Differential Protection of Synchronous Generators Implemented Using Current Transformers with Secondary-Current Biasing

A. S. Baginskaya

(Presented by Professor, Doctor of Technical Sciences I. D. Kutyavin)

Despite a number of measures aimed at increasing the selectivity of generator differential protection—increasing the relay setting to values exceeding the rated current, and using fast-saturating transformers—the percentage of its correct operations remains at an extremely low level. Thus, according to data from the Technical Administration of the Ministry of Electric Power Stations, the percentage of correct operation of longitudinal differential protection of generators in 1953 was 76.2%, whereas the percentage of correct operations of all relay protection as a whole exceeded 99%.

Analysis of the operation of longitudinal differential protection of generators shows that one of the very frequent causes of its false operations is the large unbalance currents that appear in the relay during external short circuits. Such trippings are the most dangerous, since taking a generator out of service—especially one of large capacity—during a short circuit in the system may lead to its complete disruption.

The use of fast-saturating transformers in the differential-protection scheme of hydrogenerators does not eliminate erroneous operation of the protection during external faults, since the unbalance current does not always have a sharply expressed asymmetrical character. The delay in operation of such protection, caused by the asymmetrical nature of the fault current and aggravated by low sensitivity coefficients, is a very significant drawback.

Coarsening the protection to \(I_{\mathrm{s}} = (1.3\text{--}1.5) I_{\mathrm{g}}\), as is currently accepted according to the recommendations of the Technical Administration of the Ministry of Electric Power Stations for turbogenerators, does not free the protection from false operations during external short circuits, since, as studies carried out at the Central Scientific Research Electrical Laboratory of the Ministry of Electric Power Stations [1] have shown, transient unbalance currents can reach several tens of amperes. The latter are due to the presence, in the cores of the protection current transformers at the moment of a through fault, of residual fluxes that differ in magnitude and sometimes in direction (after disconnection of an internal fault or elimination of an open circuit in the transformer secondary circuit).

In order to eliminate residual fluxes, some authors propose implementing the generator differential-protection scheme by means of special current transformers (with an air core or gap, with a small volume of steel, with biasing from an external source).

The influence of residual magnetism on the operation of generator differential protection can also be eliminated by using current transformers with secondary-current biasing for it.

Such transformers have a design analogous to that of compensated current transformers with counter-magnetization, intended for measuring purposes. Unlike the latter, the parameters of these current transformers are determined from the condition that they be magnetized by their normal operating current to an induction of 6,000–7,000 gs. Then, at a secondary current equal to 50% of nominal (operation of the generator with a smaller load is unlikely), the cores of the current transformers (for the grade of steel adopted for their manufacture) will be magnetized to an induction of 3,500–4,000 gs. The influence of residual fluxes on the operation of the differential protection, as experiments have shown, is insignificant in this case.

In analyzing the properties of generator differential protection implemented by means of current transformers with secondary-current biasing, it is important to clarify the question of the influence of the amount of steel in the transformers on the sensitivity and selectivity of the protection. Until recently, generator differential protection was implemented by means of class D current transformers with a large amount of steel. It was considered that the reduction of nominal induction thus achieved helps to reduce unbalance currents. In 1952 B. A. Nechaev recommended current transformers with a small amount of steel for generator differential protection, asserting that transient unbalance currents are thereby reduced [2]. In addition, the author shows that the response of current transformers with a small amount of steel under internal faults is no worse than the response of transformers with a large core cross-section, and that, owing to this, the sensitivity of the protection proposed by him is not lower than the sensitivity of protection using class D current transformers. This conclusion of B. A. Nechaev is highly doubtful, since, when the amount of steel is reduced, the response of the current transformers cannot remain unchanged (see below). In this connection, a further and more thorough investigation is needed of the question of the influence of the amount of steel in current transformers on the sensitivity and selectivity of differential protection. To resolve this question, we shall use a mathematical analysis of the operation of the differential-protection circuit under steady-state and transient processes, with subsequent experimental verification of its results.

An exact mathematical analysis of the protection circuit, taking into account the nonlinearity of the electrical circuits, the phenomenon of remagnetization of current transformers, etc., is impossible because of its exceptional complexity. However, relationships of a qualitative order could be established by adopting, in order to simplify the solution of the problem posed, a number of assumptions, namely: by considering the magnetizing resistance of the current transformers independent of the magnetizing current, neglecting losses in the steel and the inductances of the secondary windings of the transformers, and taking the resistance of the relay circuit equal to zero. The latter is permissible when there are no additional resistances in the relay circuit. The e.m.f. of the biasing windings in this case will represent nothing other than the e.m.f. of self-induction, and the relation of the self-induction resistance of the biasing windings to the magnetizing resistance of the current transformers will be expressed by the ratio:

\[ x_{n/m}=x_{\text{mag}}\left(\frac{\omega_{n/m}}{\omega_2}\right)^2, \tag{1} \]

where \(\omega_{n/m}\) is the number of turns in the biasing windings;

\(\omega_2\) is the number of turns of the secondary winding.

Fig. 1 shows the equivalent circuit of the differential protection of a generator, implemented according to the principle of circulation of secondary currents and using current transformers with series biasing. In it, solid lines show the directions of currents during external short circuits, and dashed lines—during short cir-

faults in the protected zone. All currents are referred to the number of turns of the secondary winding. In the diagram the following are denoted:

\(z_1\) and \(z_2\) — the impedances of the secondary circuits of the current transformers, consisting of the impedances of the secondary windings, the active resistances of the bias windings, contacts, and connecting wires;

\(\Delta z_{01}\) and \(\Delta z_{02}\) — the self-induction impedances of the bias windings (neglecting the nonlinearity of the electrical circuits);

\(z_{01}\) and \(z_{02}\) — the impedances of the magnetizing branches;

\(z_p\) — the relay impedance;

\(I'\) — the through short-circuit current;

\(I'_1\) and \(I'_2\) — the primary currents of the transformers in the event of an internal fault;

\(i_1\) and \(i_2\) — the secondary currents of the current transformers;

\(i_{01}\) and \(i_{02}\) — the magnetizing currents.

Fig. 1.

Fig. 2.

According to the equivalent circuit, the unbalance current of a differential protection using current transformers with series biasing, for an external short circuit and \(z_p \cong 0\), is:

\[ i_{nb} = I' \frac{z_{01}z_2 - z_{02}z_1}{(z_1 + z_{01} + \Delta z_{01})(z_2 + z_{02} + \Delta z_{02})}. \tag{2} \]

Assuming that an emf \(E(t) = E_m e^{j(\omega t+\alpha)} = \dot{E}_m e^{j\omega t}\) acts in the primary circuit, and passing to the operator form of writing the unbalance current, we obtain:

\[ I_{nb}(p) = \frac{p\dot{E}_m p(L_{01}r_2 - L_{02}r_1)} {(r_k + pL_k)(p - j\omega)[r_1 + p(L_1 + L_{01})][r_2 + p(L_2 + L_{02})]}, \tag{3} \]

where \(L_1, L_2\) are the self-induction coefficients of the bias windings;

\(r_k\) and \(L_k\) are the active resistance and self-induction coefficient of the short-circuit circuit.

The complex expression for the unbalance current can be found from expression (3) on the basis of the expansion theorem. Passing from it to the instantaneous value of the unbalance current, we shall have:

\[ i_{nb} = i_m \left[ a \sin(\omega t + \alpha - \varphi) + b e^{-\frac{t}{T_k}} \sin(\alpha - \varphi_k) + c e^{-\frac{t}{T_1}} \sin(\alpha - \varphi_1) + d e^{-\frac{t}{T_2}} \sin(\alpha - \varphi_2) \right], \tag{4} \]

where \(i_m\) is the amplitude of the periodic component of the primary short-circuit current, referred to the number of turns of the secondary winding of the current transformer;

\[ a = -\frac{\omega(L_{01}r_2 - L_{02}r_1)} {r_1 r_2 \sqrt{(1+\omega^2T_1^2)(1+\omega^2T_2^2)}} ; \]

\[ b = \frac{T_k(L_{01}r_2 - L_{02}r_1)} {r_1 r_2 (T_k - T_1)(T_k - T_2)} ; \]

\[ c=\frac{T_1(L_{01}r_2-L_{02}r_1)} {r_1r_2(T_1-T_k)(T_1-T_2)}; \]

\[ d=\frac{T_2(L_{01}r_2-L_{02}r_1)} {r_1r_2(T_2-T_k)(T_2-T_1)}; \]

\[ \varphi_k=\operatorname{arctg}\omega T_k,\quad \varphi_1=\operatorname{arctg}\omega T_1,\quad \varphi_2=\operatorname{arctg}\omega T_2,\quad \varphi=\varphi_k+\varphi_1+\varphi_2-\frac{\pi}{2}; \]

\[ T_k=\frac{L_k}{r_k} \]
is the time constant of the primary circuit;

\[ T_1=\frac{L_1+L_{01}}{r_1},\quad T_2=\frac{L_2+L_{02}}{r_2} \]
are the time constants of the secondary circuits.

It is known that the unbalance current assumes its maximum value at \(a=0\). For this value of the initial angle of the emf, and also for all time constants exceeding \(0.05\) sec, if it is taken into account that

\[ \frac{L_{01}}{r_1}\simeq \frac{L_{01}+L_1}{r_1}=T_1 \quad\text{and}\quad \frac{L_{02}}{r_2}\simeq \frac{L_{02}+L_2}{r_2}=T_2 \quad\text{and}\quad T_1=nT_2, \]

with \(1.1>n>1\),

\[ i_{\text{ub}}\simeq -i_m\frac{n-1}{\omega nT_2}\sin\omega t -i_m\left[ \frac{T_k-T_2(n-1)} {(T_k-nT_2)(T_k-T_2)}e^{-\frac{t}{T_k}} +\frac{T_k}{nT_2-T_k}e^{-\frac{t}{nT_2}} -\frac{T_k}{T_2-T_k}e^{-\frac{t}{T_2}} \right]. \tag{5} \]

As expression (5) shows, the periodic component of the unbalance current is inversely proportional to the time constant of the secondary circuit, directly proportional to the short-circuit current, and depends on the difference between the time constants of the secondary circuits of the current transformers, decreasing as this difference is reduced.

The dependence of the maximum value of the aperiodic component, referred to \(i_m\), on the time constant of the secondary circuit for various \(T_k\), plotted on the basis of expression (5), is given in Fig. 2. As follows from the curves in Fig. 2, the maximum value of the aperiodic component increases as \(T_2\) decreases and as \(T_k\) increases.

It is true that the dependences presented hold for comparatively large amounts of steel in the current transformers. At small \(T_2\), the mathematical dependence \(i_{\text{ub ap}}=f(T_2)\), owing to strong saturation of the transformers during short circuits, is violated. Oscillographic recording of the maximum unbalance currents for different cross sections of the current-transformer cores showed that, with sufficiently small amounts of steel, the aperiodic component in the unbalance current may be absent. This is explained by the fact that, when an aperiodic current with a considerable time constant flows through the primary windings of the transformers, owing to the sharp decrease in the magnetizing impedances, the maximum aperiodic magnetizing current of each transformer occurs in less than one half-period. From the moment the maximum is reached, the aperiodic currents become equal to the primary aperiodic current and, as a result, the aperiodic current does not enter the differential circuit. Owing to the difference in the time at which the maxima of the aperiodic components of the magnetizing currents of the two transformers occur, a surge of unbalance current is possible; however, because of its short duration it cannot in any way affect the operation of the protection. However, as a consequence of the increase in the periodic component \(i_{\text{ub}}\) when the amount of steel is reduced, the maximum operating value of the unbalance current with a small cross section of the transformer cores proves, as numerous experiments have shown, to be greater than this current in high-steel transformers with biasing.

Thus, from the standpoint of increasing the sensitivity or selectivity of differential protection, preference should be given to class D current transformers biased by secondary current.

The latter will become quite obvious if one compares the currents in the relays of protections using current transformers with a small amount of steel and class D transformers biased by secondary current, for internal short circuits.

The current in the relay for faults in the protected zone and one-sided supply is:

\[ I_p \simeq I'_1 \frac{\omega T_1}{\sqrt{1+\omega^2 T_1^2}}, \tag{6} \]

i.e., for a small time constant of the secondary circuit it depends to a considerable degree on the magnitude of this constant, decreasing as the latter decreases.

Figure 3 gives the dependences \(I_p=f(I'_1)\), obtained experimentally. Curve 1 corresponds to a protection made with current transformers having \(q=20\ \mathrm{cm}^2\), \(w_1=w_2=100\) turns and with secondary-current biasing \((w_n:M=10\) turns). The internal short circuit, when its supply was removed, was fed from one side. A measuring instrument with a very small resistance was connected in the relay circuit. Curve 2 was obtained for a protection using current transformers with \(q=1.25\ \mathrm{cm}^2\) \((B_{\mathrm{nom}}\simeq 6000\ \mathrm{gauss})\). As follows from these graphs, when protection is performed using transformers having a small core cross section, with increasing \(I'_1\), and consequently also with saturation of the transformer steel, a sharp deterioration of the output is observed.

Fig. 3.

Fig. 4.

The latter is especially strongly reduced if some resistance, especially inductive resistance, is inserted into the relay circuit. Thus, when connected into the relay circuit of a relay with setting range 250–1100 mA, having \(x_p \simeq 1.1\ \mathrm{cm}\) and \(r_p=0.6\ \Omega\), \(I_p\) decreases considerably (see curve 3 in Fig. 3).

Owing to the relatively poor output of the protection using current transformers with a small core cross section, its sensitivity coefficient has an inadmissibly low value, and the protection has an operating delay for internal faults. A protection using current transformers with secondary-current biasing is free of the indicated shortcomings and, in this connection, has unquestionable advantages over a protection made with current transformers having a small core cross section.

To increase sensitivity, and, at the accepted sensitivity, the selectivity of differential protection, the Guidelines for Relay Protection have repeatedly recommended introducing additional resistances into relay circuits. In clarifying the question of the effectiveness

the introduction into the relay circuits of a differential protection using current transformers with series magnetization of additional resistances, and also their rational type, corresponding expressions were obtained for the transient unbalance current and the minimum current in the relay during short circuits at the generator terminals. On the basis of these expressions, curves were constructed for the dependence of the amplitude of the periodic component and the maximum value of the aperiodic component of the unbalance current on the additional resistances in the relay circuit. These curves are presented in Fig. 4.

Curve 1 shows the dependence of the amplitude of the periodic component, and curve 2—the maximum value of the aperiodic component of the unbalance current, in relative units, on the additional active resistance in the relay circuit. Since the maximum of the aperiodic component considerably exceeds the amplitude of the periodic component, the maximum value of the unbalance current is determined mainly by the former component. As is seen from Fig. 4, when comparatively small active resistances are introduced into the relay circuit, the unbalance current decreases appreciably. With a further increase in resistance, the reduction of the unbalance current slows down, while the output of the transformers during internal short circuits deteriorates noticeably. Nevertheless, the sensitivity coefficient of the differential protection of generators increases when the active additional resistance in the relay circuit is increased. This is shown in Fig. 5, which presents curves of the dependence of the protection sensitivity coefficient, for equal values of the currents of internal and external short circuits, on the additional resistances in the relay circuit.

Fig. 5.

An entirely different result is obtained from introducing an inductive additional resistance into the relay circuit. Such a resistance cannot substantially affect the aperiodic component of the unbalance current and cannot noticeably reduce its magnitude (see curve \(2'\), Fig. 4). A certain reduction in the aperiodic unbalance current in the latter case is explained by the fact that, as the inductive resistance increases, the time at which the maximum unbalance current occurs increases. This leads to greater damping of the components of the aperiodic current and, consequently, to a reduction in its total magnitude. The periodic component of the unbalance current, when the inductive additional resistance is increased, decreases to a greater extent than in the presence of active resistances in the relay circuit (cf. curves \(1'\) and 1 in Fig. 4). But since its magnitude is considerably smaller than that of the aperiodic component, the reduction of the total unbalance current is determined mainly by the latter.

The decrease of the current in the relay during internal short circuits with increasing inductive additional resistance is more considerable than with active additional resistances. The latter is explained by the fact that, at sufficiently large inductive resistances, the current in the relay practically coincides in phase with the magnetizing currents of the transformers. The short-circuit current \(I_{kz}\) is then distributed between the magnetizing branches and the relay inversely proportional to the moduli of their resistances. With a purely active resistance of the relay circuit, the currents of the indicated branches are shifted in phase by an angle close to \(90^\circ\). Their arithmetic sum in this case exceeds \(I_{kz}\), i.e., the current in the relay with an active additional resistance is greater than with an inductive resistance of the same magnitude. As a result of the considerable decrease of the current in the relay during internal short circuits and the comparatively small decrease of the unbalance current, the sensitivity of the differential protection when introduced into the circuit

relay, the inductive additional resistance is reduced somewhat (Fig. 5).

The curves shown in Figs. 4 and 5 were plotted on the basis of analytically obtained expressions for $I_{nb}$, $I_r$, and $\alpha_u$ with the following initial data:

1) the time constant of the primary circuit $T_k = 0.1$ sec;

2) the time constant of the secondary circuit of one of the transformers $T_2 = 0.1$ sec;

3) the time constant of the secondary circuit of the other transformer $T_1 = nT_2 = 1.1T_2$;

4) the total active resistance of the transformer secondary circuit $r = 0.5$ ohm.

The experimental verification of the nature of the dependences obtained mathematically confirmed their validity and showed that, for comparatively small additional inductive resistances (up to 10 ohms), the limitation of the higher harmonics of the unbalance current is insignificant.

Fig. 6.

Graphs of the change in the sensitivity coefficient $\alpha'_u = f(z_r)$, obtained on the basis of experimentally determined dependences $I_{nb} = f(z_r)$ and $I_r = f(z_r)$ for the generator differential-protection circuit, are presented in Fig. 6. The protection circuit was made using laboratory current transformers with $q = 20\ \text{cm}^2$; $w_1 = w_2 = 100$ turns; $w_{pm} = 10$ turns. In carrying out the experiments, a relay with a total resistance of $(0.9 + j1.1)$ ohm was connected into the differential circuit for measuring the unbalance current, and active or inductive resistances added to it were varied. Since the constant active-inductive resistance in the relay circuit had a comparatively small value, the nature of the variation of the experimental curves closely repeats the nature of the variation of the calculated graphs (cf. Figs. 6 and 5).

Thus, in order to increase the sensitivity of generator differential protection, active additional resistances should be included in the relay circuit. With the sensitivity of generator differential protection presently adopted, there is no need to use additional resistances in the relay circuit, since biasing the transformers by secondary current leads to a considerable reduction of unbalance currents and, without question, eliminates false operation of the protection during through short circuits. Therefore, when using current transformers with series biasing, the need to include active additional resistances may arise only in connection with increased requirements for the sensitivity of generator differential protection. The additional resistance should be limited to a value of (5–10) ohms in order to avoid significant increases in voltage in the secondary circuits during internal faults.

Conclusions

1. The use of current transformers with secondary-current biasing for generator differential protection substantially increases its sensitivity or, at the accepted sensitivity, its selectivity, eliminating the main cause of the occurrence of large unbalance currents—the residual fluxes in the cores of the current transformers.

1. From the standpoint of ensuring sufficient sensitivity factors and speed of operation of the differential protection, the current transformers with series biasing recommended for it must have a steel core volume no smaller than that of class D current transformers. Reducing the steel core volume of the current transformers of differential protection leads to a significant decrease in its sensitivity factor and to delays in operation during internal faults.

2. If it is necessary to increase the sensitivity of the differential protection, active additional resistances of \(5\text{—}10\ \Omega\) should be inserted in the relay circuits.

References

1. Tsarev M. I., Differential protection of a generator with auxiliary fast-saturating transformers, Elektricheskie stantsii, 1949, No. 7.

2. Nechaev B. A., On the question of the use of current transformers with a small iron volume for differential protection of generators, dissertation, Ivanovo, 1952.