Increasing the Sensitivity and Selectivity of Differential Protection for Transformers

N. V. Lisetskii

A review of existing circuits and an analysis of their operation show that all types of differential protection for transformers fail to meet modern requirements. Some circuits have low sensitivity, others permit the possibility of nonselective operation, and still others operate too slowly.

All attempts to improve the operation of the protection amount to introducing additional elements into the circuit, which encumber it and often do not give the required results; moreover, in developing new circuits, the possibilities of improving the operation of existing ones are either not considered at all or are considered very superficially.

For example, for selective operation of the protection when a transformer is energized and during short circuits outside the protected zone accompanied by asymmetrical currents, it is proposed to shunt the relay with the magnetizing circuit of a rapidly saturating current transformer. This idea is, of course, very original and valuable, but after all, besides the rapidly saturating current transformer, the protection circuit includes a number of other transformers that also have magnetizing branches. Therefore, naturally, the question arises: is it not possible, by properly selecting and arranging the equipment of the simplest circuit, to reduce the unbalance current, to detune the protection from the action of the inrush magnetizing current, and to dispense altogether with the rapidly saturating current transformer?

The question of selecting and arranging the equipment of the protection circuit has up to the present time remained insufficiently studied and requires a number of refinements. For example, in section T—3 of the Guidelines for Relay Protection (§ 3, item 5 B) it is stated [1] that the autotransformer must be placed directly near the relay. This is justified by the fact that, with the indicated arrangement, the autotransformer will be less loaded and will give smaller errors, which is, of course, indisputable. However, this arrangement can lead to a large difference in the errors of the current transformers, will give a larger unbalance current, and may cause nonselective operation of the protection.

Further, in § 8 of the indicated section [2] it is recommended that the unbalance current be calculated as the arithmetic sum of the errors in the current of the less powerful current transformer and the autotransformer; moreover, it is recommended that the magnitude of these errors for both units be taken as identical and equal to 5% of the current flowing in the circulation circuit.

As is known from the fundamentals of electrical engineering, current-transformer errors are vector quantities and can be summed arithmetically only when their directions coincide, which may not always be the case. In addition, if the autotransformer is connected to the more powerful current transformers, then the unbalance current will be expressed not by a sum, but by a geometric ...

by the difference of these errors (see below). In this case the errors themselves, in their absolute magnitude, are equal only in special cases.

This same paragraph requires that, for faults outside the protected zone, the errors in the current of the current transformers not exceed 5%. It would be more logical to require that the errors in the current of all current transformers be the same, since in that case the reduced values of the currents in the circulation circuit would be equal, and the unbalance current caused by their difference would be zero.

To resolve these and a number of other questions, let us determine the capabilities of the simplest differential protection of a transformer with maximum-current relays. For this purpose, by means of mathematical analysis and experiments, we shall try to determine under what conditions this type of protection gives the greatest sensitivity and whether selective operation of the protection is possible without an additional time delay.

Sensitivity of the Protection

(without taking into account the unbalance current)

Figure 1 shows the schematic diagram of the protection for a single-phase transformer, or for one phase of a three-phase transformer with winding connection \(Y/\Delta\)—12. In considering the questions of protection sensitivity, we shall proceed from this circuit for a fault at point “\(K_1\).” The adopted current directions are shown in the diagram. In all derivations the notation of the complex form is omitted. The resistances and currents flowing in the circuit are denoted as follows: \(I_{k1}\) and \(i_2\)—the primary and secondary load currents (or short-circuit currents) of the power transformer, having transformation ratio \(N\), equal to the ratio of line voltages;

\(I_0\)—the no-load current of the power transformer;

\(i_1\) and \(i_2\)—the secondary currents of the current transformers of the protection, having transformation ratios \(n_1\) and \(n_2\) and resistances of the no-load branches \(Z_{01}\) and \(Z_{02}\);

\(i_{01}\) and \(i_{02}\)—the magnetizing currents of these current transformers;

\(i_{0a}\)—the magnetizing current of the autotransformer, having transformation ratio \(n_a\) and no-load-branch resistance \(Z_{0a}\).

\(i_p\)—the current flowing through the relay, which has resistance \(Z_p\);

\(Z_1\) and \(Z_2\)—the total resistances of all series-connected elements of the circulation circuit (windings of the current transformers, connecting leads, etc.).

Let us find the relation between the current in the relay and the primary current. For this purpose we write the voltage at the relay terminals:

\[ U_p = i_p Z_p = i_{0a} Z_{0a} n_a = i_{02}(Z_2 + Z_{02}) = (i_{01} Z_{01} + i_1 Z_1)n_a. \]

Assuming that the resistance of the connecting leads to the windings of the current transformers \(Z_1\) and \(Z_2\), in comparison with the resistances \(Z_{01}\), \(Z_{02}\), \(Z_{0a}\), and \(Z_p\), is small, and neglecting the voltage drop in the resistance \(Z_1\), we may write:

\[ U_p=i_p Z_p=i_{0a}Z_{0a}n_a=i_{02}Z_{02}=i_{01}Z_{01}n_a, \]
whence

\[ i_{0a}=i_p\frac{Z_p}{Z_{0a}n_a};\qquad i_{02}=i_p\frac{Z_p}{Z_{02}};\qquad i_{01}=i_p\frac{Z_p}{Z_{01}n_a}. \tag{1} \]

On the basis of Kirchhoff’s first law, for the equivalent circuit of the secondary circuit we have:

\[ \frac{I_1}{n_1}=i_{01}+i_{0a}+i_p n_a+i_{02}n_a . \tag{2} \]

Substituting into expression (2) the values of the currents \(i_{0a}\), \(i_{01}\), and \(i_{02}\) from (1), we obtain:

\[ \frac{I_1}{n_1} =i_p\frac{Z_p}{Z_{01}n_a} +i_p\frac{Z_p}{Z_{0a}n_a} +i_p n_a +i_p\frac{Z_p}{Z_{02}}\,n_a = \]

\[ =i_p n_a\left[ 1+ \frac{ Z_p\left(Z_{02}Z_{0a}n_a^2+Z_{01}n_a^2Z_{02}+Z_{01}n_a^2Z_{0a}n_a^2\right) }{ Z_{01}n_a^2\cdot Z_{0a}n_a^2\cdot Z_{02} } \right]. \tag{3} \]

Taking the first derivative of \(\left(\frac{I_1}{n_1}\right)\) with respect to \(i_p\) and setting it equal to zero, we find the condition for a minimum of \(\left(\frac{I_1}{n_1}\right)\), at which the relay will operate:

\[ \frac{d\left(\frac{I_1}{n_1}\right)}{di_p} = n_a\left[ 1+ \frac{ Z_p\left(Z_{02}Z_{0a}n_a^2+Z_{01}n_a^2Z_{02}+Z_{01}n_a^2\cdot Z_{0a}n_a^2\right) }{ Z_{01}n_a^2\cdot Z_{0a}n_a^2Z_{02} } \right] =0, \]

whence the absolute value of the relay resistance is

\[ Z_p= \frac{ Z_{01}n_a^2\cdot Z_{0a}n_a^2\cdot Z_{02} }{ Z_{01}n_a^2\cdot Z_{02}+Z_{0a}n_a^2Z_{02}+Z_{01}n_a^2Z_{0a}n_a^2 }. \tag{4} \]

Considering expression (4), it is easy to see that, in order to ensure maximum sensitivity of the protection, it is necessary to use a relay with a definite resistance. The resistance of this relay must be equal to the equivalent impedance for the parallel connection of the referred resistances of the magnetizing circuits of the current transformers and of the autotransformer.

\(Z_p\) can be found from the power consumed by the relay, which is given in catalogs or determined experimentally.

Different relays of one and the same type, made for different pickup currents, have, at the minimum pickup current on the scale \((i_{\mathrm{tr}\,\min})\), the same power consumption, which we shall denote by \(P\). Using the given quantities, the relay resistance can be found as

\[ Z_p=\frac{P}{i_{\mathrm{tr}\,\min}^{2}}. \]

The numerical value of the impedances of the magnetizing circuits can be determined experimentally or calculated by the method published in the work of Solov’ev and Fedoseev [3]. However, this value is not constant and depends on the magnetizing current. In order to determine at what magnetizing current this impedance should be taken, it is necessary to consider questions connected with the selectivity of operation of the protection.

Unbalance Currents of the Protection and Their Reduction

Unbalance currents do not lend themselves to a simple mathematical calculation, since they depend on many factors. A substantial influence on their magnitude is exerted by the difference between the characteristics of the current transformers. It is therefore expedient to determine how these characteristics should be selected so that the unbalance current caused by their difference is reduced to a minimum. For this purpose it is necessary to find at least an approximate analytical expression for the unbalance currents that takes into account only the difference in characteristics. Other causes producing unbalance current are considered in detail in the sections on differential protection of generators, and the measures eliminating them can be fully extended to the differential protection of transformers as well.

In deriving the expression for the unbalance current, we shall proceed from the circuit shown in Fig. 1, for a short circuit at point “K.” The adopted directions of the currents are indicated in the circuit. For the designations of currents and resistances we shall use the same ones as in the preceding derivations.

On the basis of Kirchhoff’s first law, for the equivalent circuit of current transformer “1” one may write:

\[ \frac{I_1}{n_1}=\frac{I_{n1}+I_o}{n_1}=i_1+i_{01}, \tag{5} \]

for the equivalent circuit of the autotransformer,

\[ i_1-i_{oa}=i_2 n_a+i_p n_a \tag{6} \]

and for the equivalent circuit of current transformer “2,”

\[ \frac{I_2}{n_2}=\frac{I_{n1}}{N n_2}=i_2+i_{02}. \tag{7} \]

The transformation ratio of the autotransformer, from the conditions of equalizing the secondary currents, must be chosen as:

\[ n_a=\frac{N n_2}{n_1}. \tag{8} \]

Solving equations (5) and (6) jointly, we obtain:

\[ \frac{I_{n1}+I_o}{n_1}-i_{01}-i_{oa}=i_2 n_a+i_p n_a. \]

If in the equality obtained we substitute \(i_2\) from expression (7) and replace \(n_a\) according to (8), we shall have:

\[ \frac{I_{n1}}{n_1}+\frac{I_o}{n_1}-i_{01}-i_{oa} = \frac{I_{n1}}{N n_2}\cdot\frac{N n_2}{n_1}-i_{02}n_a+i_p n_a. \]

Hence the current in the relay, which is the unbalance current, will be:

\[ i_p=i_{nb}=\frac{I_o}{n_1 n_a}-\frac{i_{01}}{n_a}-\frac{i_{oa}}{n_a}+i_{02}. \tag{9} \]

It is evident from expression (9) that the unbalance current of the differential protection of transformers depends on the magnitude and phase of the no-load currents of the power transformer, the current transformers, and the autotransformer.

The magnetizing current of the power transformer in the steady no-load condition is 2–5% of the rated current. Under all load conditions this current decreases (because of the decrease in magnetic flux). Thus, the component of the unbalance current caused by the magnetizing current of the power transformer, in comparison with other ...

its component is small, and in practical calculations, for the given operating conditions, it may be neglected. Then expression (9) is written as:

\[ i_p = i_{nb} = i_{o2} - \frac{i_{o1}}{n_a} - \frac{i_{oa}}{n_a}. \]

The conditions for the greatest sensitivity and selective operation of the protection require that the unbalance current be equal to zero. On this basis, for all modes there must be:

\[ i_{o2} = \frac{i_{o1}}{n_a} + \frac{i_{oa}}{n_a}. \tag{10} \]

The magnetizing currents can be expressed in terms of the electromotive forces of the current transformers and the resistances of the magnetizing circuits \(\left(i_o=\dfrac{e}{Z_o}\right)\). Then the expression obtained is written as:

\[ \frac{e_2}{Z_{o2}} = \frac{e_1}{Z_{o1} n_a} + \frac{e_a}{Z_{oa} n_a}. \]

If it is assumed that the resistances of the protection arms are balanced and that the voltage drop in the resistances \(Z_1\) and \(Z_2\) is small, it may be considered that

\[ e_1 = e'_a = \frac{e_2}{n_a}, \]

then

\[ \frac{e_2}{Z_{o2}} = \frac{e_2}{Z_{o1} n_a^2} + \frac{e_2}{Z_{oa} n_a^2} \quad \text{or} \quad \frac{1}{Z_{o2}} = \frac{1}{Z_{o1} n_a^2} + \frac{1}{Z_{oa} n_a^2}, \]

whence

\[ Z'_{oa} = Z_{oa} n_a^2 = \frac{Z_{o1} n_a^2 Z_{o2}}{Z_{o1} n_a^2 - Z_{o2}}. \tag{11} \]

If equality (11) is observed and the resistances of the protection arms are balanced, then the unbalance currents should have the smallest values. Theoretically, their magnitude should be equal to the referred value of the magnetizing current of the power transformer. The actual values of the unbalance currents will be somewhat larger because of inaccuracies in selecting the circuit parameters. However, the reduction in the unbalance currents is so substantial that selective operation of the protection during through short circuits and small relay pickup currents, as experience shows, becomes possible.

The method of selecting the parameters of current transformers that makes it possible to maintain equality (11) under almost all operating conditions is given below.

Selection of the Parameters and Characteristics of Current Transformers

At present, the selection of current transformers included in the protection circuit is reduced to choosing their transformation ratios, checking thermal and dynamic stability, and checking according to the five-percent curves [2]. However, as the analysis carried out shows, instead of selection according to the five-percent curves, it is necessary to select the resistances of the magnetizing circuits. The magnitude and the degree of variation of these resistances depend on the quantity and quality of the iron in the cores. In various current transformers the magnetic qualities of the core iron are approximately the same; therefore, the dimensions of the latter must be subject to selection.

Current transformers manufactured at the factory are usually included in the arms of the protection. Changing the amount of iron in their cores is difficult and inadvisable, since it is easier, from the magnetization-circuit characteristics of these current transformers, to select the corresponding characteristic of the autotransformer in such a way that, under all operating conditions, the resistance values of the magnetization circuits of the current transformer and of the autotransformer of one and the same phase satisfy expression (11).

As the characteristics of the current transformers it is more convenient to use the dependences of the resistance \(Z_0\) on the secondary electromotive force \([Z_0=f(e_2)]\), taken under no-load conditions of the current transformers.

It would be possible to use the dependences of \(Z_0\) on the primary or secondary current, but in order to obtain such characteristics a large load current may be required, since the curve must be taken up to values \(Z_0=Z_{0\max}\).

The dependences \(Z_0=f(e_2)\) can be obtained experimentally according to the circuit of Fig. 2. Instead of the electromotive forces \(e_2\), one may determine the voltages at the terminals of the secondary windings, since the difference between these quantities will be small.

Fig. 2.

Figure 2 shows typical curves of this kind (curves “1” and “2”). On the basis of these characteristics and expression (11), one can construct a similar curve for the autotransformer (curve “3” in Fig. 3).

Having determined, from the curve obtained, the maximum reduced resistance of the autotransformer magnetization circuit \((Z_{0a\max})\) and the corresponding electromotive force \((e'_a)\), one can find the required magnetizing current \(i_{0a}\) at \(Z_{0a}=Z_{0a\max}\):

\[ i_{0a}=\frac{e'_a n_a}{Z'_{0a\max}}. \tag{12} \]

Having specified the numbers of turns \(W_1\) and \(W_2\) to provide the transformation ratio \(n_a\), and multiplying the number of turns of the primary winding by \(i_{0a}\), one can find the magnetizing ampere-turns \((i_{0a} W_1)\) needed to obtain \(Z_{0a}=Z_{0a\max}\).

The maximum resistance of the magnetization circuit corresponds to a definite magnetic induction \((B'_m)\) and a definite specific magnetiz—

magnetizing ampere-turns \(\left(\dfrac{iW}{l}\right)'\). These quantities may be determined experimentally by constructing the curves

\[ B_m=f\!\left(\frac{iW}{l}\right) \quad\text{and}\quad z_o=\varphi\!\left(\frac{iW}{l}\right) \]

for the iron from which the core of the autotransformer will be made, or they may be taken from manufacturers’ data.

Fig. 3.

Knowing the required \((i_{oa}W_1)\) and the specific \(\left(\dfrac{iW}{l}\right)'\) ampere-turns needed to attain

\[ B_m=B'_m \quad\text{and}\quad Z_{oa}=Z_{oamax}, \]

one can find the mean length of the autotransformer core from the expression

\[ l_{\mathrm{cp}}=\frac{i_{oa}W_1}{\left(\dfrac{iW}{l}\right)'} . \tag{13} \]

If the mean length of the autotransformer core proves to be outside the structural dimensions, its size can be increased or decreased by changing the numbers of turns \(W_1\) and \(W_2\).

The cross section of the autotransformer core \((q)\) can be found from the relation:

\[ Z_{oamax}=\frac{e_a}{i_{oa}} =\frac{4.44\,f\,q\,W_1 B'_m\,10^{-8}}{i_{oa}} \]

\[ q=\frac{Z_{oamax}\,i_{oa}\,10^8}{4.44\,f\,W_1 B'_m}. \tag{14} \]

When factory-manufactured autotransformers are used, their characteristics may be adjusted by selecting the core cross section and the number of turns of the windings. Then the calculation of the autotransformer parameters may be carried out in the following order.

First, by formula (12), the magnetizing current \((i_{oa})\) required to attain \(Z_{oa}=Z_{oamax}\) is determined. Then the mean length of the autotransformer core \((l_{\mathrm{cp}})\) is measured, and the specific ampere-turns are determined.

\(\left(\dfrac{iW}{l}\right)'\) and the magnetic induction \((B'_m)\) at the maximum resistance of the magnetizing circuit.

The required number of turns of the primary winding of the autotransformer \((W_1)\) is found from the relation

\[ W_1=\frac{\left(\dfrac{iW}{l}\right)' l_{\mathrm{cp}}}{i_{oa}} . \]

The number of turns of the secondary winding \((W_2)\) is chosen from the condition of providing the required transformation ratio of the autotransformer—\(n_a\). The cross section of the autotransformer core is determined by formula (14).

In order that, under all operating conditions, the relation between the resistances \(Z_{o1}\), \(Z_{o2}\), and \(Z_{oa}\) specified by expression (11) be maintained, it is necessary that \(Z_{oa}\) vary as a function of the electromotive force of the autotransformer according to the curve obtained earlier (“3”). This will be achieved only if the cores of the current transformers and of the autotransformer are made of the same material. However, even with different core materials, the unbalance current will have smaller values when the characteristics of the current transformers and of the autotransformer are matched. This is evident from the experimental results given below.

In addition, the autotransformer should be installed near the current transformer with the larger \(Z_o\). The relay must be connected between current transformer “2” and the group consisting of current transformer “1” and the autotransformer. The resistances of the wires from the point where the relay is connected to current transformer “2” and to the group consisting of current transformer “1” and the autotransformer must be equal.

This can be achieved by the appropriate selection of the cross sections of the conductors of the protection arms.

If the resistance due to the difference in turns of the autotransformer windings is neglected, then the condition for equality of the resistances of the protection arms is written as

\[ Z_{np_1}+Z_{T_1}=Z_{np_2}+Z_{T_2}, \quad \text{where} \]

\(z_{np_1}\) and \(z_{np_2}\) are the resistances of the connecting conductors of the circulating-current circuit arms;

\(Z_{T_1}=(r_{T_1}+jx_{T_1})\) and \(Z_{T_2}=(r_{T_2}+jx_{T_2})\) are the impedances of the secondary windings of the current transformers.

Expressing the conductor resistances through the resistivity \((\rho)\), the lengths \((l)\), and the cross sections \((s)\), and neglecting the difference in leakage reactances of the secondary windings of the current transformers, we obtain

\[ \rho\left(\frac{l_1}{s_1}-\frac{l_2}{s_2}\right)=r_{T_2}-r_{T_1}. \]

The resistances of the secondary windings of the current transformers may be taken from a catalog or determined experimentally. The lengths of the protection arms are determined by the arrangement of the equipment. Consequently, the resulting relation can be used to select conductor cross sections.

Possibility of Selective Operation of the Protection

Nonselective operation of the protection, as operating experience has shown, occurs during through short circuits, energization of the transformer at no load, and restoration of voltage after a short circuit.

In the first case this occurs because a large unbalance current flows through the relay, caused by the difference between the magnetizing currents of the current transformers and the autotransformer. If the characteristics of these elements of the circuit are selected and the resistances of the protection arms are balanced, then the cause producing the large unbalance current will be eliminated, and it may be assumed that the protection will operate selectively.

In the second and third cases, nonselective tripping occurs because surge magnetizing currents flow in the relay circuit. The protection can be restrained from operating under the flow of these currents in several ways. The most reliable of these should be considered the following: 1) introduction of a time delay; 2) desensitizing the protection; 3) connecting the relay through a rapidly saturating current transformer; 4) using the main current transformers and the autotransformer as a rapidly saturating current transformer.

The first method slows the operation of the protection and, as was indicated earlier, does not meet present-day requirements.

The second method excessively desensitizes the protection, since the pickup currents have to be set within the range from 150 to 300% of the rated current. Therefore this type of restraining the protection from operation under surge magnetizing current also cannot be regarded as satisfactory.

The third method introduces an additional element into the circuit—a rapidly saturating transformer. However, the use of this method is advisable when powerful current transformers are included in both arms of the protection. The rapidly saturating transformer for this case should be selected according to the method proposed by I. D. Kutyavin [4]. The resistance of the relay circuit, according to the conditions of maximum sensitivity, must be determined by expression (4). If the parameters of the current transformers are selected, then in expression (4) \(Z_{oa} n_a^2\) may be replaced by equality (11), and then the magnitude of the relay-circuit resistance may be calculated from the formula

\[ Z_p = \frac{Z_{o2}}{2} \tag{15} \]

The value \(Z_{o2}\) depends on the loading of the current transformer by the operating current and by the fault current and, in connection with this, may vary within considerable limits; therefore \(Z_{o2}\) can be determined only for some particular mode. Such a particular mode may be taken as operation of the power transformer at rated load.

The adopted method of determining \(Z_{o2}\) is justified by the fact that the minimum of \(I_{mp} = f(Z_p)\) is not sharply expressed, and with small deviations from the optimum conditions the primary pickup current changes hardly at all [5].

When bushing or small-capacity current transformers are used for differential protection, it is advisable to restrain the protection from operation under surge magnetizing currents by the fourth method.

In bushing current transformers, the resistance of the magnetizing circuit under rated operating conditions is close to the maximum. Therefore, with comparatively small additional loads they pass into the operating mode of rapidly saturating transformers.

The magnitude of the additional load of the current transformers and the relay pickup current in each particular case should be determined experimentally when adjusting the protection; moreover, in order to maintain equality of the resistances of the protection arms, both arms must be loaded uniformly.

The resistance of the relay circuit, with selected parameters of the current transformers, may be calculated, as in the preceding case, by the for—

formula (15). Only in this formula, instead of \(Z_{o2}\), \(Z_{o2\max}\) should be substituted.

With the adopted resistance of the relay circuit and the selected parameters of the current transformers, the tap current in the magnetizing branch at the pickup current will be approximately equal to the latter. Thus, the primary pickup current will correspond to twice the relay pickup current, referred to the primary circuit,

\[ I_{mp}=2i_{mp} n_a n_1 . \tag{16} \]

The question of whether it is possible to detune the protection from the action of inrush magnetizing currents by the method considered was subjected to experimental verification in the relay-protection laboratory of the Tomsk Polytechnic Institute. The results of the experiments are given below.

Experimental verification of the theoretical conclusions

To verify the theoretical conclusions, a circuit was assembled that simulated transformer differential protection, shown in Fig. 4. The current transformers included in the circuit were designed for a nominal number of ampere-turns equal to 500, had different core iron cross sections and magnetizing-circuit resistance characteristics, as shown in Fig. 3. Instead of a relay, an active resistance and a milliammeter were connected. A resistance \(r_1=r_2=2\) ohms was introduced into the protection arms.

Fig. 4.

The investigation of the influence of the autotransformer parameters on the unbalance currents during an external short circuit was carried out as follows: various autotransformers were connected into the circuit. The switches \(K_2\) and \(K_1\) were closed, and the dependences of the unbalance current on the multiple of the primary current with respect to the nominal value were taken. The resistance of the relay circuit and of the protection arms remained constant.

The curves obtained are shown in Fig. 5. Curve “a” shows the dependence of the unbalance current on the multiple of the current of an external short-circu—

with an autotransformer selected according to the iron of the transformer being present in the circuit. Curve “б” was taken with no autotransformer in the circuit, and curve “в” with an autotransformer not selected according to the iron being present in the circuit.

Fig. 5.

On the basis of a comparison of the curves presented, it may be concluded that by properly selecting the parameters of the autotransformer, a considerable reduction in unbalance currents can be achieved.

Also of interest is the question of the influence of the relay resistance and of the magnitude of the through short-circuit current on the value of the unbalance currents.

Fig. 6.

To clarify this question, curves were taken of the dependence of the unbalance current on the relay resistance. The curves were taken at nominal current and at fourfold current. In this case the parameters of all the other elements of the circuit remained constant. The curves obtained are shown in Fig. 6. In this figure curve “а” corresponds to the nominal primary current, and curve “б” to the fourfold current.

From examination of the obtained curves it is evident that introducing additional resistance into the relay circuit, although it reduces the absolute value

of the unbalance current; however, the factor by which it increases during through short circuits remains almost unchanged; therefore, in order to reduce the unbalance current, it is inexpedient to introduce additional resistance into the relay circuit. The resistance of the relay circuit must remain that required by the conditions of maximum sensitivity, while it is more advisable to increase the selectivity of the protection only by increasing the pickup current.

Next, the question of the possibility of selective operation of the protection was considered. For this purpose, in the circuit of Fig. 4, oscillograph loops were connected instead of ammeters, and oscillograms of the currents were taken when the transformer was switched on at no load and during a short circuit outside the protection zone.

Fig. 7.

Fig. 8.

The obtained oscillograms are shown in Figs. 7, 8, 9, and 10. The upper curves of the oscillograms show the currents in the primary circuit, and the lower ones show the currents in the relay circuit. Figs. 7 and 8 give oscillograms of the currents when the transformer is switched on at no load. In the first case the parameters of the protection circuit were matched, and in the second they were not matched. The current scales in both figures are taken to be the same.

It is seen from the oscillograms presented that, with matched parameters of the circuit, the currents in the relay circuit, when the transformer is switched on, do not exceed the values of the steady-state current. Hence it follows that nonselective tripping of the protection under this mode of operation is impossible.

The oscillograms of Figs. 9 and 10 show the currents when the transformer is switched on to a short circuit outside the protection zone, respectively with matched and unmatched parameters of the latter. The magnitude of the short-circuit current was chosen such that its steady-state value corresponded to three times the rated current. The current scales in both figures are the same.

From consideration of the oscillograms presented it is seen that the unbalance currents have a small aperiodic component. Its relative magnitude is approximately the same for both cases. However, the total magni-

...the magnitude of the unbalance current in the first case is considerably smaller than in the second. In this case, the maximum value of the unbalance current in the oscillogram of Fig. 9 does not reach the value of the steady-state unbalance current in the oscillogram of Fig. 10.

Fig. 9.          Fig. 10.

Conclusions

On the basis of all the above, the following conclusions may be drawn:

1. By special selection of the characteristic of the autotransformer included in the differential-protection circuit of a power transformer, and of its place of installation in the circuit, it is possible to reduce considerably the magnitude of the unbalance currents.

2. For differential protections in which bushing-type or low-power current transformers are used, by special selection of the circuit parameters it is possible, simultaneously with reducing the unbalance currents, to make the current transformers and the autotransformer perform the role of a quick-saturating transformer and thereby detune the protection from the action of the inrush magnetizing current.

3. If the protection is made with the aid of powerful current transformers, then for its selective operation it is necessary to use an additional quick-saturating current transformer.

4. The question should be put before the factories manufacturing measuring transformers of supplying, as a set, selected current transformers and an autotransformer for the differential protection of power transformers. In the absence of a complete set, the selection of current transformers on site is possible.

Selection of current-transformer parameters for differential protection of three-phase transformers

In all the theoretical derivations made earlier, the schematic diagram of differential protection for a single-phase or...

of one phase of a three-phase transformer having the winding connection group $\Delta/Y-12$. In practice, three-phase power transformers with other winding-connection arrangements are often encountered. This causes a certain phase displacement between the primary and secondary currents. To compensate for this displacement in the protection circuits, the secondary windings of the current transformers in one of the arms must be connected in delta. In this case, all the formulas derived earlier for selecting the parameters of the current transformers will be somewhat modified.

Below are the formulas for selecting the parameters and characteristics of current transformers for various connections of their secondary windings. The derivation of these formulas is carried out analogously to those given earlier.

Basic formulas for selecting the parameters of current transformers for differential protection of power transformers

I. For the protection circuit shown in Fig. 11.

1) Transformation ratio of the autotransformer

\[ n_a=\frac{N n_2}{n_1}. \]

2) Reduced resistance of the magnetizing circuit of the autotransformer

\[ Z'_{oa}=Z_{oa} n_a^2=\frac{Z_{01} n_a^2\cdot Z_{02}}{Z_{01} n_a^2-Z_{02}}. \]

3) To determine $Z_{oa\,max} n_a^2$, it is necessary to construct the following dependencies (Fig. 3):

\[ Z_{01} n_a^2=f_1(e_1 n_a)\quad \text{and}\quad Z_{02}=f_2(e_2). \]

Fig. 11.

4) The magnetizing current of the autotransformer for attaining $Z_{oa}=Z_{oa\,max}$

\[ i_{oa}=\frac{e'_a n_a}{Z'_{oa\,max}}. \]

5) Mean length of the core of the autotransformer

\[ l_{cp}=\frac{i_{oa} W_1}{\left(\dfrac{iW}{l}\right)}. \]

6) Cross section of the core of the autotransformer

\[ q=\frac{Z_{oa\,max} i_{oa}\,10^8}{4.44\,f\,W_1\,B''_m}. \]

7) Resistance of the relay circuit

\[ Z_p=\frac{Z_{02\,max}}{2}. \]

II. For the protection circuit shown in Fig. 12.

1) Transformation ratio of the autotransformer

\[ n_a=\frac{N n_2}{n_1}. \]

2) The reduced resistance of the autotransformer magnetizing circuit

\[ Z_{oa}'=Z_{oa} n_a^2=\frac{1}{3}\,\frac{Z_{o1} n_a^2 Z_{o2}}{Z_{o1} n_a^2-Z_{o2}}. \]

3) To determine \(Z_{oa\,max} n_a^2\), it is necessary to construct the following dependences:

\[ Z_{o1} n_a^2=f_1(e_1 n_a)\quad \text{and}\quad Z_{o2}=f_2(e_2). \]

4) The magnetizing current of the autotransformer for achieving \(Z_{oa}=Z_{oa\,max}\)

\[ i_{oa}=\frac{e_a n_a}{\sqrt{3}\,Z_{oa\,max}}. \]

5) The mean length of the autotransformer core:

\[ l_{cp}=\frac{i_{oa} W_1}{\left(\dfrac{iW}{l}\right)}. \]

6) The cross section of the autotransformer core

\[ q=\frac{Z_{oa\,max}\, i_{oa}\,10^8}{4.44\,f\,W_1\,B_m}. \]

7) Resistance of the relay circuit

\[ Z_p=\frac{Z_{o2}}{6}. \]

III. For the protection circuit shown in Fig. 13.

1) Transformation ratio of the autotransformer

\[ n_a=\frac{N n_2}{\sqrt{3}\,n_1}. \]

2) The referred resistance of the magnetizing circuit of the autotransformer

\[ Z'_{oa}=Z_{oa} n_a^2=\frac{Z_{o1} n_a^2 Z_{o2}}{3 Z_{o1} n_a^2-Z_{o2}} . \]

3) To determine \(Z_{oa\,max} n_a^2\), it is necessary to construct the following dependences:

\[ Z_{o1} n_a^2=f_1(e_1\sqrt{3}\,n_a) \quad \text{and} \quad Z_{o2}=f_2(e_2). \]

4) The magnetizing current of the autotransformer for attaining \(Z_{oa}=Z_{oa\,max}\)

\[ i_{oa}=\frac{e'_a n_a}{\sqrt{3}\,Z'_{oa\,max}} . \]

Fig. 14.

5) Mean length of the autotransformer core

\[ l_{ocp}=\frac{l_{oa} W_1}{\left(\dfrac{iW}{l}\right)} . \]

6) Cross section of the iron of the autotransformer core

\[ q=\frac{Z_{oa\,max}\, i_{oa}\,10^8}{4.44 f W_1 B'_m} . \]

7) Resistance of the relay circuit

\[ Z_p=\frac{Z_{o2\,max}}{4.73}. \]

IV. For the protection circuit shown in Fig. 14.

1) Transformation ratio of autotransformer “1”

\[ n_{a1}=\frac{N_2}{n_3}\,n_2 . \]

2) Transformation ratio of autotransformer “2”

\[ n_{a2}=\frac{N_1}{\sqrt{3}\,n_1}\,n_2 . \]

3) Referred resistance of the magnetizing circuits of autotransformers “1” and “2”

\[ Z'_{oa1}=Z_{oa1} n_{a1}^2=\frac{1}{3}\,\frac{Z_{o3} n_{a1}^2 Z_{o2}}{Z_{o3} n_{a1}^2-Z_{o2}}, \]

\[ Z'_{oa2}=Z_{oa2} n_{a2}^2=\frac{Z_{o1} n_{a2}^2 Z_{o2}}{6Z_{o1} n_{a2}^2-Z_{o2}} . \]

4) To determine \(Z_{oa1\,max} n_{a1}^{2}\), it is necessary to construct the following dependences:

\[ Z_{o3} n_{a1}^{2}=f_{1}(e_{3} n_{a1}) \quad \text{and} \quad Z_{o2}=f_{2}(e_{2}). \]

5) To determine \(Z_{oa2\,max} n_{a2}^{2}\), it is necessary to construct the dependences:

\[ Z_{o1} n_{a2}^{2}=f_{3}(e_{1}\sqrt{3}\,n_{a2}) \quad \text{and} \quad Z_{o2}=f_{2}(e_{2}). \]

6) The magnetizing currents of autotransformers “1” and “2” for achieving \(Z_{oa}=Z_{oa\,max}\)

\[ i_{oa1}=\frac{e'_{a1} n_{a1}}{\sqrt{3}\,Z'_{oa1\,max}}, \]

\[ i_{oa2}=\frac{e'_{a2} n_{a2}}{\sqrt{3}\,Z'_{oa2\,max}}. \]

7) The mean length of the cores of autotransformers “1” and “2”

\[ l_{cp1}=\frac{i_{oa1} W_{1a1}}{\left(\dfrac{iW}{l}\right)^{1}}, \]

\[ l_{cp2}=\frac{i_{oa2} W_{1a2}}{\left(\dfrac{iW}{l}\right)^{\prime}}. \]

8) Cross sections of the cores of autotransformers “1” and “2”

\[ q_{1}=\frac{Z_{oa1\,max}\, i_{oa1}\,10^{8}}{4.44\,f\,W_{1a1}\,B_{m}}, \]

\[ q_{2}=\frac{Z_{oa2\,max}\, i_{oa2}\,10^{8}}{4.44\,f\,W_{1a2}\,B_{m}}. \]

9) Resistance of the relay circuit

\[ Z_{p}=\frac{Z_{o3\,max}}{9.46}. \]

LITERATURE

1. Guiding instructions on relay protection. Journal Electric Stations, No. 3, 1940.

2. Guiding instructions on relay protection. Journal Electric Stations, Nos. 5–6, 1940.

3. L. E. Solov′ev and A. M. Fedoseev.—Relay Protection, Part 1, ONTI, 1938.

4. I. D. Kutyavin.—A fast-saturating current transformer for increasing the sensitivity of differential protections. Journal Electric Stations, No. 8, 1946.

5. G. V. Zvezek.—Differential protections of electrical machines. Journal Electric Stations, No. 1, 1940.