ON THE DESIGN AND ADJUSTMENT OF DIFFERENTIAL PROTECTION FOR TRANSFORMERS

I. D. Kutyavin
Associate Professor, Candidate of Technical Sciences

The design and commissioning tests of differential protection for transformers at the present time amount only to the selection of the main equipment and to a very approximate determination of the principal protection parameters¹. In this, the main difficulty is the determination of the pickup current, the magnitude of which must be coordinated with the maximum value of the protection unbalance current.

The present article gives a method for an approximate determination of this value and ways of reducing the unbalance current of differential protection for transformers.

Derivation of the basic expressions for the unbalance current

Figure 1 shows, for one phase, the schematic diagram of differential protection for a transformer with winding connection Y/Y-12, carried out with a relay of type ET. The adopted direction of the currents in the circuit is indicated by arrows; the following designations are used for the currents:

Fig. 1

\(I_1\) and \(I_2\) — primary and secondary currents of the power transformer, having transformation ratio \(N\), equal to the ratio of the no-load line voltages;

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

\(i_1\) and \(i_2\) — secondary currents of the protection current transformers, having transformation ratios \(n_1\) and \(n_2\);

\(i_{01}\) and \(i_{02}\) — no-load currents of the current transformers;

\(i'_1\) and \(i'_{01}\) — secondary current and no-load current of an autotransformer with transformation ratio \(n_a\).

In all derivations the notation of complex form is omitted. The current in the relay, on the basis of Fig. 1, is

\[ i_p = i'_1 - i_2. \tag{1} \]

The secondary current of the first current transformer is related to the primary current (for like polarity) by the following relation:

\[ i_1 = \frac{I_1}{n_1} - i_{01}. \]

Secondary current of the autotransformer

\[ i'_1=\frac{i_1}{n_a}-i'_{01}=\frac{I_1}{n_1 n_a}-\frac{i_{01}}{n_a}-i'_{01}. \tag{2} \]

Secondary current of the power transformer

\[ I_2=\frac{I'_2}{N}, \]

where \(I'_2\) is the secondary current of the power transformer, referred to the primary side.

The secondary current of the current transformer installed on the high-voltage side of the power transformer is

\[ i_2=\frac{I_2}{n_2}-i_{02}=\frac{I'_2}{N n_2}-i_{02}. \tag{3} \]

On the basis of (1), (2), and (3), the current in the relay is

\[ \begin{aligned} i_p &=\left(\frac{I_1}{n_1 n_a}-\frac{i_{01}}{n_a}-i'_{01}\right) -\left(\frac{I'_2}{N n_2}-i_{02}\right) \\ &=\left(\frac{I_1}{n_1 n_a}-\frac{I'_2}{N n_2}\right) +\left(i_{02}-i'_{01}-\frac{i_{01}}{n_a}\right). \end{aligned} \tag{4} \]

The transformation ratio of the autotransformer must be selected of such a magnitude that the denominators of the first term on the right-hand side of expression (4) are equal, i.e.,

\[ n_a=N\frac{n_2}{n_1}. \tag{5} \]

When the windings of the power transformer are connected in the group \(Y/Y\)-12 and the windings of the current transformers in delta (the second case), the current in the relay is

\[ \begin{aligned} i_p &=\left[\left(\frac{I_1}{n_1 n_a}-\frac{i_{01}}{n_a}\right)\sqrt{3}-i'_{01}\right] -\left(\frac{I'_2}{N n_2}-i_{02}\right)\sqrt{3} \\ &=\left(\frac{I_1}{n_1 n_a}-\frac{I'_2}{N n_2}\right)\sqrt{3} +\left(i_{02}\sqrt{3}-i'_{01}-\frac{i_{01}\sqrt{3}}{n_a}\right). \end{aligned} \tag{6} \]

The transformation ratio of the autotransformer is determined by expression (5).

When the windings of the power transformer are connected in the group \(\Delta/Y\)-11, and the windings of the current transformers on the delta side are connected in star, and on the star side in delta (the third case), the current in the relay is

\[ \begin{aligned} i_p &=\left(\frac{I_1}{n_1 n_a}-\frac{i_{01}}{n_a}-i'_{01}\right) -\left(\frac{I'_2}{N n_2}-i_{02}\right)\sqrt{3} \\ &=\left(\frac{I_1}{n_1 n_a}-\frac{I'_2\sqrt{3}}{N n_2}\right) +\left(\sqrt{3}i_{02}-i'_{01}-\frac{i'_{01}}{n_a}\right). \end{aligned} \tag{7} \]

The transformation ratio of the autotransformer for this case is

\[ n_a=\frac{N n_2}{\sqrt{3}\,n_1}. \tag{8} \]

In the normal operating mode of the power transformer, or in the event of a short circuit outside the protection zone, the difference between the primary current

\(I_1\) and the secondary referred \(I'_2\) is the no-load current of the power transformer \(I_0\). Taking this into account, and also on the basis of (4) and (5), for the first case

\[ i_p=i_{\text{unb}}=\left(\frac{I_0}{n_1 n_a}+i_{02}\right)-\left(\frac{i_{01}}{n_a}+i'_{01}\right), \tag{9} \]

where \(i_{\text{unb}}\) is the unbalance current flowing through the relay under the conditions indicated above.

On the basis of (6) and (5), for the second case

\[ i_p=i_{\text{unb}}=\left(\frac{I_{01}\sqrt{3}}{n_1 n_a}+i_{02}\sqrt{3}\right)-\left(\frac{i_{01}\sqrt{3}}{n_a}+i'_{01}\right). \tag{10} \]

On the basis of (7) and (8), for the third case

\[ i_p=i_{\text{unb}}=\left(\frac{I_{01}}{n_1 n_a}+i_{02}\sqrt{3}\right)-\left(\frac{i_{01}}{n_a}+i'_1\right). \tag{11} \]

With reverse feeding of the power transformer, the unbalance current for the first case is

\[ i_{\text{unb}}=\left(\frac{I_{02}}{n_2}+\frac{i_{01}}{n_a}+i'_{01}\right)-i_{02}; \tag{12} \]

for the second case

\[ i_{\text{unb}}=\left(\frac{I_{02}\sqrt{3}}{n_2}+\frac{i_{01}\sqrt{3}}{n_a}+i'_{01}\right)-i_{02}\sqrt{3}; \tag{13} \]

for the third case

\[ I_{\text{unb}}=\left(\frac{I_{02}\sqrt{3}}{n_2}+\frac{i_{01}}{n_a}+i'_{01}\right)-i_{02}\sqrt{3}. \tag{14} \]

In deriving expressions (9)—(14) it is assumed that the autotransformer is connected into the protection circuit according to Fig. 1.

It follows from expressions (9)—(14) that the unbalance current of transformer differential protection depends on the magnitude and phase of the no-load currents of the current transformers, the autotransformer, and the power transformer. And the no-load currents, in turn, depend on the quality of the iron of these apparatuses and on their loading.

Determination of no-load currents

As is known, the dependence of the specific magnetizing ampere-turns for current transformers and autotransformers on the maximum induction (within the range 0—7000 gauss) can be expressed with sufficient accuracy by the equation (according to data of the Elektroapparat plant) \(aw_{\text{ud}}=cB_m^\alpha\) (15), where \(c=0.0023+0.0045\) (depending on the grade of iron), \(\alpha=0.6+0.7\).

The magnitude of the total magnetizing current on the basis of (15) is

\[ i_n=\frac{c\cdot l_{\text{av}}}{w}B_m^\alpha, \tag{16} \]

where \(l_{\text{av}}\) is the length of the mean line of force in cm;
\(w\) is the secondary number of turns of the current transformer or autotransformer.

The magnitude of the maximum induction for the transformer is

\[ B_m=\frac{e\cdot 10^8}{4.44fqW} \tag{17} \]

The active component of the no-load current may be neglected; then, on the basis of (16) and (17),

\[ i_0=\frac{e^{\alpha}}{\dfrac{(4.44fq)^{\alpha}\cdot W^{(1+\alpha)}}{10^{8\alpha}\cdot C\cdot l_{\mathrm{cp}}}} =\frac{e^{\alpha}}{A}, \tag{18} \]

where the constant¹

\[ A=\frac{(4.44fq)^{\alpha}\cdot W^{(1+\alpha)}}{10^{8\alpha}\cdot C\cdot l_{\mathrm{cp}}}. \tag{19} \]

The constant \(A\) for current transformers may be taken from Section C–1 of the Guidelines for Relay Protection, Vol. II.

The magnitude of the secondary e.m.f. of the current transformer (or autotransformer) is

\[ e=iZ, \tag{20} \]

where \(i\) is the current in the secondary winding of the current transformer (or autotransformer).

The resistance of the secondary circuit of current transformers connected in star, when an autotransformer is present in the circuit, is

\[ Z_y=Z_t+Z_{\mathrm{pr}}+\frac{1}{n_a^2}\left(Z_{\mathrm{at}}+Z'_{\mathrm{pr}}\right). \tag{21} \]

The same, but connected in delta, is

\[ Z_{\Delta}=Z_t+3\left[Z_{\mathrm{pr}}+\frac{1}{n_a^2}\left(Z_{\mathrm{at}}+Z'_{\mathrm{pr}}\right)\right]. \tag{22} \]

The resistance of the secondary circuit of current transformers in the absence of an autotransformer is calculated from expression (21) or (22) with

\[ Z_{\mathrm{at}}=0 \quad \text{and} \quad Z'_{\mathrm{pr}}=0. \]

The resistance of the secondary circuit of the autotransformer is

\[ Z_a=\frac{Z_{\mathrm{at}}}{2}+Z'_{\mathrm{pr}}, \tag{23} \]

where \(Z\) is the resistance of the secondary winding of the current transformer;
\(Z_{\mathrm{pr}}\) is the resistance of the leads to the autotransformer (or to the tapping point on the relay);
\(Z_{\mathrm{at}}\) is the resistance of the autotransformer, referred to its secondary side;
\(Z'_{\mathrm{pr}}\) is the resistance of the leads from the autotransformer to the relay.

When relays with a restraining action are used, the resistance of the restraining coil must also be included in expressions (21), (22), and (23).

Figures 2 and 3 give the dependence \(i_0=f(e,A)\) according to expression (18) for \(\alpha=2/3=0.667\). The curves are plotted, for convenience of use, on two scales. Along the abscissa axis the e.m.f. is plotted in volts, and along the ordinate axis—the no-load current in milliamperes. The curves are marked with different values of \(A\).

The no-load current of power transformers for the normal operating condition may be taken from the catalog. In the event of a short circuit beyond the transformer, the e.m.f. of the transformer, depending on the power of the supply source, lies within the limits \((25 \div 50)\%\ U_n\). In accordance with this

¹ See L. E. Solov’ev and D. M. Fedoseev, Relay Protection, Part 1, 1938.

and the no-load current for the given operating condition may be taken within the limits of \((25\text{—}50)\%\, I_{0n}\).

However, the adjustment of transformer differential protection and the measurement of the unbalance current are usually carried out during a normal short circuit of the transformer; in this case the calculated no-load current of the transformer can be roughly determined from the following expression:

Fig. 2

\[ I_{op}=\frac{e_k\%}{100}\, I_{0n}, \tag{24} \]

where \(e_k\%\) is the short-circuit voltage of the transformer,

Fig. 3

\(I_{0n}\)—the no-load current at rated voltage and under rated conditions.

Methods for Reducing the Unbalance Current

On the basis of (9), (18), and (24), the expression for the unbalance current in the first case is

\[ i_{\mathrm{ub}} \cong \left(\frac{I_{01}}{n_1 n_a}+\frac{e_2^\alpha}{A_2}\right) - \left(\frac{e_1^\alpha}{A_1 n_a}+\frac{e_a^\alpha}{A_a}\right), \tag{25} \]

where \(e_1\), \(e_2\), and \(e_a\) are the e.m.f.’s of the current transformers and the autotransformer;

\(A_1\), \(A_2\), and \(A_a\) are their constants.

All quantities entering expression (25) are substituted in accordance with the operating condition under which the protection is adjusted.

The condition of greatest sensitivity of the protection requires that the unbalance current according to expression (25) be equal to zero; then

\[ \left(\frac{I_{01}}{n_1 n_a}+\frac{e_2^\alpha}{A_2}\right) - \left(\frac{e_1^\alpha}{A_1 n_a}+\frac{e_a^\alpha}{A_a}\right)=0. \tag{26} \]

The values of the e.m.f.’s entering equation (26), on the basis of (5) and (20), are

\[ e_1=\frac{I_1}{n_1} Z_1; \]

\[ e_a=\frac{I_1}{n_1 n_a} Z_a; \]

\[ e_2=\frac{I_2'}{N n_2} Z_2 \cong \frac{I_1}{n_1 n_a} Z_2. \]

If the no-load current of the power transformer \(I_{01}\) is neglected and the values of the e.m.f.’s are substituted, then equation (26) assumes the following form:

\[ \frac{I_1^\alpha \cdot Z_2^\alpha}{(n_1 n_a)^\alpha A_2} - \left[ \frac{I_1^\alpha Z_1^\alpha}{n_1^\alpha n_a A_1} + \frac{I_1^\alpha Z_a^\alpha}{(n_1 n_a)^\alpha A_a} \right] =0, \]

\[ \frac{Z_2^\alpha}{A_2} - \left( \frac{Z_1^\alpha}{A_1 n_a^{(1-\alpha)}} + \frac{Z_a^\alpha}{A_a} \right) =0, \tag{27} \]

where \(Z_1\), \(Z_2\), and \(Z_a\) are the total impedances of the secondary circuits of the current transformers and the autotransformer.

Equations (26) and (27) express the condition of equilibrium of the arms of the transformer differential protection; moreover, equation (27) indicates that the unbalance current of the protection will have its least value for a quite definite ratio of the secondary impedances \(Z\) and the constants \(A\) of the current transformers and autotransformers.

In the event that the parameters of the selected protection equipment do not satisfy the equilibrium condition (26) and (27), the following measures may be applied¹:

1. Insert autotransformers in the circuit of the underloaded current transformers (with the larger constant \(A\)).

1. Increase the cross-section of the connecting wires of the overloaded current transformers (with a small constant \(A\)).

2. Increase the transformation ratio of the overloaded current transformers, taking into account the limits of the transformation ratios of the autotransformers.

3. Use more powerful current transformers instead of the overloaded ones.

4. Connect two current transformers in series (when bushing-type transformers are used); in this case the no-load current is

\[ i_0=\frac{\left(\dfrac{e}{2}\right)^\alpha}{A}, \tag{28} \]

and the value of the secondary resistance is

\[ Z_\Delta=2Z_T+3Z_{\mathrm{pr}}+\ldots . \tag{29} \]

In this case, if the measures indicated above do not make it possible to satisfy equations (26) and (27), an additional adjustable resistance \(Z_d\) may be inserted into the circuit of the underloaded current transformers (in the break of the connecting wires). If, for example, in the circuit of Fig. 1 the first current transformer is underloaded, then the additional resistance that must be inserted in its circuit, on the basis of (21) and (26), is

\[ Z_d=-\frac{n_1}{I_4} \left[ \left( \frac{I_{01}}{n_1 n_a} +\frac{e_2^\alpha}{A_2} -\frac{e_a^\alpha}{A_a} \right) A_1 n_a \right]^{\frac{1}{\alpha}} -Z_T-Z_{\mathrm{pr}}-\ldots . \tag{30} \]

Instead of inserting the additional resistance \(Z_d\) in the break of the connecting wires, when this resistance proves to be large, the secondary circuit of the underloaded current transformer may be shunted by an adjustable resistance. When the current transformers and shunting resistances are connected in star,

\[ Z_{\mathrm{sh}\Delta}=\frac{e-iZ}{i_{\mathrm{nb}}n_a}, \tag{31} \]

where the numerator is the voltage at the point where the shunting resistance is connected;

\(i_{\mathrm{nb}}\) is the unbalance current, determined from expression (9), (10), or (12), (13), depending on the direction of supply;

\(Z_{\mathrm{sh}}\) is the shunting resistance.

When the current transformers and shunting resistances are connected in delta,

\[ Z_{\mathrm{sh}\Delta}=\frac{(e-iZ)\sqrt{3}}{n_a i_{\mathrm{nb}}}, \tag{32} \]

where \(i_{\mathrm{nb}}\) is the unbalance current, determined from expression (11) or (14).

The shunting resistance must be made in the form of an iron-core coil with the same saturation as the current transformers; then this resistance will have a current characteristic in accordance with expression (18).

The ratio between the active component of the no-load current and the reactive component of the current transformers of the same phase may be different; therefore, additional balancing of the active components may sometimes be required by shunting the underloaded current transformer with an active adjustable resistance.

Conclusion

The above analysis of differential protection for power transformers shows that its sensitivity and selectivity depend mainly on the proper selection and arrangement of the principal equipment of the circuit.

It follows from equations (26) and (27) that, in order to implement the protection, current transformers with constants \(A\) in a definite ratio must be used. In particular, when \(Z_1 = Z_2 = Z_a\) and \(n_a = 1\), on the basis of (27),

\[ A_2 = \frac{A_1 A_a}{A_1 + A_a}. \tag{33} \]

The Elektroapparat plant should be asked to consider the complete supply of current transformers and autotransformers for the differential protection of transformers, taking account of the constants \(A\).

It also follows from expressions (9)—(14) that autotransformers must be connected into the protection circuit when current transformers with the same constant \(A\) are used on the supply side, and in all other cases—in the circuit of the current transformers with the larger \(A\).

Expressions (9)—(14) may be used for an approximate determination of the unbalance current in the case where artificial balancing by means of \(Z_d\) or \(Z_{\mathrm{sh}}\) is not used. When \(Z_d\) or \(Z_{\mathrm{sh}}\) is used, the unbalance current can be determined only during adjustment of the protection. From the known value of the unbalance current under rated conditions, its maximum value during a through short circuit can be approximately determined from the following expression:

\[ i_{\mathrm{nbm}} = i_{\mathrm{nbn}} \left( \frac{k I_{k3}}{I_{nt}} \right)^{\alpha}, \tag{34} \]

where \(k\) is the impact coefficient, taking into account the aperiodic component of the short-circuit current;

\(I_{k3}\) is the periodic component of the short-circuit current at the initial instant of time under the maximum condition in the system;

\(I_{nt}\) is the rated current of the power transformer.

Noted Misprints

1. Guiding instructions on relay protection. Section T-3, Electric Stations. 1940, Nos. 5–6. ↩↩