ELECTROMAGNETIC RELAY RESPONDING TO CHANGES IN THE AREA OF THE LINE-VOLTAGE TRIANGLE

N. V. LISETSKY, L. V. SHMOILOV

(Tomsk Polytechnic Institute)

In modern relay-protection and automation devices, line voltages are monitored by two types of circuits:

1) a circuit with two or three high-speed relays connected to the line voltages;

2) a circuit with a negative-sequence voltage filter relay and a relay connected to one of the phase-to-phase voltages.

These circuits meet operational needs. However, voltage changes during disturbances of the normal operating mode can be transformed by means of a contactless functional converter—for example, a Spinelli frequency tripler. An electromagnetic relay can be connected at the output of such a converter in series with a capacitor (Fig. 1, a) and can record all types of faults.

The proposed device is high-speed. It can be manufactured in two versions:

1) responding to all types of faults associated with a decrease in the area of the line-voltage triangle;

2) sensitive to breaks in voltage circuits and relatively insensitive to faults on the protected object.

If the relay is intended for monitoring the serviceability of voltage circuits, then the capacitor is excluded from the circuit.

A similar relay, implemented on an induction system, is described in [1]. That relay was intended for blocking distance protection in the event of breaks in voltage circuits. It was slow-acting and operated not only during breaks but also during faults on the protected object. At present such relays are not manufactured by industry.

Single-phase transformers of the Spinelli frequency tripler, as is known, operate in the saturated-steel mode. Therefore, the sinusoidal voltages applied to the tripler are transformed to the secondary side in distorted form. The voltages of the secondary windings are added, and at the tripler terminals a voltage is obtained equal to the geometric sum of the distorted phase voltages.

In three-pole metallic short circuits, the indicated voltages are symmetrical, and the voltage at the tripler output will contain only the sum of the third and multiples-of-third harmonic components; the 1st, 5th, 7th, 11th, etc., components, when added geometrically, give zero.

Fig. 1.

a)

b) Vector diagram with labels \(A\), \(B\), \(C\), \(A_k\), \(B_k\), \(C_k\), \(O_k\), \(O\);
\(\dot U_{ABn}\), \(\dot U_{ABk}\), \(\dot U_{BCn}\), \(\dot U_{BCk}\), \(\dot U_{CAn}\), \(\dot U_{CAk}\);
\(\dot U_{COn}\), \(\dot U_{COk}\), \(\dot U_{AOn}\), \(\dot U_{AOk}\);
\(\dot U_{BOn}\), \(\dot U_{BOk}\);
\(\dot U_{Ck}\), \(\dot U_{Ak}\), \(\dot U_{Bk}\);
\(\delta_n\), \(\delta_k\), \(\alpha_k\), \(\beta_k\), \(\gamma_k\), \(\varphi_k\).

c) Vector diagram with labels \(A\), \(B\), \(C\), \(A_k\), \(B_k\), \(C_k\), \(O_k\), \(K\);
\(\dot U_{ABn}\), \(\dot U_{ABk}\), \(\dot U_{BCn}\), \(\dot U_{BCk}\), \(\dot U_{CAn}\), \(\dot U_{CAk}\);
\(\dot U_{COn}\), \(\dot U_{COk}\), \(\dot U_{AOn}\), \(\dot U_{AOk}\);
\(\dot U_{BOn}\), \(\dot U_{BOk}\);
\(\delta_n\), \(\delta_k\), \(\alpha_k\), \(\beta_k\), \(\gamma_k\), \(\varphi_k\).

In unsymmetrical three-phase and two-phase short circuits, the voltages supplied to the frequency tripler are unsymmetrical. As a result, saturation of the steel of the transformers of the different phases will be unequal, and at the output voltage of the tripler, in addition to the third harmonic components and components that are multiples of it, an unbalance voltage will appear, representing the geometric sum of the 1st, 5th, 7th, etc., harmonic components.

In order to establish the relation between the voltage at the output of the frequency tripler and the area of the line-voltage triangle, let us analyze the operation of the device for various types of faults. In carrying out this analysis we shall make the following assumptions:

1) the tripler operates at no load;
2) harmonic components that are multiples of three, beginning with the ninth and higher, are not taken into account;
3) the geometric sum of the voltages of the 1st, 5th, 7th, etc., harmonic components arising in the device during unsymmetrical short circuits will be taken into account as the unbalance voltage.

In a three-phase symmetrical short circuit, the area of the line-voltage triangle changes from its nominal value to zero, depending on the electrical distance from the fault point to the busbars of the electrical installation. In this case the system of phase and line vectors remains symmetrical (Fig. 1, б). The area of the line-voltage triangle for this case can be expressed as

\[ S_{\Delta}^{(3)}=\frac{1}{2}U_{BCK}U_{ABK}\sin\delta_k =\frac{\sqrt{3}}{4}U_k^2. \tag{1} \]

The voltage at the output of the frequency tripler for this case is written as

\[ \dot U_3=U_3=\frac{1}{k}\left(U_{AK}e^{-j3\alpha_K}+U_{AK}e^{j3\gamma_K}\right) =\frac{\sqrt{3}}{4}U_k, \tag{2} \]

where \(k\) is the transformation coefficient of the frequency tripler for the third harmonic component. According to [2] and [4], this coefficient is very close to a constant value.

The vector diagram of the voltages on the busbars in a short circuit between phases B and C on the outgoing line is shown in Fig. 1в. The area of the line-voltage triangle for this case is written as

\[ S_{\Delta}^{(2)}=S_{\Delta AKB_K}+S_{\Delta AKC_K} =\frac{\sqrt{3}}{4}U_lU_k\cos\varphi, \tag{3} \]

where \(U_l\) is the line voltage of normal operation;
\(U_k\) is the residual voltage on the busbars between the short-circuited phases;
\(\varphi\) is the angle equal to the difference between the angles of the short-circuit impedance and of the impedance outgoing from the connection busbars.

The vector of the geometric sum of the third harmonic components of the phase voltages on the secondary side of the frequency tripler can be expressed as follows:

\[ U_3=\frac{1}{k}\left[U_{AK}+U_{BK}\cos3\alpha_k+U_{CK}\cos3\gamma_k +j\left(U_{CK}\sin3\gamma_k-U_{BK}\sin3\alpha_k\right)\right] \]

\[ =\frac{1}{k}\left\{U_{AK}+U_{BK}(4\cos^3\alpha_k-3\cos\alpha_k) +U_{CK}(4\cos^3\gamma_k-3\cos\gamma_k)\right. \]

\[ \left.+\,j\left[U_{CK}(3\sin\gamma_k-4\sin^3\gamma_k) -U_{BK}(3\sin\alpha_k-4\sin^3\alpha_k)\right]\right\}. \tag{4} \]

From the vector diagram of Fig. 1, c,

\[ U_{AK}=\frac{U_l}{\sqrt{3}}; \]

\[ U_{BK}=\sqrt{OK^{2}+BK^{2}-2OK\cdot BK\cos(90^\circ+\varphi)} = \sqrt{\frac{0.25U_l^{2}}{3}+ \frac{U_k^{2}}{4}+ \frac{U_lU_k}{2\sqrt{3}}\sin\varphi}; \]

\[ U_{CK}=\sqrt{OK^{2}+KB_k^{2}-2OK\cdot KB_k\cos(90^\circ-\varphi)} = \sqrt{\frac{0.25U_l^{2}}{3}+ \frac{U_k^{2}}{4}- \frac{U_lU_k}{2\sqrt{3}}\sin\varphi}; \]

\[ U_{ABK}=\sqrt{AK^{2}+KB_k^{2}-2AK\cdot KB_k\cos(90^\circ+\varphi)} = \sqrt{\frac{2.25U_l^{2}}{3}+ \frac{U_k^{2}}{4}+ \frac{\sqrt{3}\,U_lU_k}{2}\sin\varphi}; \]

\[ U_{CAK}=\sqrt{AK^{2}+KC_k^{2}-2AK\cdot KC_k\cos(90^\circ-\varphi)} = \sqrt{\frac{2.25U_l^{2}}{3}+ \frac{U_k^{2}}{4}- \frac{\sqrt{3}\,U_lU_k}{2}\sin\varphi}; \]

\[ \alpha_k = \arccos\frac{U_{AK}^{2}+U_{BK}^{2}-U_{ABK}^{2}}{2U_{AK}\cdot U_{BK}} = \arccos \frac{ -\dfrac{U_l}{2\sqrt{3}}-\dfrac{U_k}{2}\sin\varphi }{ \sqrt{ \dfrac{0.25U_l^{2}}{3} +\dfrac{U_k^{2}}{4} +\dfrac{U_lU_k}{2\sqrt{3}}\sin\varphi } }; \]

\[ \gamma_k = \arccos\frac{U_{AK}^{2}+U_{CK}^{2}-U_{CAK}^{2}}{2U_{AK}\cdot U_{CK}} = \arccos \frac{ -\dfrac{U_l}{2\sqrt{3}}+\dfrac{U_k}{2}\sin\varphi }{ \sqrt{ \dfrac{0.25U_l^{2}}{3} +\dfrac{U_k^{2}}{4} -\dfrac{U_lU_k}{2\sqrt{3}}\sin\varphi } }; \]

\[ \delta_k = \arcsin \frac{ \dfrac{U_k}{2}\cos\varphi }{ \sqrt{ \dfrac{0.25U_l^{2}}{3} +\dfrac{U_k^{2}}{4} +\dfrac{U_lU_k}{2\sqrt{3}}\sin\varphi } }; \]

\[ \gamma_k = \arcsin \frac{ \dfrac{U_k}{2}\cos\varphi }{ \sqrt{ \dfrac{0.25U_l^{2}}{3} +\dfrac{U_k^{2}}{4} -\dfrac{U_lU_k}{2\sqrt{3}}\sin\varphi } }. \]

Substituting the values written above into expression (4), we obtain

\[ U_z = \frac{ U_lU_k^{2}\cos^{2}\varphi }{ \sqrt{3}\,k \left[ \left( \frac{U_l^{4}}{12}+\frac{U_k^{4}}{4} \right) - \frac{U_l^{2}U_k^{2}}{12}\sin^{2}\varphi \right] } - \left( \frac{U_l^{2}}{12} + \frac{U_k^{2}}{4}\cos 2\varphi - j\frac{U_k^{2}}{4}\sin 2\varphi \right). \tag{5} \]

The magnitude of this vector is

\[ |\dot U_3|=\frac{4U_lU_k^2\cos^2\varphi}{\sqrt{3}\,k\sqrt{\frac{U_l^4}{9}+\frac{2}{3}U_l^2U_k^2\cos 2\varphi+U_k^4}}. \tag{6} \]

As is seen from equation (6), in the case of a two-phase metallic short circuit in the immediate vicinity of the busbars \((U_k=0)\), the voltage of the third harmonic component becomes zero. However, the total voltage will not be equal to zero. In this case, a current twice as large as the currents in the other two windings will flow through one of the primary windings of the tripler. The saturation of the core steel will be different, and an unbalance voltage will appear, representing the geometric sum of the 1st, 5th, 7th, etc., harmonic components.

If one phase of the voltage transformer is open-circuited (for example, phase C), the area of the line-voltage triangle will be equal to zero, since of the three line voltages only one will remain. The voltage at the output of the frequency tripler will also be equal to zero:

\[ U_{AB}=U_l;\qquad U_{AO}=\frac{U_l}{2};\qquad U_{BO}=\frac{U_l}{2}e^{-j180^\circ};\qquad U_{CO}=0. \]

The voltage vector of the third harmonic components for this case is

\[ \dot U_3=\frac{1}{k_3}\left(U_{AO}+U_{BO}e^{-j3\alpha}+U_{CO}e^{j3\gamma}\right) =\frac{1}{k_3}\left(\frac{U_l}{2}+\frac{U_l}{2}e^{-j3\cdot180^\circ}+0\right)= \]

\[ =0. \]

For the fifth harmonic components, the voltage vector is written as:

\[ \dot U_5=\frac{1}{k_5}\left(U_{AO}+U_{BO}e^{-j5\alpha}+U_{CO}e^{j5\gamma}\right) =\frac{1}{k_5}\left(\frac{U_l}{2}+\frac{U_l}{2}e^{-j5\cdot180^\circ}+0\right)= \]

\[ =0. \]

Analogous equations can be written for all other higher-order harmonic components.

The analysis carried out made it possible to obtain parametric dependences of the no-load voltage of the frequency tripler on the area of the line-voltage triangle (1, 2, 3, 6) and on the residual voltage between the faulted phases (2, 6). In Fig. 2, a and b, these dependences are plotted in relative units for \(k=1\) and \(\varphi=0\). In plotting them, the line voltage of the normal operating condition was taken as the base quantity. The designations of the curves in Fig. 2 are: T—for a three-phase short circuit, D—for a two-phase short circuit. Experimental curves are also shown in the same figure:

TX and DX—for no-load operation of the frequency tripler;

TR and DR—when the tripler is loaded only by the relay;

TC and DC—when the tripler is loaded by a series-connected capacitor and relay.

In the experiments a frequency tripler of power 1 VA, an ET-522/0.2 relay, and a capacitor of capacitance 15 μF were used.

Figure 3 shows curves of the current flowing through the relay as a function of the residual-voltage magnitude under the same conditions.

The waveform of the established voltages and currents is shown in the oscillograms of Figs. 4 and 5 for the absence and presence of a capacitor in the relay circuit.

Fig. 2

Fig. 3

Comparing the experimental curves of Fig. 2 with the theoretical ones, it may be noted that the nature of their variation is similar.

The TX curves at small residual voltages lie below the calculated ones. This is explained by the weak saturation of the steel of the transformers of the frequency tripler and, consequently, by the weak distortion of the waveform of the voltages in the secondary windings.

Fig. 4

Annotations in the figure:

• With symmetrical line voltages equal to 100%
  Voltage at the relay
• Current in the relay winding
• In the presence of reverse residual voltages at the point of installation of the short circuit
• Voltage at the relay
• Current in the relay winding

Fig. 5

Annotations in the figure:

• With symmetrical line voltages equal to 100%
  Voltage at the relay
• Current in the relay winding
• In the presence of angular residual voltages at the point of installation of the short circuit
• Voltage at the relay
• Current in the relay winding
• Currents at the relay
• Voltage at the relay
• Current in the relay winding

For large values of the residual voltages, the TX curves lie above the calculated ones. This is explained by the fact that the actual output voltages of the tripler, in addition to the third harmonics, contain a significant portion of higher harmonics that are multiples of three, which we did not take into account in the theoretical calculations. Moreover, for three-phase symmetrical sh—

short-circuits all these higher harmonics coincide in phase with the third harmonic.

The DP curves over the entire range lie above the calculated ones because of the unbalance voltages not taken into account by expression (6). Moreover, at small residual voltages the 1st, 5th, 7th, etc. harmonics predominate in the unbalance voltage, which is clearly seen in Figs. 4 and 5. At large residual voltages the line-voltage triangle becomes more symmetrical. Therefore the unbalance voltage due to the 1st, 5th, 7th, etc. harmonic components decreases, but the 3rd and other harmonics that are multiples of it appear, which make the voltage at the output of the tripler greater than the calculated value.

The shape of the TR and DR curves repeats the shape of the TX and DX curves, the difference in ordinates being the voltage loss in the resistances of the secondary windings of the frequency tripler when the relay current flows through them.

The TC and DC curves, beginning with a residual voltage \(U_k = 0.3\) and an area of the line-voltage triangle \(S_\Delta = 0.14\), are located above the corresponding TR and DR curves. Such an increase in the voltage on the relay when capacitances are connected in series with its winding occurs because of the geometric difference between the voltage of the frequency tripler \((U_y)\) and the voltage drop across the capacitive resistance \((U_c)\) when the relay current \((I_p)\) flows in the circuit, as shown in the vector diagram of Fig. 2, \(a\).

At small residual voltages the DC curves are located below the DR curves. This is caused by the capacitance smoothing the peaks of the unbalance voltage, which can be traced from the oscillogram curves in Figs. 4 and 5.

In our opinion, the most important dependences are the curves in Fig. 3. The shape of the curves in this figure basically repeats the shape of the analogous curves in Fig. 2, \(b\). This is natural, since the curves in Fig. 2, \(b\) and Fig. 3 are related to each other by Ohm’s-law relation. However, at small residual voltages the DR curve of Fig. 3 differs sharply from the analogous curve of Fig. 2, \(b\)—the current in the relay depends to a lesser degree on the magnitude of the residual voltage. This can be explained as follows.

In two-phase metallic short-circuits close to the location where the protection is installed, the voltage supplied to the relay has a fairly large 1st harmonic component (Fig. 4). The inductive resistance of the circuit for the 1st harmonic component is equal to \(\omega L\), and for the 3rd it is \(3\omega L\). Therefore, in nearby two-phase short-circuits the fundamental component of the relay current is the 1st harmonic component, for which the circuit resistance is small (Fig. 4). As the short-circuit point moves farther away, the voltage at the output of the frequency tripler changes its form: the 1st harmonic component decreases and the 3rd increases. As a result, the equivalent resistance of the relay circuit increases. Along with the increase in resistance, the voltage also increases. Since the voltage on the relay grows faster than the resistance of its circuit, a certain rise of the DR curve in Fig. 3 is observed.

As can be seen from the curves in Fig. 3, with a relay setting of \(50\,ma\) or less and with no capacitance in its circuit, the relay will not operate for all two-phase and remote three-phase short-circuits, and will operate distinctly when the secondary voltage circuits are broken.

To determine the character of the changes in currents and voltages during transient processes and the operating time of the relay, oscillograms were taken for a three-phase short-circuit (Fig. 6, \(a\)) and a two-phase one (Fig. 6, \(b\)). The indicated oscillograms were taken for sudden short-circuits giving an established residual voltage

Fig. 6

between the damaged phases equal to 67 percent of the relay setting (providing a sensitivity factor of 1.5).

From the oscillograms presented it is clear that, despite the presence of capacitance in the relay circuit, the change in voltages and currents during transients has a rapidly damped character. The relay reliably closes its contacts for three-phase short circuits after 1.5 periods and for two-phase short circuits after 2.5 periods of the industrial-frequency current. At lower residual voltages the operating speed of the relay increases.

In addition to the tests indicated above, the proposed relay was switched on for residual voltages in single-phase and two-phase ground faults. In single-phase faults the relay did not operate. In two-phase ground faults it operated with a smaller sensitivity margin than in a two-phase metallic fault.

Conclusions

1. The proposed relay can be made in two versions, namely:

a) operating for all types of faults associated with a decrease in the area of the line-voltage triangle;

b) operating only in the event of an open circuit in the secondary voltage circuits and close-in three-pole short circuits, and not operating for all other faults.

1. All existing devices for monitoring the serviceability of the secondary circuits of voltage transformers may operate falsely during close-in two-phase short circuits; the proposed relay is free of this drawback.

2. The frequency tripler with a power of 1 VA and the capacitor can be

mounted in the relay housing, then a single relay replaces an entire voltage-monitoring circuit.

1. As the responsive element of a device monitoring line voltages, not only an ET-522/0.2 relay may be used, but any other relay as well. By using a relay with lower power consumption, it is possible to reduce the power of the frequency tripler and the power consumed from the voltage transformer to a value of less than 1 VA.

REFERENCES

1. M. F. Kostrov, I. I. Solov'ev, A. M. Fedoseev. Fundamentals of Relay Protection Engineering. GEI, 1944.
2. A. M. Bamdas, V. A. Kulinich, S. V. Shapiro. Static Electromagnetic Converters of Frequency and Number of Phases. GEI, 1961.
3. A. M. Fedoseev. Fundamentals of Relay Protection. GEI, 1961.
4. L. N. Bessonov. Electric Circuits with Steel. GEI, 1948.