ON THE CALCULATION OF PROTECTION AGAINST GROUND FAULTS FOR A UNIT GENERATOR OPERATING WITH NEUTRAL GROUNDING THROUGH AN ARC-SUPPRESSION COIL

P. T. ANOKHIN, R. A. VAINSTEIN

(Presented by the scientific seminar of the Department of Electric Power Stations)

For protection against ground faults in the stator windings of a unit generator, zero-sequence voltage protection is used. The calculation procedure [1, 2] provides for detuning from the effective value of the triple-frequency voltage \(U_{03}\), caused by distortion of the sinusoidal form of the generator phase-voltage curve, and from the fundamental-frequency voltage \(U_{0 \,\mathrm{el.st}}\), caused by electrostatic induction between the windings of the unit transformer during a fault on the high-voltage side. The setting is usually taken to be the minimum relay pickup voltage \(U_{\mathrm{sr}} = 15\ \mathrm{V}\). To exclude detuning from the voltage \(U_{0 \,\mathrm{el.st}}\), a time delay is provided that exceeds the operating time of the zero-sequence current protection of the unit transformer.

Medium- and high-power hydrogenerators, as a rule, have capacitive ground-fault currents \(J_c\) greater than \(5\ \mathrm{A}\); therefore, arc-suppression coils connected in the generator neutrals are used for their compensation. Operation of the coils with resonance tuning or with slight detuning is recommended [2].

Compensation of the capacitive current was intended to keep the generator in operation in the presence of a single-phase ground fault in the generator circuit and thereby to increase the reliability of the power supply. In this case the protection must operate to signal, and a time delay on the order of several seconds is of no fundamental importance.

At some hydroelectric power stations, where a ground fault of the stator winding is considered dangerous, the protection acts to trip the generator with a small time delay. At the same time, blocking of the zero-sequence current protection of the unit transformer is removed. This fundamentally changes the operating mode corresponding to the purpose of the arc-suppression coils, as a result of which the question arises of the advisability of their use.

This article presents certain generalizations in calculating the generator neutral-displacement voltage \(U_{0 \,\mathrm{el.st}}\), taking the transient process into account. The influence of detuning and losses on the effectiveness of operation of the arc-suppression coil is considered. When determining the time delay of the protection acting to trip the generator, in addition to the selectivity condition, the possibility of using the arc-suppression properties of the coil under intermittent ground faults is taken into account.

Determination of the Voltage \(U_{0\ \mathrm{el.st}}\)

The equivalent circuit is shown in Fig. 1, where

\[ U_0^{(v)} \]

is the zero-sequence voltage during a fault on the high-voltage side of the unit;

\(K\) is a coefficient taking into account the distribution of the voltage \(U_0^{(v)}\) along the high-voltage winding of the transformer (with an isolated neutral \(K=1.0\), and with a grounded neutral \(K=0.5\)) [2];

\(C_{\mathrm{t.mo}}\) is the capacitance between the high- and low-voltage windings of one phase of the unit transformer;

\(C_{\mathrm{g}}\) and \(C_{\mathrm{t}}\) are the capacitances of one phase of the generator and transformer with respect to ground;

\(R_{\mathrm{u}}\) is the active leakage resistance of one generator phase with respect to ground;

\(L_{\mathrm{k}}\) is the inductance of the arc-suppression coil;

\(R_{\mathrm{k}}\) is the active resistance, allowing for copper and core losses in the coil.

The calculation circuit has been drawn up on the assumption that the protection is connected to the secondary winding of the arc-suppression coil. In addition, in compiling the zero-sequence equivalent circuit, the assumptions listed below have been adopted.

1. The three-phase system is assumed to be symmetrical.

2. The inductance \(L_{\mathrm{k}}\) of the arc-suppression coil is linear.

3. The capacitances \(C_{\mathrm{t}}\) are neglected, assuming \(C_{\mathrm{t}} \ll C_{\mathrm{g}}\).

4. Active losses due to the equivalent leakage resistance

\[ R_{\mathrm{ue}}=\frac{R_{\mathrm{u}}}{3} \]

are not taken into account. For generators with gas cooling they are negligibly small, while for generators with direct water cooling of the stator winding they can, where necessary, be allowed for additionally.

Fig. 1. Equivalent circuit for determining the voltage \(U_{0\ \mathrm{el.st}}\), caused by electrostatic induction between transformer windings during a ground fault on the high-voltage side.

1. The zero-sequence impedance of the generator is very small in comparison with the impedance of the arc-suppression coil.

Introduce the following abbreviated notation:

\[ K_1=\frac{1}{3\omega^2 L_{\mathrm{k}} C_{\mathrm{g}}} \]

—the tuning degree of the oscillatory circuit \(R_{\mathrm{k}}, L_{\mathrm{k}}, 3C_{\mathrm{g}}\);

\[ V_1=1-K_1 \]

—the detuning degree of the same circuit,

\[ d_{\mathrm{k}}=\frac{R_{\mathrm{k}}}{\omega L_{\mathrm{k}}} \]

—a coefficient characterizing the losses in the arc-suppression coil (the coil damping coefficient).

Taking into account that \(d_{\mathrm{k}}^2 \ll 1\) and \(C_{\mathrm{t.mo}} \ll C_{\mathrm{g}}\), the voltage \(U_{0\ \mathrm{el.st}}\) in the steady-state mode can be represented as

\[ U_{0\ \mathrm{el.st}} = \frac{C_{\mathrm{t.mo}}}{C_{\mathrm{g}}} \cdot \frac{K\cdot U_0^{(v)}}{\sqrt{d_{\mathrm{k}}^2+V_1^2}} . \tag{1} \]

For networks operating with solidly grounded neutrals, the value \(U_0^{(v)}\) is determined by the expression

\[ U_0^{(\mathrm{v})}=K_0\cdot U_\phi^{(\mathrm{v})}, \tag{2} \]

where

\[ K_0=\frac{Z_0}{Z_1+Z_2+Z_0}, \tag{3} \]

\(Z_1\), \(Z_2\), and \(Z_0\) are the total positive-, negative-, and zero-sequence impedances relative to the fault point;

\(U_\phi^{(\mathrm{v})}\) is the phase voltage of the high-voltage winding of the unit transformer.

It is known that, other conditions being equal, the greatest value of the voltage \(U_0^{(\mathrm{v})}\) on the transformer will occur for a fault near the high-voltage buses. Our analysis shows that, when the relative number of transformers having grounded neutral points is equal to \(2/3\) or more, the coefficient \(K_0\) varies comparatively little and its average value is approximately equal to 0.2.

Consequently, for practical calculations one may take

\[ U_0^{(\mathrm{v})}=0.2U_\phi^{(\mathrm{v})}. \tag{4} \]

Substituting in (1), instead of \(U_0^{(\mathrm{v})}\), its value (4), after some transformations we obtain

\[ U_{0\,\mathrm{el.st}}= \frac{K_p}{\sqrt{d_k^2+V_1^2}}\cdot U_\phi, \tag{5} \]

where

\[ K_p=0.2\cdot K\cdot K_{\mathrm{st}}, \tag{6} \]

\[ K_{\mathrm{st}}=\frac{C_{\mathrm{t.mo}}}{C_g}\cdot \frac{U_\phi^{(\mathrm{v})}}{U_\phi}; \tag{7} \]

\(U_\phi\) is the phase voltage of the generator.

Let us determine the order of magnitude of \(K_{\mathrm{st}}\) using as an example a hydrogenerator of type SV \(\frac{1160}{170}-72\). In Table 9 [2] its parameters are given: \(S=90\) MVA, \(U_n=13.8\) kV, \(C_g=0.83\) μF/phase. We take the unit transformer to be of the same power at \(U_n=110\) kV. From the curves in Fig. 65 [2] we find \(C_{\mathrm{t.mo}}=0.005\) μF. Substituting in (7) \(\frac{U_\phi^{(\mathrm{v})}}{U_\phi}=\frac{121\cdot\sqrt{3}}{13.8\cdot\sqrt{3}}\), as well as the values of \(C_g\) and \(C_{\mathrm{t.mo}}\), we have \(K_{\mathrm{st}}=0.0528\).

If it is assumed that the generator operates in a unit with a transformer at \(U_n=220\) kV, then we obtain \(K_{\mathrm{st}}=0.095\).

For a unit with other parameters, the coefficient \(K_{\mathrm{st}}\) will deviate upward or downward. However, for the purpose of simplifying the calculations, we shall settle on the value \(K_{\mathrm{st}}=0.1\).

A higher value of \(U_{0\,\mathrm{el.st}}\) will be obtained for a unit in which the neutral of the transformer is isolated (\(K=1.0;\ K_p=0.02\)).

The dependence (5) is shown graphically in Fig. 2 in the form of curves 1 and 2, which are constructed for \(K_p=0.02\) and \(d_k=0.02\) and 0.03. Conversion to other values of \(K_p\) is performed by multiplying the ordinates of curves 1 and 2 by the ratio

\[ n_p=\frac{K_p}{0.02}. \]

Thus, for example, at \(K_{\mathrm{st}}=0.1\), in the case of grounding of the neutral of the unit transformer (\(K=0.5\)), we have \(K_p=0.01\), and \(n_p=0.5\). As a result we obtain curves 3 and 4 (Fig. 2).

Thus, if the neutral of the unit transformer is not grounded, then, with small losses of the arc-suppression coil and its operation with a tuning close to resonance (curve 1, Fig. 2), the voltage \(U_{0\,\mathrm{el.st}}\), although

and for a short time, but it may reach the value \(U_\phi\). At the same time, for values of \(|V_1|\) on the order of 0.1 and higher, the magnitude of the voltage \(U_{0\,\mathrm{el.st}}\) and its dependence on the degree of detuning and on losses are substantially reduced.

When the arc-suppression coil operates with some detuning, for example with overcompensation \((V_1 < 0)\), free oscillations arise in the transient regime with angular frequency \(\omega'\), close in magnitude to the industrial angular frequency \(\omega\). Therefore the function \(U_{0\,\mathrm{el.st}}(t)\) has the character of beats. The beat period \(T_{\mathrm{B}}\) is found from the formula

\[ T_{\mathrm{B}}=\frac{2\pi}{\omega'-\omega} =\frac{0.02}{\sqrt{1-V_1}-1}. \tag{8} \]

Fig. 2. Curves characterizing the dependence of the neutral displacement voltage during a ground fault on the high-voltage side of the unit transformer

The maximum value of the envelope \(U_{0\,\mathrm{el.st.m}}\) may be determined approximately at

\[ t=\frac{0.01}{\sqrt{1-V_1}-1} \]

and written in the form

\[ U^{(\mathrm{B})}_{0\,\mathrm{el.st.m}} = \frac{K_p\cdot U_\phi}{\sqrt{d_k^2+V_1^2}} \cdot n^{(\mathrm{B})}_{0\,\mathrm{el.st}}, \tag{9} \]

where

\[ n^{(\mathrm{B})}_{0\,\mathrm{el.st}} = 1+\sqrt{1-V_1}\cdot e^{-\frac{1.57\,d_k}{\sqrt{1-V_1}-1}}. \tag{10} \]

The dependence \(n^{(\mathrm{B})}_{0\,\mathrm{el.st}}(V_1)\) for \(d_k=0.02\) and \(0.03\) is shown in Fig. 2 (curves 5 and 6). The same figure also gives curve 7, characterizing the depen-

dependence \(U_{0\text{ el.st.m}}^{(\mathrm{B})}(V_1)\) at \(K_{\rho}=0.02\) and \(d_k=0.02\). The voltage \(U_{0\text{ el.st.m}}^{(\mathrm{B})}\) for other values of \(K_{\rho}\) is determined in the same way as for the established regime.

Consequently, in the transient regime as well, \(U_{0\text{ el.st}}\) decreases as the coil detuning increases.

Figure 2 also gives curves 8, 10, and 9, which, unlike curves 1, 3, and 5, were constructed for \(d_k=0.05\).

It is easy to see that, as the power of the unit transformer and its rated voltage increase, the coefficient \(K_{\text{st}}\) increases. However, at a voltage of 500 kV, when the neutrals of all transformers are grounded, we have \(K=0.5\) and \(K_0<0.2\). Therefore, for high-power generators the voltage \(U_{0\text{ el.st}}\) will in practice not appreciably exceed the value \(0.5U_{\phi}\). In other cases, a reduction of \(U_{0\text{ el.st}}\) can be provided by means of some detuning of the arc-suppression coil.

Specific recommendations on limiting the voltage \(U_{0\text{ el.st}}\) by increasing the coil detuning are absent from the Guidelines [2]. At the same time it is known that, although in the presence of detuning the coil retains its arc-suppression properties [3], the effectiveness of its use decreases. Moreover, with substantial detuning of the coil, the same arc overvoltages are possible as with an isolated neutral [4].

For unit generators, recommendations should be available for regulating the coil setting with allowance for various factors. Thus, for example, we believe that if, for any reason, the coil operates with considerable detuning, then, in evaluating its effectiveness, it is necessary to take into account the factor of active losses in the oscillatory circuit.

Influence of active losses and detuning of the arc-suppression coil on the effectiveness of its operation

In the case of a sustained fault in the generator-voltage circuit, at the point of the ground fault there will flow a residual current caused by active losses and by detuning of the coil.

In the case of an unstable fault, at the instant when the resultant current at the fault point passes through zero, the arc is extinguished. Arc extinction and the development of overvoltages in arc faults are described in sufficient detail in works [3, 4, 5]. At the initial instant after ignition of the arc, the transient process is practically independent of the presence of the coil [5].

After interruption of the arc, the transient process may conventionally be divided into two stages. First, the charges between the capacitances of the faulted and unfaulted phases are equalized. This stage proceeds in the same way as without an arc-suppression coil and is accompanied by the appearance on the faulted phase of a certain voltage value—the extinction peak \(U_{\text{pg}}\). In this case the maximum neutral displacement voltage \(U_{\text{sm}}\) is approximately equal to \(1.2U_{\phi}\) [4]. If immediate reignition of the arc does not occur, then the second stage begins, when the capacitances of all three phases begin to discharge through the inductance of the coil \(L_k\). This process has the character of damped oscillations with angular frequency \(\omega'\), close to \(\omega\). We shall not take into account the voltage \(U_{\text{pg}}\), since it does not depend on the coil parameters, and we write the process of decay of the voltage \(U_{\text{sm}}\) in the form

\[ U_{\text{sm}}(t)=U_{m\phi}e^{-\delta t}\cos \omega't, \tag{11} \]

where

\[ \omega' \simeq \omega_0=\frac{1}{\sqrt{L_k3\omega C_{\Gamma}}}. \]

The voltage on the damaged phase rises from zero:

\[ U_{\mathrm{d}}(t)=U_{m\phi}\left(\cos \omega t-e^{-\delta t}\cdot \cos \omega' t\right). \tag{12} \]

The expression for the envelope of the amplitude values of function (12) is obtained in the form

\[ U_{\mathrm{d}}^{(\mathrm{B})}(t)=U_{m\phi}\sqrt{\,1-e^{-2\delta t}-2e^{-\delta t}\cdot \cos\left(\sqrt{1-\vartheta_1}-1\right)\omega t\,}. \tag{13} \]

Figure 3 shows curves 1 and 2, constructed from expression (13) for \(d_k=0.02\) and, respectively, \(\vartheta_1=0\) and \(\vartheta_1=-0.1\). Also shown there is curve 3, characterizing the dependence \(U_{\mathrm{cm}}(t)\).

It is evident that, when the coil is detuned (curve 2), the rate of rise of the voltage on the damaged phase and its amplitude are considerably higher than with resonant tuning, as a result of which

Fig. 3. Curves characterizing the rate of rise of the voltage on the damaged phase (1, 2, 4, 5, 7) and the decay of the neutral displacement voltage (3, 6) under an unstable ground fault

the probability of a repeated breakdown, with increasing detuning of the coil, naturally increases.

Assume that repeated ignition of the arc occurs at the moment \(t=0.15\ \mathrm{s}\), when \(U_{\mathrm{d}}=1.5\,U_\phi\) and \(U_{\mathrm{cm}}\approx 0.6\,U_\phi\). If, as the current passes through zero, the arc is then interrupted, the conditions for its final extinction will be less favorable than before. The neutral displacement voltage takes the value \(U'_{\mathrm{cm}}>U_\phi\), while the maximum voltage on the damaged phase reaches a higher value than in the preceding case. If the arc does not assume a stable character, then, as a result of overvoltages, the probability of breakdown at another point of the stator winding increases.

Figure 3 gives curves 4, 5, and 6, which, unlike curves 1, 2, and 3, were constructed for \(d_k=0.05\). It is evident that, with resonant tuning of the coil, active losses affect only the rate of rise of the voltage on the faulted phase (curves 1 and 4). In the presence of coil detuning (curves 2 and 5), losses have practically no effect on the rate of voltage rise at the initial instant, but they do affect the maximum value of the voltage \(U_{\mathrm{dm}}\).

Thus, in the case of operation with appreciable detuning of the arc-suppression coil, at higher losses in the oscillatory circuit the conditions for arc extinction in intermittent faults will be more favorable.

Figure 3 also shows curve 7, constructed for \(d_k=0.05\) and \(\delta_1=-0.04\). Comparing curves 4 and 7, on the one hand, and curve 2, on the other, one may conclude that at higher active losses, but with resonant or nearly resonant tuning of the coil, the effectiveness of its use proves to be appreciably higher than at lower losses but with substantial detuning.

In practice, with detuning of the order of 10% and higher, an increase in losses in the oscillatory circuit should be regarded as a positive factor. In this case the total losses should be approximately 5–6%, and by what resistances they are caused is of no fundamental significance.

For generators with direct water cooling of the stator winding, it is necessary to take into account the losses due to the equivalent active leakage resistance \(R_{y\mathrm{e}}\) in the form of the damping factor

\[ d_c=\frac{1}{3R_{y\mathrm{e}}\omega C_g}. \tag{14} \]

In this case the attenuation coefficient is

\[ \delta=\frac{d}{2}\omega, \tag{15} \]

where

\[ d=d_k+d_c. \tag{16} \]

Determination of the Protection Time Delay

The pickup voltage of the protection in Fig. 2 is shown as horizontal line 11, whose ordinate is \(\dfrac{U}{U_\phi}=0.15\). It is evident that tuning to the voltage \(U_{0\ \mathrm{el.-st}}\) would in a number of cases lead to a considerable coarsening of the protection.

In connection with the introduction of protection for tripping the generator with a small time delay, it is necessary to note the conditions under which the greatest effectiveness of coil use can be ensured.

In the case of a sustained fault, after the free component of the capacitive current has decayed, the transient process at the fault point is determined by the free component of the current of the arc-suppression coil. The maximum value of this current is approximately equal to the capacitive current of the established regime. In the absence of a coil, the capacitive current of the generator flows at the fault point; with protection having a small time delay, this current cannot cause serious damage to the stator steel. In this case the use of a coil to compensate the capacitive current gives no obvious advantage.

In the case of an unstable fault, after extinction of the arc under the action of the neutral-displacement voltage (for example, curve 3, Fig. 3), the relay will keep the armature attracted until the voltage drops to the value ensuring its release, \(U_{* \mathrm{rel}}\). Taking \(U_{* \mathrm{op}} = 0.15\), we obtain

\[ U_{* \mathrm{rel}}=\frac{K_{\mathrm{r}}\cdot U_{* \mathrm{op}}}{K_{\mathrm{n}}}\approx 0.1, \tag{17} \]

where \(K_{\mathrm{r}}=0.8\) is the return coefficient,
\(K_{\mathrm{n}}=1.2\) is the reliability coefficient.

This voltage is shown in Fig. 3 as horizontal line 8, whose ordinate is

\[ \frac{U}{U_{\phi}}=0.1. \]

It is not difficult to see that, if the time delay of the protection is chosen without taking into account the decay time of the neutral-displacement voltage, then, despite the short-term nature of the fault with subsequent successful arc extinction, the protection may trip the generator. In this case the arc-suppression capability of the coil will not be utilized, as a result of which the effectiveness of its application becomes questionable.

The speed of operation of the protection, while ensuring that the generator remains in operation under unstable ground faults, can be increased by increasing the losses in the oscillatory circuit.

The value of the calculated decay time \(t_{\mathrm{d,calc.1}}\) will be found from the expression characterizing the decay process from

\[ \frac{U_{\mathrm{sm}}}{U_{\phi}} \]

to \(U_{* \mathrm{rel}}\), at which the protection returns to its initial state,

\[ \frac{U_{\mathrm{sm}}}{U_{\phi}}\cdot e^{-t_{\mathrm{d,calc.1}}}=U_{* \mathrm{rel}}. \tag{18} \]

Assuming the maximum neutral-displacement voltage after arc extinction to be \(U_{\mathrm{sm}}=1.2U_{\phi}\) [4] and substituting \(U_{* \mathrm{rel}}=0.1\) (17) and \(\delta=\dfrac{d}{2}\omega\) (16), after certain transformations we obtain

\[ t_{\mathrm{d,calc.1}}=\frac{1.58}{d}\,10^{-2}. \tag{19} \]

The second condition for determining the time delay is the selectivity of the protection for ground faults on the high-voltage side of the unit. The delay is necessary because, after the blocking is removed, the process of decay of the neutral-displacement voltage and the action of the protection will proceed analogously to what was considered above.

In most cases the voltage \(U_{0\,\mathrm{el.st}}<U_{\mathrm{sm}}=1.2U_{\phi}\); therefore the calculated decay time determined by the second condition, \(t_{\mathrm{d,calc.2}}\), will be less than \(t_{\mathrm{d,calc.1}}\), obtained according to (19). Therefore, as the time delay of the protection, while ensuring that the generator remains in operation under unstable ground faults, one may take

\[ t_{\mathrm{op}}=t_{\mathrm{d,calc.1}}=\frac{1.58}{d}\cdot 10^{-2}. \]

If the calculation is carried out without taking \(t_{\mathrm{d,calc.1}}\) into account, then it is necessary to determine the maximum value of the voltage \(U_{0\,\mathrm{el.st.m}}^{(\mathrm{B})}\) from expression (9), or graphically with the aid of the curves of Fig. 2, and then find \(t_{\mathrm{d,calc.2}}\). It follows

It should be noted that an increase in losses has an exceptionally favorable effect on reducing \(t_{\text{calc.2}}\). Thus, for example, at \(d \geq 0.05\) it will practically not exceed 0.2 sec.

Conclusions

1. It has been shown that, when a unit transformer operates with a grounded neutral, the voltage \(U_{0\,\text{el.st}}\) should not appreciably exceed the value \(0.5\,U_{\phi}\). However, in the absence of accurate data on the capacitance \(C_{\text{t.mo}}\), an experimental verification of this conclusion is required.

2. If it is necessary to limit \(U_{0\,\text{el.st}}\), preference should be given to operation of the arc-suppression coil with a setting as close as possible to resonance, but with higher active losses. In this case the effectiveness of the coil will be greater.

3. The arc-suppression properties of the coil can be realized only if, when determining the time delay of the protection acting to trip the generator, the transient process after arc extinction is taken into account.

References

1. A. M. Fedoseev. Fundamentals of Relay Protection. Moscow–Leningrad, Gosenergoizdat, 1961.

2. Guidelines for Relay Protection, issue 5. Protection of generator–transformer and generator–autotransformer units. Moscow–Leningrad, “Energiya,” 1963.

3. R. Wilheim and M. Waters. Neutral Grounding in High-Voltage Systems. Moscow–Leningrad, Gosenergoizdat, 1959.

4. A. I. Dolginov. High-Voltage Engineering in Electric Power Systems. Moscow, “Energiya,” 1968.

5. P. V. Borisoglebskii, L. F. Dmokhovskaya et al., under the general editorship of D. V. Razevig. High-Voltage Engineering. Moscow–Leningrad, “Energiya,” 1964.