OPERATING PRINCIPLES AND CALCULATION OF GENERATOR PROTECTION AGAINST EXTERNAL AND INTERNAL INTERPHASE SHORT CIRCUITS USING DIFFERENTIAL CURRENT TRANSFORMERS

Part I. Circuit of Combined Protection of Generators Against External and Internal Interphase Short Circuits

N. A. DULZON

(Presented by the scientific seminar of the Department of Electric Stations and Networks)

The schematic diagram of this protection is shown in Fig. 1. In this case, the middle limb of the core of the DTT, as a rule, is made with an air gap. On the outer limbs there are two secondary windings; in one of them a comparatively sensitive relay for overcurrent protection is connected, and in the other—a coarser relay for protection against internal faults.

Fig. 1.

In normal operation and during through overcurrents, the currents in the primary windings of the DTT are equal and opposite. As a result, the magnetic fluxes in the right and left halves of the cross section of the core will be directed oppositely and will be forced to close through the middle limb with an air gap. In this mode the DTT may be mentally represented as two separate current transformers with an air gap, for which the dependence of the current in the relay on the primary current is a straight line with positive slope, the angle of which depends on the magnitude of the air gap (Fig. 2, curve 1). The setting of the more sensitive—

Fig. 2.

... of the relay (relay \(M\) in Fig. 1). \(I_{\text{set}m}\) is chosen so that it operates at the required primary operating current of the overcurrent protection, i.e., approximately at \(1.4 I_{\text{nom}}\). The setting of the second relay, \(I_{\text{set}d}\), is selected with a definite margin so that this relay does not operate under maximum through overcurrents.

In internal interphase short circuits, the primary currents in the DCT are not equal to one another. With respect to the difference of these currents, the DCT is an ordinary current transformer with a solid core and with two secondary windings. For such a transformer, the dependence of the secondary currents on the primary current has the form of curve 2 in Fig. 2. In an internal short circuit, both relays operate. In this case relay \(D\) trips the generator without time delay, while relay \(M\) (with a time delay) performs the function of backup. The sensitivity of the protection against internal faults with respect to the primary current \(I_{\text{av}d}\) depends on the steepness of curve 2, which must be taken into account in calculating the DCT.

Fig. 3.

The equivalent circuit of the protection under consideration is shown in Fig. 3. In the figure:

\[ \dot Z_3=\frac{1}{\dot Y_3};\qquad \dot Z_4=\frac{1}{\dot Y_4} \]

are, respectively, the resistances of the circuit of the overcurrent protection and of the protection against internal faults;

\[ \dot Z_{01}=\frac{1}{\dot Y_{01}};\qquad \dot Z_{02}=\frac{1}{\dot Y_{02}} \]

are the magnetizing resistances of the left and right halves of the core;

\[ \dot Z_{0s1}=\frac{1}{\dot Y_{0s1}};\qquad \dot Z_{0s2}=\frac{1}{\dot Y_{0s2}} \]

are the magnetizing resistances corresponding to the leakage fluxes of the primary current;

\[ \dot Z_{0\mathrm{sh}}=\frac{1}{\dot Y_{0\mathrm{sh}}} \]

is the magnetizing resistance corresponding to the gap between the primary conductors, including the middle limb with an air gap.

For the system of equations describing this equivalent circuit, the principal determinant is

\[ \Delta= \left[ \frac{\dot Y_{0\mathrm{sh}}}{\dot Y_{0s1}} + \frac{\dot Y_{0\mathrm{sh}}}{\dot Y_{0s2}} +1 \right] (\dot Y_{01}+\dot Y_3)(\dot Y_{02}+\dot Y_4) + \dot Y_{0\mathrm{sh}}(\dot Y_{01}+\dot Y_{02}+\dot Y_3+\dot Y_4). \tag{1} \]

The partial determinants for calculating the secondary and magnetizing currents are:

\[ \Delta_3= \dot I_1\dot Y_3 \left[ \frac{\dot Y_{0\mathrm{sh}}}{\dot Y_{0s2}}(\dot Y_4+\dot Y_{02}) + (\dot Y_4+\dot Y_{02}) + \dot Y_{0\mathrm{sh}} \right] - \dot I_2\dot Y_3 \left[ \frac{\dot Y_{0\mathrm{sh}}}{\dot Y_{0s2}}(\dot Y_4+\dot Y_{02}) + \dot Y_{0\mathrm{sh}} \right]; \tag{2} \]

\[ \Delta_4= \dot I_2\dot Y_4 \left[ \left( \frac{\dot Y_{0\mathrm{sh}}}{\dot Y_{0s1}}+1 \right) (\dot Y_3+\dot Y_{01}) + \dot Y_{0\mathrm{sh}} \right] - \]

\[ -\,\dot{i}_{1}\dot{Y}_{4} \left[ \frac{\dot{Y}_{\mathrm{sh}}}{\dot{Y}_{0s1}} (\dot{Y}_{3}+\dot{Y}_{01})+\dot{Y}_{\mathrm{sh}} \right]; \tag{3} \]

\[ \Delta_{01}=\frac{\Delta_{3}\dot{Y}_{01}}{\dot{Y}_{3}}; \qquad \Delta_{02}=\frac{\Delta_{4}\dot{Y}_{02}}{\dot{Y}_{4}}. \tag{4} \]

The design condition for an external short circuit is the equality of the primary currents \(\dot{I}_{1}=\dot{I}_{2}=\dot{I}_{\mathrm{kz}}\). In this case the currents in the circuit will be:

\[ \dot{i}_{3}= \frac{\dot{i}_{1}\dot{Y}_{3}(\dot{Y}_{4}+\dot{Y}_{02})}{\Delta}; \tag{5} \]

\[ \dot{i}_{4}= \frac{\dot{i}_{1}\dot{Y}_{4}(\dot{Y}_{3}+\dot{Y}_{01})}{\Delta}. \tag{6} \]

For an internal short circuit the smallest secondary currents will occur at \(\dot{I}_{2}=0\). Then

\[ \dot{i}_{3}= \frac{ \dot{i}_{1}\dot{Y}_{3} \left[ \left( \frac{\dot{Y}_{\mathrm{sh}}}{\dot{Y}_{0s2}}+1 \right) (\dot{Y}_{4}+\dot{Y}_{02})+\dot{Y}_{\mathrm{sh}} \right] }{\Delta}, \tag{7} \]

\[ \dot{i}_{4}= \frac{ -\,\dot{i}_{1}\dot{Y}_{4} \left[ \frac{\dot{Y}_{\mathrm{sh}}}{\dot{Y}_{0s1}} (\dot{Y}_{3}+\dot{Y}_{01})+\dot{Y}_{\mathrm{sh}} \right] }{\Delta}; \tag{8} \]

\[ \dot{i}_{01}=\dot{i}_{3}\frac{\dot{Y}_{01}}{\dot{Y}_{3}}; \qquad \dot{i}_{02}=\dot{i}_{4}\frac{\dot{Y}_{02}}{\dot{Y}_{4}}. \tag{9} \]

From expressions (5) and (6) it is evident that, with a single-phase arrangement and with an open circuit in the internal-fault protection circuit \((\dot{Y}_{4}=0)\), the current in the maximum-protection circuit decreases sharply.

If, when considering formulas (7) and (8), the leakage fluxes of the primary windings are neglected \((\dot{Y}_{0s1}=\dot{Y}_{0s2}=\infty)\), they take the form:

\[ \dot{i}_{3}= \frac{ \dot{i}_{1}\dot{Y}_{3}(\dot{Y}_{4}+\dot{Y}_{02}+\dot{Y}_{\mathrm{sh}}) }{ (\dot{Y}_{01}+\dot{Y}_{3})(\dot{Y}_{02}+\dot{Y}_{4}) +\dot{Y}_{\mathrm{sh}}(\dot{Y}_{01}+\dot{Y}_{02}+\dot{Y}_{3}+\dot{Y}_{4}) }; \tag{10} \]

\[ \dot{i}_{4}= \frac{ -\,\dot{i}_{1}\dot{Y}_{4}\dot{Y}_{\mathrm{sh}} }{ (\dot{Y}_{01}+\dot{Y}_{3})(\dot{Y}_{02}+\dot{Y}_{4}) +\dot{Y}_{\mathrm{sh}}(\dot{Y}_{01}+\dot{Y}_{02}+\dot{Y}_{3}+\dot{Y}_{4}) }. \tag{11} \]

From the last expressions it is evident that, in the event of an internal short circuit in the generator, an open circuit in the maximum-protection circuit \((\dot{Y}_{3}=0)\) leads to an increase in the sensitivity of the protection against internal faults, since the current \(I_{4}\) increases in this mode. It is clear, however, that an open circuit in any circuit leads to failure of the protection with the broken circuit. Taking this into account, this protection cannot be recommended for a single-phase arrangement.

In order to save conductor, the single-phase circuit of Fig. 1 may be made as shown in Fig. 4, a, where the secondary windings of the differential current transformers are connected cumulatively, or as in Fig. 5, a, where these windings are connected differentially. The first schematic diagram corresponds to the substitution circuit of Fig. 4, b.

For circuit 4, b, with through currents in the generator \((I_1 = I_2)\), the current in the maximum-protection circuit is:

\[ \dot I_3 = \frac{\dot I_1 \dot Y_3 \left[\dot Y_4(\dot Y_{01}+\dot Y_{02})+\dot Y_{\mathrm{opr}}(\dot Y_4+\dot Y_{02})\right]} {\Delta}. \tag{12} \]

The current in the circuit of protection against internal interphase faults is

\[ \dot I_4 = \frac{\dot I_1 \dot Y_4 \left[\dot Y_3(\dot Y_{01}+\dot Y_{02})+\dot Y_{\mathrm{opr}}(\dot Y_3+\dot Y_{01})\right]} {\Delta}. \tag{13} \]

Fig. 4.

The current in the neutral conductor is

\[ \dot I_{\mathrm{opr}} = \frac{\dot I_1 \dot Y_{\mathrm{opr}}\left[\dot Y_3(\dot Y_4+\dot Y_{02})-\dot Y_4(\dot Y_3+\dot Y_{01})\right]} {\Delta}. \tag{14} \]

In the last expressions:

\[ \Delta = \left( \frac{\dot Y_{\mathrm{osh}}}{\dot Y_{os1}} + \frac{\dot Y_{\mathrm{osh}}}{\dot Y_{os2}} +1 \right) \left[ \dot Y_{01}\dot Y_{02}(\dot Y_3+\dot Y_4) + \dot Y_3\dot Y_4(\dot Y_{01}+\dot Y_{02}) + \right. \]

\[ \left. + \dot Y_{\mathrm{opr}}(\dot Y_3+\dot Y_{01})(\dot Y_4+\dot Y_{02}) \right] + \dot Y_{\mathrm{osh}} \left[ (\dot Y_{01}+\dot Y_{02})(\dot Y_3+\dot Y_4) + \dot Y_{\mathrm{opr}}(\dot Y_{01}+\dot Y_{02}+\dot Y_3+\dot Y_4) \right]. \]

\[ \dot Y_3 \]
— conductance of the maximum-protection circuit;

\[ \dot Y_4 \]
— conductance of the protection circuit against internal interphase faults;

\[ \dot Y_{\mathrm{opr}} \]
— conductance of the neutral conductor.

Fig. 5.

For the second basic protection circuit (Fig. 5, a), the equivalent electrical circuit shown ... can easily be obtained.

in Figs. 5, 6. Calculation of this circuit gives the following values of the currents in it:

\[ \dot I_3=\frac{\Delta_3}{\Delta};\qquad \dot I_4=\frac{\Delta_4}{\Delta};\qquad \dot I_{01}=\frac{\Delta_{01}}{\Delta};\qquad \dot I_{02}=\frac{\Delta_{02}}{\Delta};\qquad \dot I_{\mathrm{op}}=\frac{\Delta_{\mathrm{op}}}{\Delta}, \tag{15} \]

where

\[ \begin{aligned} \Delta_3={}& \dot I_2 \dot Y_3 \left\{ \frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s2}} \left[\dot Y_4 \dot Y_{02}+\dot Y_{\mathrm{op}}(\dot Y_4+\dot Y_{02})\right] +\left(\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s1}}+1\right)\dot Y_{01}\dot Y_4 \right.\\ &\left. +\dot Y_{\mathrm{err}}(\dot Y_{\mathrm{op}}+2\dot Y_4) \right\} -\dot I_1 \dot Y_3 \left\{ \left(\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s2}}+1\right) \left[\dot Y_4\dot Y_{02}+\dot Y_{\mathrm{op}}(\dot Y_4+\dot Y_{02})\right] \right.\\ &\left. +\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s1}}\dot Y_{01}\dot Y_4 +\dot Y_{\mathrm{err}}(\dot Y_{\mathrm{op}}+2\dot Y_4) \right\}; \end{aligned} \]

\[ \begin{aligned} \Delta_4={}& \dot I_1 \dot Y_4 \left\{ \frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s1}} \left[\dot Y_3\dot Y_{01}+\dot Y_{\mathrm{op}}(\dot Y_3+\dot Y_{01})\right] +\left(\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s2}}+1\right)\dot Y_{02}\dot Y_3 \right.\\ &\left. +\dot Y_{\mathrm{err}}(\dot Y_{\mathrm{op}}+2\dot Y_3) \right\} -\dot I_2 \dot Y_4 \left\{ \left(\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s1}}+1\right) \left[\dot Y_3\dot Y_{01}+\dot Y_{\mathrm{op}}(\dot Y_3+\dot Y_{01})\right] \right.\\ &\left. +\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s2}}\dot Y_{02}\dot Y_3 +\dot Y_{\mathrm{err}}(\dot Y_{\mathrm{op}}+2\dot Y_3) \right\}; \end{aligned} \]

\[ \begin{aligned} \Delta_{01}={}& \dot I_2\dot Y_{01} \left\{ \frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s2}} \left[(\dot Y_3+\dot Y_4+\dot Y_{\mathrm{op}})\dot Y_{02} +(\dot Y_3+\dot Y_{\mathrm{op}})\dot Y_4\right] \right.\\ &\left. -\left(\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s1}}+1\right)\dot Y_3\dot Y_4 +\dot Y_{\mathrm{err}}(\dot Y_3+\dot Y_4+\dot Y_{\mathrm{op}}) \right\}\\ &-\dot I_1\dot Y_{01} \left\{ \left(\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s2}}+1\right) \left[(\dot Y_3+\dot Y_4+\dot Y_{\mathrm{op}})\dot Y_{02} +(\dot Y_3+\dot Y_{\mathrm{op}})\dot Y_4\right] \right.\\ &\left. -\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s1}}\dot Y_3\dot Y_4 +\dot Y_{\mathrm{err}}(\dot Y_3+\dot Y_4+\dot Y_{\mathrm{op}}) \right\}; \end{aligned} \]

\[ \begin{aligned} \Delta_{02}={}& \dot I_1\dot Y_{02} \left\{ \frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s1}} \left[(\dot Y_3+\dot Y_4+\dot Y_{\mathrm{op}})\dot Y_{01} +(\dot Y_4+\dot Y_{\mathrm{op}})\dot Y_3\right] \right.\\ &\left. -\left(\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s2}}+1\right)\dot Y_3\dot Y_4 +\dot Y_{\mathrm{err}}(\dot Y_3+\dot Y_4+\dot Y_{\mathrm{op}}) \right\}\\ &-\dot I_2\dot Y_{02} \left\{ \left(\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s1}}+1\right) \left[(\dot Y_3+\dot Y_4+\dot Y_{\mathrm{op}})\dot Y_{01} +(\dot Y_4+\dot Y_{\mathrm{op}})\dot Y_3\right] \right.\\ &\left. -\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s2}}\dot Y_3\dot Y_4 +\dot Y_{\mathrm{err}}(\dot Y_3+\dot Y_4+\dot Y_{\mathrm{op}}) \right\}; \end{aligned} \]

\[ \begin{aligned} \Delta_{\mathrm{op}}={}& \dot I_2\dot Y_{\mathrm{op}} \left\{ \frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s2}}(\dot Y_4+\dot Y_{02})\dot Y_3 -\left(\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s1}}+1\right)(\dot Y_3+\dot Y_{01})\dot Y_4 \right.\\ &\left. +\dot Y_{\mathrm{err}}(\dot Y_3-\dot Y_4) \right\} -\dot I_1\dot Y_{\mathrm{op}} \left\{ \left(\frac{\dot Y_{\mathrm{err}}}{\dot Y_{0s2}}+1\right)(\dot Y_4+\dot Y_{02})\dot Y_3 - \right. \end{aligned} \]

\[ -\frac{\dot{Y}_{\mathrm{osh}}}{\dot{Y}_{0s1}} \left[ (\dot{Y}_{3}+\dot{Y}_{01})\dot{Y}_{4} +\dot{Y}_{\mathrm{osh}}(\dot{Y}_{3}-\dot{Y}_{4}) \right], \]

The main determinant is

\[ \Delta = \left( \frac{\dot{Y}_{\mathrm{osh}}}{\dot{Y}_{0s1}} + \frac{\dot{Y}_{\mathrm{osh}}}{\dot{Y}_{0s2}} +1 \right) \left[ \dot{Y}_{01}\dot{Y}_{02}(\dot{Y}_{3}+\dot{Y}_{4}) +\dot{Y}_{3}\dot{Y}_{4}(\dot{Y}_{01}+\dot{Y}_{02}) + \right. \]
\[ \left. +\dot{Y}_{\mathrm{opr}}(\dot{Y}_{3}+\dot{Y}_{01})(\dot{Y}_{4}+\dot{Y}_{02}) \right] + \dot{Y}_{\mathrm{osh}} \left[ (\dot{Y}_{01}+\dot{Y}_{02})(\dot{Y}_{3}+\dot{Y}_{4}) + \right. \]
\[ \left. +\dot{Y}_{\mathrm{opr}}(\dot{Y}_{01}+\dot{Y}_{02}+\dot{Y}_{3}+\dot{Y}_{4}) \right]. \]

When the primary currents are equal \((I_{1}=I_{2})\), the currents in the relay of the maximum-current protection, in the protection against internal interphase faults, and in the neutral wire, respectively, will be

\[ \dot{I}_{3}= \frac{ \dot{I}_{1}\dot{Y}_{3} \left[ \dot{Y}_{4}(\dot{Y}_{01}-\dot{Y}_{02}) -\dot{Y}_{\mathrm{opr}}(\dot{Y}_{4}+\dot{Y}_{02}) \right] }{\Delta}; \tag{16} \]

\[ \dot{I}_{4}= \frac{ \dot{I}_{1}\dot{Y}_{4} \left[ \dot{Y}_{3}(\dot{Y}_{02}-\dot{Y}_{01}) -\dot{Y}_{\mathrm{opr}}(\dot{Y}_{3}+\dot{Y}_{01}) \right] }{\Delta}; \tag{17} \]

\[ \dot{I}_{\mathrm{opr}}= -\frac{ \dot{I}_{1}\dot{Y}_{\mathrm{opr}} \left[ (\dot{Y}_{3}+\dot{Y}_{01})\dot{Y}_{4} +(\dot{Y}_{4}+\dot{Y}_{02})\dot{Y}_{3} \right] }{\Delta}. \tag{18} \]

Analysis of the last two circuits (Figs. 4a, 5a) shows that, with through currents in the generator and with a break in one of the secondary DTT circuits, as in the preceding case (Fig. 1), the current in the undamaged circuit decreases sharply (12, 13, 16, 17). Consequently, the circuits under consideration likewise cannot be recommended for single-phase implementation.

Let us consider possible variants of protection against external and internal interphase faults in a three-phase implementation.

The three-phase protection circuit corresponding to Fig. 1 is shown in Fig. 6a.

In external interphase short circuits, a break in the circuit of the protection against internal faults of one of the phases leads to a decrease of the current in the maximum-current protection circuit of that same phase. However, failure of the maximum-current protection to operate will not occur, owing to the operation of the two remaining healthy phases.

With analogous damage to the secondary circuit and an internal interphase short circuit, there will likewise be no failure of the protection to operate, for the same reasons.

At the same time, when there is a break in the secondary circuits, a phase current will flow in the neutral wire of the corresponding protection, which makes it possible, by means of a relay installed in the neutral wire (relay KC, Fig. 6a), to provide reliable circuit supervision. In addition, owing to the reasons stated above, a current will also flow in the neutral wire of the undamaged protection, which may cause the second circuit-supervision relay to operate.

As we see, the circuit considered satisfies the requirements imposed on it and can be used. In this circuit, an eight-core control cable is required to connect the differential current transformers to the relay panel.

In the neutral wire of the three-phase protection circuit corresponding to Fig. 4a, with through currents in the generator and a break in one of the secondary circuits, only a small current will flow, equal to the current in the healthy circuit of the damaged phase, since no current flows from the undamaged phases in the neutral wire (with symmetrical DTT), as is evident from (14). This causes certain difficulties in carrying out supervision of the soundness of the secondary circuits.

It is clear from Fig. 5 that, when the three-phase scheme is used (Fig. 6, b), no such difficulty arises in monitoring the circuits. This scheme meets all requirements; its operation differs little from the variant according to

a)

b)

Fig. 6.

Fig. 6, a, and at the same time it saves one conductor of the secondary connections. Therefore, this scheme can be recommended for use.