On the Use of Longitudinal Differential Protection of Generators to Protect Them Also Against Ground Faults

I. D. Kutyavin

At the present time, for the protection of generators against ground faults, differential zero-sequence protection, shown in Fig. 1 (scheme 1), is beginning to find application. As has been shown by a number of researchers (G. I. Atabekov, A. M. Fedoseev, and others), this scheme, in its qualities, is not inferior to maximum-power zero-sequence protection with compensation of the unbalance current.

This circumstance indicates the possibility of further simplifying generator protection by using longitudinal differential protection, made according to scheme 2 or 3, Fig. 1, also for protecting them against ground faults.

scheme 1   scheme 2   scheme 3

Fig. 1

ground faults. To obtain the indicated schemes it is necessary to connect, into the relay circuit of normal differential protection, as shown in Fig. 1, one additional relay in each phase (scheme 2), or only in the neutral wire (scheme 3). The operating circuit of these relays is blocked by overcurrent protection. It is therefore of interest to compare the three schemes indicated above with respect to their sensitivity and other qualities.

Some considerations on the magnitude of the unbalance currents of the indicated schemes

The schemes under consideration are not equivalent with respect to the magnitude of the unbalance current. Scheme 2, in our opinion, will have the smallest unbalance current, since only two transformers, located under identical magnetic conditions, participate in its creation, if one does not take into account the possib—

the possibility of the appearance in their cores of different residual magnetization. This residual magnetization in transformers of the same phase is created by the same through overcurrent and therefore should be the same. In practice, however, it is necessary to reckon with the possibility of a decrease of residual induction in individual transformers owing to temperature fluctuations, mechanical vibrations, etc.

The unbalance current in the relay of circuit 3 will be equal to the sum of the unbalance currents of the individual phases of circuit 2. Suppose that the unbalance currents in the phases of circuit 2 are positive and in the vector diagram of Fig. 2a are represented by the vectors $i_{nb1}$, $i_{nb2}$, and $i_{nb3}$; then the current in the neutral conductor (and consequently also in the relay of circuit 3) will be represented by the vector $i_{nb}^{(3)}$. Such a relationship of the unbalance currents will occur in the case when all the transformers of the circuit are in the same magnetic state and when one of the transformer sets (the zero-sequence set or the external one) has a higher accuracy class, or when one of the sets is more heavily loaded than the other. Under these conditions the current $i_{nb}^{(3)}$ will have the smallest values. In the particular case when the currents $i_{nb1}$, $i_{nb2}$, and $i_{nb3}$ constitute a symmetrical system of vectors, the current $i_{nb}^{(3)}$ will be equal to zero.

With equal accuracy and loading of the transformer sets, the unbalance currents in the phases will be smaller and, with equal probability, may have a plus or minus sign; therefore the most probable situation is that two of these currents will have one sign, while the third current will have the opposite sign (Fig. 2b). In this case the current $i_{nb}$ will be approximately twice as large as the unbalance currents in the phases. Such is the relationship between the unbalance currents flowing through the relay in circuits 2 and 3.

In creating the unbalance current of circuit 1, four transformers take part. The external transformers generate an unbalance current containing all harmonics. The magnetic state of these transformers will be different, despite the symmetry of the generator load, because of the appearance in their cores, under the action of through asymmetrical overcurrents, of residual magnetization. The magnitude of this magnetization in individual transformers will be different and will change after each instance of through overcurrent flow; in connection with this, the unbalance current of the external transformers will sharply change its magnitude and phase.

Fig. 2

The zero-sequence transformer will generate into the protection circuit only harmonics that are multiples of three. Since the circuit under consideration is connected for circulation of third harmonics, the current of the zero-sequence transformer will compensate part of these harmonics contained in the unbalance current of the external transformers (the part of the third harmonics that is transformed from the primary circuit), and will take upon itself part of the current of the remaining harmonics. It follows from this that the zero-sequence transformer reduces the unbalance current of the external transformers entering the relay.

Let us now try to establish the relationship between the unbalance currents of circuits 1, 2, and 3. The relationship between the unbalance currents of circuits 1 and 3 can be established most simply. Indeed, circuit 3 may be regarded as consisting of two filters of zero-sequence currents (analogous to the circuit of external transformers of circuit 1), connected with respect to the relay for circulation. Each of these filters has its own unbalance current, commensurate with the unbalance current of the external transformers of circuit 1.

If we denote the unbalance currents of the individual filters of circuit 3 by \(i'_{ub}\) and \(i''_{ub}\), then the current in the relay will vary within the following limits:

\[ i_p = i_{ub}^{(3)} = i'_{ub} + i''_{ub}, \]

with both limits having the same probability.

Let us now assume that the unbalance currents of the individual filters of circuits 1 and 3 are identical; then the current in the relay of circuit 3 will be twice as large as the current in the relay of circuit 1. But above we obtained the same ratio for the unbalance currents in the relays for circuits 2 and 3. On this basis it may be concluded that, with respect to unbalance currents, circuits 1 and 2 are approximately equivalent. In practice, however, it may be expected that circuit 2 will have the smallest unbalance current of all the circuits considered.

Dependence of the Unbalance Current of the Protection Circuits Considered on the Relay Resistance

The dependence of the unbalance current on the relay resistance in calculating protections against ground faults is not presently taken into account; meanwhile, as will be shown below, this dependence is very significant and cannot be neglected.

The above-mentioned dependence for longitudinal differential protection of generators (circuit 2, Fig. 1) can be obtained from the known expression for the unbalance current of this circuit¹

\[ i_{ub} = \frac{I_1\left(z_{n1}z_2 - z_{n2}z_1\right)} {z_p\left(z_{n1}+z_1+z_{n2}+z_2\right)+\left(z_{n1}+z_1\right)\left(z_{n2}+z_2\right)} \tag{1} \]

The no-load impedances of the transformers entering the denominator of expression (1) may be considered equal:

\[ z_{n1}+z_1 = z_{n2}+z_2 = z_x . \tag{2} \]

Then expression (1) will take the following form:

\[ i_{ub} = \frac{I_1\left(z_{n1}z_2 - z_{n2}z_1\right)} {z_x\left(z_x+2z_p\right)} . \tag{2} \]

On the basis of this expression, the sought dependence \(i_{ub}=f(z_p)\) for the circuit under consideration can be represented in the following form:

\[ i_{ub1}=i_{ub2}\frac{z_x+2z_{p2}}{z_x+2z_{p1}} . \tag{3} \]

An analogous dependence can also be obtained for the three-phase zero-sequence current filter shown in Fig. 3a.

Using the superposition method, let us assume that in the primary circuit current flows only in phase \(A\). Then the equivalent circuit of the filter will take the form shown in Fig. 3b.

Fig. 3

To simplify the solution of the problem, we shall assume (as above) that the total no-load impedances of the transformers are equal:

\[ z_{na}+z_{2a}=z_{nb}+z_{2b}=z_{nc}+z_{2c}=z_x . \tag{4} \]

The secondary current of phase \(A\), on the basis of the indicated equivalent circuit, will be equal to:

\[ i_{2a}=-\frac{i_a z_{na}}{z_{na}+z_{2a}+\dfrac{z_x z_p}{z_x+2z_p}} = \frac{i_a z_{na}(z_x+2z_p)}{z_x(z_x+3z_p)} . \tag{5} \]

The current in the relay from phase \(A\):

\[ i_{pa}=i_{2a}\frac{z_x}{z_x+2z_p} \tag{6} \]

or, taking expression (5) into account:

\[ i_{pa}= \frac{i_a z_{na}(z_x+2z_p)}{z_x(z_x+3z_p)} \cdot \frac{z_x}{z_x+2z_p} = -\frac{i_a z_{na}}{z_x+3z_p}. \tag{7} \]

Expressions similar to (7) can also be obtained for the currents in the relay \(i_{pb}\) and \(i_{pc}\), if the primary current is assumed to flow successively in phases \(B\) and \(C\).

The total current in the relay of the circuit of Fig. 2a will be equal to the sum of the currents \(i_{pa}\), \(i_{pb}\), \(i_{pc}\) and will be the unbalance current of this circuit:

\[ i_{nb}=i_{pa}+i_{pb}+i_{pc} = \frac{i_a\left(z_{na}+a^2 z_{nb}+a z_{nc}\right)}{z_x+3z_p}, \tag{8} \]

where \(a\) is the three-phase-current operator.

On the basis of expression (8), the required dependence \(i_{nb}=f(z_p)\) can be represented in the following form:

\[ i_{nb1}=i_{nb2}\frac{z_x+3z_{p2}}{z_x+3z_{p1}} . \tag{9} \]

Using this method, one can obtain the dependence \(i_{nb}=f(z_p)\) also for circuit 1 of Fig. 1, which has the following form (under the condition of equality of the no-load impedances of all four transformers):

\[ i_{nb2}=i_{nb1}\frac{z_x+4z_{p2}}{z_x+4z_{p1}} . \tag{10} \]

Similarly, for circuit 3 of Fig. 1:

\[ i_{nb1}=i_{nb2}\frac{z_x+6z_{p2}}{z_x+6z_{p1}} . \tag{11} \]

The obtained dependence \(i_{nb}=f(z_p)\) can be expressed in the general form:

\[ i_{nb1}=i_{nb2}\frac{z_x+m z_{p2}}{z_x+m z_{p1}}, \tag{12} \]

where \(m\) is the number of transformers participating in the generation of the given unbalance current.

The conclusions given above with respect to the dependence \(i_{nb}=f(z_p)\) agree very well with experience.

Fundamentals of Calculating the Protections under Consideration

Let us first consider the general principles for calculating the protections under consideration. In view of the fact that the sensitivity of the protection is often limited by the maximum permissible current in the arc at the point of insulation breakdown (150–200 A), the protection calculation must be based on attaining optimal conditions under which, for a specified dead zone, there corresponds the minimum primary pickup current. This condition is satisfied when the resistance of the relay winding is equal to the sum of the no-load resistances of the current transformers feeding this relay¹

\[ z_p = \sum (z_x). \tag{13} \]

The indicated sum \(\sum (z_x)\) represents the total resistance of the protection circuit to the current that would be produced by an emf introduced into the circuit in place of the relay. Thus, for example, for circuit 1, Fig. 1, \(\sum (z_x)=z_x:4\); for circuit 2, \(\sum (z_x)=z_x:2\); and for circuit 3, \(\sum (z_x)=z_x:6\).

The relay pickup current must be greater than the unbalance current at the blocking current in the primary circuit:

\[ i_{pT} = k_n i_{nbb}. \tag{14} \]

The unbalance current at the blocking current (if impedance angles are neglected when adding resistances) is:

\[ i_{nbb} = I_{nbn}\,\frac{z_n + m z_p}{z_n + m z_p}\cdot \left(\frac{I_b}{I_{nT}}\right)^\alpha, \tag{15} \]

where
\(k_n\) — reliability coefficient;
\(i_{nbn}\) — unbalance current in the relay circuit at a primary current equal to the rated current of the transformers \(I_{nT}\);
\(z_n\) — resistance of the transformer magnetizing circuit at rated current in its primary circuit;
\(z_p\) — resistance of the measuring instrument by which the unbalance current \(i_{nbn}\) was measured;
\(I_b\) — primary blocking current.

In the right-hand side of expression (15), the unbalance current measured at some generator current \(I_g\) may be substituted; then this current must also be substituted into the denominator of the fraction.

The current in the relay for a ground fault in the protection zone is:

\[ i_p = \frac{I_{zm} b k_0}{n_T} - i_{nbr}. \tag{16} \]

With \(i_p=i_{pT}\), expression (16) can be used to determine the current \(I_{zm}\) or the magnitude of the dead zone \(b_0\):

\[ I_{zm} = \frac{(i_{pT}+i_{nbr})\,n_T}{b_0 k_0}, \tag{17} \]

where
\(I_{zm}\) — maximum ground-fault current ensuring the required sensitivity of the protection;
\(k_0\) — residual-current coefficient;
\(n_T\) — transformation ratio of the current transformers;
\(b_0\) — magnitude of the dead zone and

\(i_{\mathrm{unb},r}\) is the unbalance current in the relay circuit under the calculation condition, determined from the following expression:

\[ i_{\mathrm{unb},r}=i_{\mathrm{unb},n}\frac{z_{xp}+mz_n}{z_{xp}+mz_p}\left(\frac{I_p}{I_{nT}}\right)^\alpha, \tag{18} \]

where \(I_p\) is the calculated primary load current of the generator at which the sensitivity of the protection is determined. This current lies within the limits of the generator rated current,

\(z_{xp}\) is the impedance of the transformer magnetizing circuit at the calculated current in the primary circuit.

It should be noted here that most authors and the 1948 Guiding Instructions for Relay Protection recommend substituting into expression (17) the unbalance current at the blocking current, which leads to an unjustified overstatement of \(I_{\mathrm{g.f.}}\).

The bias coefficient is determined from the following expression:

\[ k_o=\frac{z_{xp}}{z_{xp}+mz_p}, \tag{19} \]

where \(m\) is the number of current transformers (with identical \(z_x\)) feeding the relay. Expression (19) is easily obtained if one uses the method employed to derive expression (7).

We shall now determine the calculated operating condition of the generator for which the current \(I_{\mathrm{g.f.}}\) has the greatest value. To this end, transform expression (17), taking into account (18) and (19):

\[ \frac{I_{\mathrm{g.f.}}}{n_T}=i_{\mathrm{g.f.}}= \frac{i_{pT}}{b_o}\left(1+m\frac{z_p}{z_{xp}}\right) +\frac{i_{\mathrm{unb},n}}{b_o}\left(1+m\frac{z_n}{z_{xp}}\right) \left(\frac{I_p}{I_{nT}}\right)^\alpha . \tag{20} \]

In expression (20), the impedance \(z_{xp}\) is also a function of the current \(I_p\), being directly dependent on it. As a result, the first term and the first factor in parentheses of the second term of the right-hand side of this expression, as functions of \(\left(\frac{I_p}{I_{nT}}\right)\), are represented graphically by decreasing curves. The second factor of the second term, with increasing \(\left(\frac{I_p}{I_{nT}}\right)\), increases without bound. Therefore expression (20), as a function \(I_{\mathrm{g.f.}}=f\left(\frac{I_p}{I_{nT}}\right)\), for positive values of the independent variable, has no maximum but only one minimum. It follows from this that the calculated operating condition of the generator for determining the sensitivity of the protection may be one of the limiting conditions: no-load or rated load, since all intermediate conditions must give a current \(I_{\mathrm{g.f.}}\) smaller than under one of these limiting conditions.

Expression (20) for generator no-load operation is:

\[ I_{\mathrm{g.f.}0}=\frac{i_{pT}n_T}{b_o}\left(1+m\frac{z_p}{z_o}\right). \tag{21} \]

The same for generator rated load:

\[ I_{\mathrm{g.f.}n}=n_T\left[ \frac{i_{pT}}{b_o}\left(1+m\frac{z_p}{z_n}\right) +\frac{i_{\mathrm{unb},n}}{b_o}\left(1+m\frac{z_n}{z_n}\right) \left(\frac{I_{ng}}{I_{Tn}}\right)^\alpha \right], \tag{22} \]

where \(I_{ng}\) is the rated current of the generator,

\(z_o\)—the resistance of the magnetizing circuit of the current transformer at no load of the generator, and

\(z_n\)—the same at rated load of the generator.

If relays with the optimum winding resistance, obeying expression (13), are used to implement the protection, namely:

\[ z_p=\frac{z_n}{m}, \tag{23} \]

then expressions (21) and (22) are somewhat simplified and take the following form:

a) for no load of the generator

\[ I_{zmo}=\frac{i_{eT} n_T}{v_o}\left(1+\frac{z_n}{z_o}\right); \tag{24} \]

b) for rated load of the generator:

\[ I_{zmn}=\frac{n_T}{b_o}\left[2I_{pT}+i_{nbn}\left(1+\frac{z_n}{z_p}\right)\left(\frac{I_{ng}}{I_{nT}}\right)^{\alpha}\right]. \tag{25} \]

The expressions obtained above, (14), (15), (21), (22), and (23), are the basic calculation formulas, identical for all three circuit arrangements under consideration. In the case where condition (23) can be satisfied, formulas (21) and (22) may be replaced by formulas (24) and (25).

To illustrate the recommended calculation method and to compare the three circuit arrangements under consideration, an example of calculating the protection in all three variants for generators with rated currents of 344, 825, and 2860 A is given below in tabular form. In this case, for generators with rated current 2860 A having protection according to circuit 2 or 3, the calculation was made in two variants: for current transformers with a transformation ratio of \(3000/5\) A (the calculation results are entered in the first line of each item) and \(3000/10\) A (the calculation results are entered in the second line of each item).

As can be seen from items 11 and 15 of the indicated table, the maximum ground-fault current \(I_{zm}\) for all three circuits, with the same transformation ratio and optimum relay parameters, is practically identical, which indicates the identical sensitivity of the circuit arrangements under consideration.

As can be seen from item 5, in all variants of the circuits the same maximum unbalance current was adopted. This circumstance cannot serve as grounds for doubting the possibility of such a comparison of the circuits, since the actual unbalance current can be reduced practically to the same value for any of the circuits of Fig. 1.

Conclusions

From comparison of the calculation results given in the table it is evident that, when relays with optimum parameters are used, the sensitivity of all three variants of the circuits is practically identical (see items 11 and 15) and is quite sufficient for the entire range of generator ratings (from 3 to 50 MW) that can operate in parallel at generator voltage. In this connection, the need for a special protection device for generators against ground faults disappears, since its role can be performed by ordinary longitudinal differential protection according to circuit 2 or 3. This will considerably simplify generator protection and will make it possible to save, in its implementation, four current transformers and a control-and-measuring cable.

The use of circuit 2 with optimum resistances of the auxiliary relays is practically impossible because of the large magnitude of these resistances.

resistances, since in the case of phase-to-phase faults in the zone of differential protection the voltage at the terminals of these relays may reach dangerous values. However, this scheme is easy to implement by means of three wattmeter relays of type IM-143 (see Fig. 4), whose voltage windings must be supplied with the corresponding phase voltages from measuring voltage transformers, while the current windings are connected in series with the main protection relays. In this case the calculated operating current of the relay is determined by the following formula:

\[ i_{pT}=\frac{p_{pT}\sqrt{3}\,n_n}{U_{ng}}, \tag{26} \]

where \(p_{pT}\) is the operating power of the relay at which it is used,
\(U_{ng}\) is the rated voltage of the generator,
\(n_n\) is the transformation ratio of the voltage transformer.

The calculated resistance of the relay winding will be equal to the resistance of the current winding for the selected connection of its sections.

Fig. 4

The use of wattmeter relays in this scheme will make it somewhat more complicated, but will increase its sensitivity in comparison with that obtained in the calculation given above.

Scheme 3 can be implemented by means of a simple current relay of high sensitivity. Such a relay has been developed in the relay laboratory of the Tomsk Polytechnic Institute. It consumes, at the minimum operating current, no more than \(0.002\) VA.

In conclusion it should be noted that the calculation method proposed by us in the present article is more accurate and simpler in comparison with that recommended by the Guidelines for Relay Protection, published in 1948.

1. See the article by G. V. Zeveke, “Differential Protections of Electrical Machines,” Electric Stations, No. 1, 1940. ↩↩