, equation (24) proved to be completely valid for \(A = 6\).

5. Calculation of the sensitivity and selectivity of AEG-type protection when the generator neutral is grounded through an iron resistance.

When a ground fault occurs outside the protected zone, the false power at the relay terminals, on the basis of equation (7), is

\[ P_{\mathrm{rl}} = \frac{X U_{\phi}}{n_{\mathrm{n}}} J_{\mathrm{nbn}} - \frac{X U_{\phi}}{n_{\mathrm{n}}} \left( \frac{J_3}{n_{\mathrm{tv}}} - \frac{J_3}{n_{\mathrm{to}}} \right) \tag{25} \]

where \(n_{\mathrm{n}}\) is the transformation ratio of the zero-sequence voltage transformer, and \(J_3\) is the ground-fault current according to equation (24).

Let us determine \(X\) for which the resultant false power at the relay terminals reaches a maximum:

\[ \frac{d P_{\mathrm{rl}}}{d X} = \frac{U_{\phi}}{n_{\mathrm{n}}} J_{\mathrm{nbn}} - \frac{A U_{\phi}}{n_{\mathrm{n}}} \frac{ \left( \frac{J_{3m}}{n_{\mathrm{tv}}} - \frac{J_{3m}}{n_{\mathrm{to}}} \right) \left[2 + X^2(A^2 - 1)\right] }{ \left[(1 + X^2(A^2 - 1))\right]^{3/2} } = 0 . \]

Reducing the expression obtained by \(\dfrac{U_{\phi}}{n_{\mathrm{n}}}\), we make the substitution

\[ 1 + X^2(A^2 - 1) = y; \tag{26} \]

then we obtain

\[ J_{\mathrm{nbn}} = \frac{ A \left( \frac{J_{3m}}{n_{\mathrm{tv}}} - \frac{J_{3m}}{n_{\mathrm{to}}} \right) (1 + y)\sqrt{y - 1} }{ \sqrt{A^2 - 1}\, y^{3/2} } \]

or

\[ y^{3/2} = \sqrt{\frac{A}{A^2 - 1}} \cdot \left( \frac{ \frac{J_{3m}}{n_{\mathrm{tv}}} - \frac{J_{3m}}{n_{\mathrm{to}}} }{ J_{\mathrm{nbn}}} \right) (1 + y)\sqrt{y - 1}. \tag{27} \]

We shall take as an additional condition that

\[ \frac{A}{\sqrt{A^{2}-1}}\cdot \left( \frac{\dfrac{J_{3m}}{n_{\mathrm{tv}}}-\dfrac{J_{3m}}{n_{\mathrm{to}}}}{J_{\mathrm{nbn}}} \right)=1 \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots (28) \]

Since the first factor of condition (28) is very close to unity, this condition corresponds to the basic idea of the device of this protection—to compensate the unbalance current in the secondary circuit of the protection in the event of an external ground fault. Then equation (27), after raising to the second power, is written in the form: \(Y^{3}=Y^{3}+Y^{2}-Y-1\),

whence \(Y=\dfrac{1+\sqrt{5}}{2}\), and the value of \(X\), upon substituting \(Y\) into equation (26), will be:

\[ X=\frac{\sqrt{Y-1}}{\sqrt{A^{2}-1}}= \frac{0.786}{\sqrt{A^{2}-1}} \]

Substitute the obtained value of \(X\) and the value of \(J_{3}\) from (24) into equation (25); then we obtain

\[ P_{\mathrm{rl}\,\max}= \frac{0.786\,U_{\mathrm{f}}\,J_{\mathrm{nbn}}}{n_{\mathrm{n}}\sqrt{A^{2}-1}} - \frac{0.62\,A\,U_{\mathrm{f}} \left(\dfrac{J_{3}}{n_{\mathrm{tv}}}-\dfrac{J_{3m}}{n_{\mathrm{to}}}\right)} {n_{\mathrm{n}}(A^{2}-1)\sqrt{1+0.62}} \]

Taking condition (28) into account, the expression obtained can be simplified

\[ P_{\mathrm{rl}\,\max}= \frac{0.3\,U_{\mathrm{f}}\,J_{\mathrm{nbn}}}{n_{\mathrm{n}}\sqrt{A^{2}-1}} \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots (29) \]

The pickup power must be greater than the maximum false power at the relay terminals

\[ P_{\mathrm{rt}}= K_{\mathrm{n}}P_{\mathrm{rl}\,\max} = K_{\mathrm{n}}\frac{0.3\,U_{\mathrm{f}}\,J_{\mathrm{nbn}}}{n_{\mathrm{n}}\sqrt{A^{2}-1}} \qquad \ldots \ldots \ldots \ldots \ldots (30) \]

where \(K_{\mathrm{n}}\) is the reliability coefficient.

The ratio of the transformation coefficients of the zero-sequence and external current transformers can be determined on the basis of condition (28)

\[ n_{\mathrm{to}}= \frac{A\,J_{3m}\,n_{\mathrm{tv}}} {A_{3m}-n_{\mathrm{tv}}J_{\mathrm{nbn}}\sqrt{A^{2}-1}} \qquad \ldots \ldots \ldots \ldots \ldots (31) \]

Let us also determine the percentage of unprotected turns. The sensitivity equation for protection of type \(AEG\), in the case of operation of one generator and a ground fault inside the protection zone, has the form:

\[ \frac{X\,U_{\mathrm{f}}}{n_{\mathrm{n}}} \left( \frac{J_{3}}{n_{\mathrm{to}}}-J_{\mathrm{nbn}} \right)=P_{\mathrm{rt}} \qquad \ldots \ldots \ldots \ldots \ldots (32) \]

Substitute into equation (32) the value of \(J_{3}\) from (24) and \(P_{\mathrm{rt}}\) from (30); then we obtain:

\[ \frac{X^{2}\,U_{\mathrm{f}}\,A\,J_{3m}} {n_{\mathrm{n}}\,n_{\mathrm{to}}\sqrt{1+X^{2}}\,(A^{3}-1)} = \frac{X_{\mathrm{f}}\,U_{\mathrm{f}}\,J_{\mathrm{nbn}}}{n_{\mathrm{n}}} = \frac{0.3\,K_{\mathrm{n}}\,U_{\mathrm{f}}\,J_{\mathrm{nbn}}} {n_{\mathrm{n}}\sqrt{A^{2}-1}} \]

Reducing the obtained expression by \(\dfrac{U_{\mathrm{f}}}{n_{\mathrm{n}}}\) and making the substitution \(X=\dfrac{\alpha_{0}}{100}\),

then we obtain:

\[ \frac{J_{z m}}{n_{\mathrm{to}} J_{\mathrm{nbn}}} = \frac{1}{A} \left( 1+\frac{30 K_{\mathrm{n}}}{a_0 \sqrt{A^2-1}} \right) \sqrt{\frac{10^4}{a_0^2}+(A^2-1)} \qquad (33) \]

In equation (33) we shall make one more substitution, of \(n_{\mathrm{to}}\) from (31); then

\[ \frac{J_{z m}}{n_{\mathrm{tv}} J_{\mathrm{nbn}}} = \frac{1}{A} \left[ \sqrt{A^2-1} \left( 1+\frac{30 \cdot K_{\mathrm{n}}}{a_0^2 \sqrt{A^2-1}} \right) \sqrt{\frac{10^4}{a_0^2}+(A^2-1)} \right] = m \qquad (34) \]

Solving this equation with respect to \(a_0\) does not appear possible; therefore Fig. 9 gives a dependence of the form \(m=f(a_0\%)\) for various values of \(A\).

Having determined \(m\) from

\[ m=\frac{J_{z m}}{n_{\mathrm{tv}} J_{\mathrm{nbn}}} \qquad \cdots \qquad (35) \]

from Fig. 9, for a definite value of \(A\), one can determine the percentage of unprotected turns, or conversely.

For comparison of the advantage of grounding the neutral point of generators through iron resistances, Fig. 10 gives the dependence of the maximum ground-fault current on the generator rated current, according to equation (34), for \(A=6\), and for the secondary rated current of the transformers \(J_{2\mathrm{n}}=5a\).

As is seen from Fig. 10, when the generator neutral is grounded through an iron resistance, the maximum ground-fault current may be reduced

Fig. 9.

Fig. 10.

by a factor of 4 and is obtained within the limits of permissible values. But the use of iron resistances for this purpose will in practice be possible only when the unbalance current is reduced to a few milliamperes, since for large \(J_{z m}\) the device for grounding the generator neutral becomes bulky and expensive. Iron resistances must be made in evacuated or gas-filled cylinders for direct connection into the neutral or through a step-down transformer.

6. Increasing the sensitivity of AEG-type protection by replacing the external transformers with a Bushing-type current transformer

The author carried out an investigation of a Bushing-type current transformer under conditions of its operation in the ground-fault protection scheme. The test was carried out in order to determine the magnitude of possible unbalance currents in the secondary circuit of the transformer for the sum of primary ampere-

turns, equal to zero. The magnitude of the unbalance current was measured with the loop of an oscillograph, which was used as a vibration galvanometer. The sensitivity of the measurement was 11.6 mm of spot deflection per 1 ma at a distance of 4 m from the loop to the screen.

The test results are given in Fig. 11, and the circuit according to which the tests were carried out is shown in Fig. 12.

Fig. 11.

Figure 11 shows the dependence of the unbalance current in the secondary circuit of the transformer on the magnitude of the primary current for different loop widths \(d\), with the number of secondary turns \(W_2 = 120\) and the resistance of the secondary circuit \(Z_2 = 2.9\ \Omega\)—solid curves, and for \(W_2 = 240\) and \(Z_2 = 3.4\ \Omega\)—dashed curves.

Fig. 12.

Further, Fig. 13 gives the dependence of the unbalance current on the loop width \(d\) in mm at a primary current \(J_1 = 1200\) a, with \(W_2 = 120\) and \(Z_2 = 2.9\ \Omega\). In this case one conductor was fixed at the center of the core, while the other was moved in the radial direction. The last reading (\(d = 80\) mm) was taken with the movable conductor pressed against the core.

From an analysis of the results it is evident that for Bushing-type current transformers the magnitude of the unbalance current corresponds to that for the best-quality transformers connected according to the Holmgreen circuit. Therefore, of great interest is the replacement, in accordance with the circuit of Fig. 14, of the external protection transformers of the AEG type by a Bushing-type current transformer. Such a replacement will make it possible to increase the sensitivity of the protection and reduce its cost.

Fig. 13.

Fig. 14.

For comparison with the preceding examples, Fig. 15 gives the dependence of the maximum ground-fault current on the nominal current of the generators, according to equation (10), with the number of unprotected turns \(\alpha_0 = 20\%\), the Bushing transformation ratio \(n_{\mathrm{tv}} = 120\), and the unbalance current according to Fig. 16 (the curve for \(d = 15\) mm), on which the curves of Fig. 11 are continued up to a primary current of 5000 a. The curve for \(d = 15\) mm was taken because the thickness of the insulation between the cable conductors is usually insignificant.

The relay pickup power and the transformation ratio of the zero-sequence current transformer can be determined from the corresponding equations for normal protection of type AEG.

When such protection is implemented, it is necessary to ground the lead sheath and armor of the generator cable only at the generator end, in order to avoid the possibility of currents flowing in them and, consequently, incorrect operation of the protection.

Fig. 15.

Fig. 16.

7. Selectivity of AEG-Type Protection

The relay pickup power, determined according to equation (9), ensures selectivity of the protection only at the rated load of the external transformers and in the event of a ground fault outside the protection zone, since in deriving the basic equations for calculating the protection the unbalance current of Holmgreen’s circuit is usually taken as constant and equal to that at rated load. But in the case of external short circuits with simultaneous grounding, the increased unbalance current may prove to be in phase with the voltage at the relay terminals, and the protection may operate nonselectively, as is confirmed by operating practice with this protection. Therefore, at a number of power stations in the Union, in order to increase selectivity, the AEG ground-fault protection is blocked by the maximum generator protection.

It is of interest to determine at what secondary current of the external transformers the protection, having a pickup power according to equation (9), will operate nonselectively. For this purpose we equate equations (7) and (9):

\[ P_{\mathrm{r.l}} = P_{\mathrm{r.t}} = \left[J_{\mathrm{nb}} - \left(J_{\mathrm{nb}} - (J_{\mathrm{v}} - J_{\mathrm{n}})\right)\right] U_{\mathrm{r}} = 0.005 \sqrt{K_{\mathrm{n}}}\, J_{\mathrm{nbn}} U_{\mathrm{rm}} a_{0}. \]

On the basis of (16) one may write the following relation:

\[ \left(\frac{J_{\mathrm{nb}}}{J_{\mathrm{nbn}}}\right) = \left(\frac{J_{2}}{J_{2\mathrm{n}}}\right)^{3/5}, \]

where \(J_{\mathrm{nbn}}\) is the unbalance current at the rated secondary current of the transformers \(J_{2\mathrm{n}}\).

In addition, the substitution can be made

\[ J_{\mathrm{v}} = \frac{J_{\mathrm{z m}}}{n_{\mathrm{tv}}}\frac{U_{\mathrm{r}}}{U_{\mathrm{rm}}}, \qquad J_{\mathrm{n}} = \frac{J_{\mathrm{z m}}}{n_{\mathrm{to}}}\frac{U_{\mathrm{r}}}{U_{\mathrm{rm}}}. \]

and \(n_{t0}=q\,n_{\mathrm{tv}}\). Then equation (36) takes the form:

\[ \left(\frac{J_2}{J_{2\mathrm{n}}}\right)^{3/5} = \frac{J_{3\mathrm{m}}(q-1)U_{\mathrm{p}}}{q n_{\mathrm{tv}} J_{\mathrm{nbn}} U_{\mathrm{pm}}} + 0.005\sqrt{K_{\mathrm{n}}}\,\alpha_0\,\frac{U_{\mathrm{pm}}}{U_{\mathrm{p}}} \ldots \tag{37} \]

If, in addition, the value of \(J_{3\mathrm{m}}\) from (10) is substituted, then

\[ \left(\frac{J_2}{J_{2\mathrm{n}}}\right)^{3/5} = (1+\sqrt{K_{\mathrm{n}}})\,\frac{q-1}{q}\,\frac{U_{\mathrm{p}}}{U_{\mathrm{pm}}}\,\frac{100}{\alpha_0} + 0.005\sqrt{K_{\mathrm{n}}}\,\alpha_0\,\frac{U_{\mathrm{pm}}}{U_{\mathrm{p}}} \ldots \tag{38} \]

In the case of a two-phase short circuit outside the protection zone with simultaneous grounding, the initial voltage at the relay terminals is

\[ U_{\mathrm{p\,init}} = 0.5\,U_{\mathrm{pm}}, \ldots \tag{39} \]

and, for the steady-state condition during a short circuit,

\[ U_{\mathrm{py}}\simeq 0.2\,U_{\mathrm{pm}}. \ldots \tag{40} \]

Substituting into equation (38) the value of \(U_{\mathrm{p}}\) from (39) or (40), one can determine the secondary current of the external transformers at which the protection will operate nonselectively.

In Fig. 17 the dependence

\[ \frac{J_2}{J_{2\mathrm{n}}}=f(\alpha_0\%) \]

has been plotted according to equation (38) for \(U_{\mathrm{p\,init}}\) and \(U_{\mathrm{py}}\), which gives the limit of selective operation of the protection for the case being analyzed. In constructing it, the following were taken: \(K_{\mathrm{n}}=1.25\) and \(q=1.36\). From Fig. 17 it is evident that, for the case under consideration, the \(AEG\) protection with sensitivity up to 95% will operate nonselectively, and only at a sensitivity from 95 to 100% will selectivity be obtained. Moreover, the multiple of the secondary current limiting the zone of selective operation, for sensitivities of 50–80% encountered in practice, varies within the limits \((1.2—2)J_{2\mathrm{n}}\).

Fig. 17.

8. Conclusion.

In conclusion it must be said that the simplest and most effective measures for increasing the sensitivity of \(AEG\)-type protection should be considered to be:

1. Reducing the transformation ratio of the external current transformers while simultaneously increasing the cross section of the secondary winding to the maximum possible limits.
2. Grounding the neutral of generators through iron resistances.
3. Mandatory selection of external current transformers under factory conditions.

Carrying out these measures will make it possible, with protection sensitivity of 80–90%, to reduce the maximum ground-fault current to several amperes.

It must be borne in mind that an excessive reduction of the transformation ratio of the external transformers, with large impedances of the relay current coils (the \(AEG\) relay 8 ohms, and the KhEMZ relay 50, 13, and 3.5 ohms, corresponding to operating powers of 0.11, 0.22, and 0.44 watt), may so in-

increase the shunting-off of the secondary ground current by external transformers, so that the sensitivity of the scheme may prove negligible (the quantitative accounting of shunting is sufficiently covered in the literature (L 7).

Recently proposals have appeared in the literature (L 7) to reduce the shunting-off of the secondary ground current by external transformers by using, for Holmgreen’s scheme, transformers with additional magnetization by an extraneous current. It is recommended that the additional magnetization be brought up to 5000–6000 gauss, since at this value of magnetic induction the iron used for transformers has maximum magnetic permeability.

It must be borne in mind that external current transformers with additional magnetization by alternating current can create larger unbalance currents than ordinary transformers, since the magnetizing current will create in the secondary circuit of the protection its own unbalance current, which will be summed from all three transformers. In this case the normal operation of the protection will also depend on the serviceability of one additional circuit, which is generally undesirable.

The simplest method of reducing shunting must be considered to be the lowering of the impedance of the current coils of the wattmeter-type ground relays. It may even be necessary to abandon the induction-relay type if this type, at the required sensitivity, does not permit reducing the impedance of the current coil.

A ferrodynamic wattmeter relay of extremely high sensitivity can be manufactured. The principle of construction of similar other devices is known. It is only necessary to note that such a relay, in order to create the corresponding torque, will not require a large number of ampere-turns of the windings; moreover, to increase the sensitivity of this relay, an iron resistance or an ordinary tungsten lamp may be connected into the voltage winding instead of an additional resistance.

II. Ground-fault protection of the Ferranti type.

Protection of the Ferranti type according to the scheme of Fig. 18, which theoretically is very perfect, has not yet found wide application in practice because of the absence of sufficient investigations under the operating conditions, in the protection scheme, of the Bushing-type current transformer necessary for its implementation. The investigations carried out by the author are not exhaustive; however, on the basis of the results obtained one can judge the character of the increase of the unbalance current in the secondary circuit of the current transformer as a function of the magnitude of the primary current, and consequently the basic parameters of the protection can be determined. The relay pickup current can be determined from the condition

Fig. 18.

\[ J_{\mathrm{rt}} = K_{\mathrm{n}} J_{\mathrm{nb}\ \max}, \ldots \ldots \ldots \ldots \ldots \ldots (41) \]

where \(K_{\mathrm{n}} = 1.25\) is the reliability coefficient of the protection, and
\(J_{\mathrm{nb}\ \max} = f(J_{\mathrm{k.z.}})\) is the unbalance current in the secondary circuit of the current transformer during a two-phase short circuit outside the protection zone, which can be determined according to the equation

\[ J_{\mathrm{nb}} = \left( \sqrt{73 + 0.1173\,J_{2\mathrm{n}} - 8.55} \right) 10^{-3}\ \mathrm{amp} \ldots \ldots \ldots (42) \]

upon substitution, instead of the rated current of the generator, of the value of the initial short-circuit current at the generator terminals. Equation (42) corresponds exactly to the curve for \(d = 15\) mm, Fig. 15.

The magnitude of the maximum ground-fault current can be determined from the equation

\[ J_{\mathrm{g.m}} = J_{\mathrm{rt}}\, n_{\mathrm{tb}} \frac{100}{a_0} \qquad \ldots \ldots \ldots \ldots \ldots \ldots \ldots (43) \]

Fig. 19 gives the dependence of the maximum ground-fault current on the rated current of the generator at \(a_0 = 20\%\), \(n_{\mathrm{tb}} = 120\), and at a tenfold initial short-circuit current of the generator.

In this case, the lead sheath and armor of the cables must likewise be grounded only at the generator end.

Fig. 19.

III. Ferranti-Type Differential Protection for Generators.

Ferranti-type differential protection has likewise not become widely used in practice because of the absence of corresponding studies in the field of the design of this protection. The Guidelines for Relay Protection (Energoizdat, 1933) make no mention at all of the possibility of applying it; meanwhile this protection, being theoretically ideal, can completely solve the problem of generator protection. To implement such protection, special current transformers with two primary windings are required. Until now such current transformers have been made either multi-turn or single-turn with adjacent electrodes. Therefore such protection was used only for generators of small capacity, since for large generators the differential current transformers proved to have impractical dimensions.

The author has investigated a special design of a through-type single-turn differential current transformer with concentric primary electrodes, shown in Fig. 20. Such a transformer, with uniform distribution of current over the cross section of the electrodes and with the sum of the primary ampere-turns equal to zero, should have no unbalance current in the secondary circuit, but in practice there will always be some asymmetry of the actual current density, since the separate current paths in the electrodes and terminals will have different resistances. The centers of current distribution and the cross section of the conductor have no special influence on the magnitude of the unbalance current.

Fig. 20.

To achieve sufficient sensitivity of the protection, the transformation ratio of the transformer may be taken within the range of 100–150.

in the electrodes will be displaced, and this displacement will be greater for the outer electrode, since it has larger geometrical dimensions. Therefore, the theory of such a transformer may be reduced to the theory of a transformer with non-concentric parallel electrodes (Fig. 21-a).

Fig. 21.

The primary currents \(J_{\mathrm{n}}\) (of the outer electrode) and \(J_{\mathrm{v}}\) (of the inner one), equal in magnitude but opposite in phase, create fluxes \(\Phi_{\mathrm{n}}\) and \(\Phi_{\mathrm{v}}\), which induce in the corresponding parts of the secondary winding opposing electromotive forces, the difference of which will determine the magnitude and phase of the unbalance current in the secondary circuit of the transformer. In this case, the magnitude of the difference of the secondary electromotive forces will be proportional to the arithmetic difference of the flux linkages formed by the magnetic fields of the electrodes (Fig. 21-c).

\[ E_{\mathrm{nb}} \equiv \Phi_{w\mathrm{n}} - \Phi_{w\mathrm{v}}, \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (44) \]

where \(\Phi_{w\mathrm{n}}\) is the number of flux linkages formed by the field of the outer electrode, and \(\Phi_{w\mathrm{v}}\) by the inner one.

The magnitude of the unbalance current is

\[ J_{\mathrm{nb}} = \frac{E_{\mathrm{nb}}}{Z_2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (45) \]

where \(Z_2\) is the total resistance of the secondary circuit of the transformer, and its phase will depend on the relative magnitude of the flux linkages \(\Phi_{w\mathrm{n}}\) and \(\Phi_{w\mathrm{v}}\) (Fig. 21-c) and can vary only within insignificant limits, depending on the magnitude and character of the generator load.

The results of tests carried out by the author are presented in Figs. 22 and 23. Fig. 22 gives the dependence of the magnitude of the unbalance current on the primary current for various diameters \(d_{\mathrm{mm}}\) of the outer electrode, with the number of secondary turns \(W_2 = 120\) and with the resistance of the secondary circuit of the transformer \(Z_2 = 2.9\ \Omega\), and Fig. 25—with \(W_2\) 240 and \(Z_2 = 3.4\ \Omega\).

In addition, it was found that:

1) the unbalance current is created by the magnetic flux of the outer electrode in accordance with the vector diagram (Fig. 21-c);

2) the magnitude of the unbalance current is inversely proportional to the resistance of the secondary circuit of the transformer, according to Eq. (45), and

3) the angle \(\alpha\) (Fig. 21-c), determining the phase of the unbalance current, is close to zero.

The measurement of the magnitude and the determination of the phase of the unbalance current were carried out by means of a loop oscillograph, according to the circuit of Fig. 12.

Fig. 22.

As can be seen from the test results, the magnitude of the unbalance current of the transformer of the proposed design is quite insignificant and also depends on the ratio of the diameters of the inner and outer electrodes. According to Figs. 22 and 23, for an outer electrode with a diameter of 75 mm the unbalance current proved to be the smallest.

grounding transformer (Gestelldrossel) connected to the station buses. The voltage winding of the relay is connected to the grounding transformer.

Such a protection arrangement will make it possible to assign to the grounding transformer the compensation of the capacitive ground-fault current and to obtain constancy of the quality of protection (absence of a ground-fault current supplied from other generators).

Fig. 27.

Fig. 28.

Protection of the generator against a ground fault before it is connected to the station buses in this case is of no particular importance, since the neutral of the generator is not grounded, and the capacitive ground-fault current due to the capacitance of the generator is insignificant.

In conclusion it is necessary to say that the combined generator protection proposed by the author is, in sensitivity and selectivity, almost ideal and extremely simple; moreover, it requires only three current transformers, the cost of which will be only insignificantly higher than the cost of ordinary transformers, whereas when ordinary ground-fault protection of the AEG type is used, four current transformers are required and six for differential protection, i.e., ten in all; the circuit thus becomes extremely cumbersome and the protection unreliable.

LIST OF REFERENCES USED

1. Ivanov V. I. “Relays and Relay Protection.” Moscow, 1932.
2. Ivashchenko N. D. “Relays.” Moscow, 1929.
3. Prof. N. F. Morgolin and Eng. A. A. Shiferson. “Protection of Generators and Generator-Transformer Units.” Moscow, 1932.
4. Glavenergo. “Guidelines on Relay Protection.” Moscow, 1933.
5. A. M. Fedoseev and A. A. Filshtinsky. “Ground-Fault Protection of Generators Operating Directly on the Busbars.” Elektricheskie stantsii journal, 1933, No. 3.
6. Ibid. “Commissioning and Operation of Generator Protections at Power Stations of the Union during the Period 1929–1933.” Elektricheskie stantsii journal, 1933, No. 9.
7. A. M. Fedoseev. “Current Distribution in Protection Schemes with Allowance for the Impedances of Current Transformers.” Elektricheskie stantsii journal, 1934, No. 11.
8. V. Yu. Gessen and Ya. D. Rizin. “On the Question of Permissible Generator Ground-Fault Currents.” Elektricheskie stantsii journal, 1934, No. 12.

the unbalance currents can be reduced almost to zero. In an external two-phase short circuit with simultaneous grounding, the individual unbalance currents of the transformers will increase, and so will their sum, but its magnitude will hardly be sufficient for relay operation. Therefore such protection may be considered fully selective.

The relay pickup current may be taken to be of the order of several milliamperes; in this case the maximum ground-fault current, for 80% of the protected turns, will be, according to Fig. 27, 3.5 A.

With such a relay connection, the sensitivity of the protection will be reduced owing to the shunting of the grounding current in the secondary circuit of the protection by the non-operating transformers. To avoid the influence of shunting on the sensitivity of the protection, such protection can be arranged with three wattmetric ground relays, the current windings of which may be connected in series with the differential relays of Fig. 25, while the voltage windings may be connected in parallel to a common zero-sequence voltage transformer. With this arrangement the protection will have high sensitivity and absolute selectivity.

In an external two-phase short circuit, with simultaneous grounding, the unbalance current of the current transformer together with the voltage at the terminals \(U_{\mathrm{p}}\), according to Fig. 21, \(b\), will create negative (braking) power, and the relay will not operate.

In the event of a ground fault within the protection zone bounded by the transformers, the grounding current in the secondary circuit of the protection will, together with the voltage at the relay terminals, produce positive power (Fig. 26), and the relay will operate.

The relay pickup power may be determined, according to Figs. 21 and 26, from the following equation

\[ P_{\mathrm{pt}} \simeq (J_{3} - J_{\mathrm{nbn}})\,U_{\mathrm{p}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (46) \]

where \(J_{3}\) is the grounding current in the secondary circuit of the protection, and

\(J_{\mathrm{nbn}}\) is the unbalance current at the nominal load of the transformers.

The relay pickup current on the basis of equation (46) is

\[ J_{\mathrm{pt}} \simeq J_{3} - J_{\mathrm{nbn}} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (47) \]

Fig. 26.

The maximum ground-fault current when the number of unprotected turns is \(a_{0}\%\) and the transformation ratio of the current transformers is \(n_{\mathrm{t}}\) can be determined from the equation

\[ J_{3\mathrm{m}} = \frac{100}{a_{0}}\, n_{\mathrm{t}} J_{3} \simeq \frac{100}{a_{0}}\, n_{\mathrm{t}}(J_{\mathrm{pt}} + J_{\mathrm{nbn}}) \ldots \ldots \ldots \ldots \ldots (48) \]

where \(J_{3}\) is according to equation (47).

In Fig. 29 curves are plotted as a function \(J_{3\mathrm{m}} = f(J_{3\mathrm{n}})\), according to equation (48), for \(a_{0} = 20\%\) and \(n_{\mathrm{t}} = 120\), for relay pickup currents \(J_{\mathrm{pt}} = 5\) and 10 mA.

The unbalance current is taken from Fig. 28 for \(d = 100\) mm, on which the curves (Fig. 24) have been extended to a primary current of 5000 A.

In networks with a large capacitive grounding current (modern factory and urban networks), the proposed protection can be arranged without grounding the generator neutral. As a source of the wattless ground-fault current necessary for operation of the protection, a special ...