One-hundred-percent protection against frame faults
Abstract Generated abstract
This paper examines whether third-harmonic currents can be used to improve generator frame-fault protection for generators operating in parallel on generator-voltage busbars. It analyzes an AEG-type protection scheme with an insulated neutral and active earth-fault current, proposes series capacitance in the relay voltage circuit to amplify the otherwise small third-harmonic component, and derives conditions for compensation, relay voltage, selectivity, and sensitivity. The calculations show that the resulting grounding currents are comparable to those of existing protections while potentially eliminating the dead zone, although the author notes that practical significance would require operating tests.
Full Text
ONE-HUNDRED-PERCENT PROTECTION AGAINST FRAME FAULTS
I. D. Kutyavin
Associate Professor, Candidate of Technical Sciences
General considerations
The present work aims to investigate the possibility of using third-harmonic currents to increase the sensitivity of protection against frame faults for generators operating in parallel on generator-voltage busbars.
The circuit shown in Fig. 1 was subjected to investigation. This circuit is an AEG protection scheme with an insulated neutral point and with the creation of an active earth-fault current by means of a special grounding transformer connected to the operating terminals of the generator. In Fig. 2A the distribution of zero-sequence currents of the first harmonic is presented for a frame fault at the generator terminal, and in Fig. 2B—the third-harmonic currents for a frame fault at the generator neutral. A comparison of these figures shows that the use of third-harmonic currents to increase the sensitivity of the protection under consideration is possible in principle, but the power at the relay terminals from the third harmonic will be negligibly small, since the magnitude of the third-harmonic voltage in the composition of the phase voltage of generators is 3–5%. In this case, the maximum power of the third harmonic will amount to only 0.09–0.25% of the maximum power of the first harmonic.
Fig. 1
However, the necessary increase in the power of the third harmonic can be obtained by amplifying the voltage of this harmonic at the relay terminals by connecting the relay voltage winding in series with the capacitance \(C\), according to Fig. 1.
In this case, the value of the required capacitance can be determined from the following conditions.
- The maximum voltage of the first harmonic at the relay terminals \(U_{\mathrm{pm1}}\) must be equal to the rated voltage of the relay \(U_{\mathrm{pn}}\).
- The maximum voltage of the third harmonic on relay \(U_{\mathrm{r}3}\) must be \(k'\) times greater than the rated voltage of the relay \(U_{\mathrm{rn}}\),
\[ \frac{U_{\mathrm{r}3}}{k'}=U_{\mathrm{r}1}=U_{\mathrm{rn}}. \tag{1} \]
- The displacement of the currents of the first and third harmonics in the voltage winding of the relay relative to the corresponding voltages at the terminals must be the same, and the relay must have sufficient torque when an active grounding resistance is applied. The latter condition can be
Fig. 2
fulfilled if overcompensation for the third harmonic is applied in the relay voltage circuit:
\[ 3X_{\mathrm{r}}<\frac{X_{\mathrm{c}}}{3}; \]
the preceding inequality can be represented in the following general form:
\[ X_{\mathrm{c}}=nX_{\mathrm{r}}, \tag{2} \]
where \(X_{\mathrm{c}}\) is the capacitive reactance in the relay voltage circuit for the first harmonic, and \(X_{\mathrm{r}}\) is the inductive reactance of the relay voltage winding for the first harmonic.
The degree of overcompensation must be such that ferroresonance at the third harmonic is impossible and, in addition, that the internal displacement of the relay be as small as possible. On the basis of these conditions, the ratio \(n \geq 12\).
In Fig. 3 vector diagrams are given for two cases.
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In Fig. 3a a vector diagram is given of the voltage, current, and magnetic fluxes of the relay for the first harmonic at \(X_{\mathrm{c}}=0\).
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In Fig. 3b a vector diagram is given of the voltage, current, and magnetic fluxes for the third harmonic at \(X_{\mathrm{c}}=12X_{\mathrm{r}}\).
Comparison of these vector diagrams shows that the internal displacement of the relay will remain almost unchanged and will have only the opposite sign. Experiments carried out on relay IM-43 showed that the internal displacement under overcompensation by the value \(X_{\mathrm{r}}\) increases only by \(5\text{–}7^\circ\). These experiments also showed that at \(n=10\) the phenomenon of ferroresonance at the third harmonic begins to appear.
It is further necessary to determine the degree of amplification of the third-harmonic voltage by relating to one another the three conditions indicated above. For this purpose
denote the voltage of the first harmonic between points \(ab\) on the grounding resistance (Fig. 1) by \(U_1\), the voltage of the third harmonic by \(U_3\), and the ratio between them by
\[ \frac{U_3}{U_1}=k''. \]
Then the voltage of the first harmonic at the relay terminals can be determined as follows:
\[ U_{pm1}=\frac{U_1 Z_{p1}}{\sqrt{R_p^2+(X_p-nX_p)^2}}, \tag{3} \]
and the voltage of the third harmonic at the relay terminals
\[ U_{pm3}=\frac{U_3 Z_{p3}}{\sqrt{R_p^2+\left(3X_p-\frac{n}{3}X_p\right)^2}}, \tag{4} \]
where \(Z_{p1}\) and \(Z_{p3}\) are the total impedances of the relay voltage winding for the first and third harmonics.
\(R_p\) is the active resistance of the voltage winding. Substituting into relation (1) the values of the voltages \(U_{pm1}\) from (3) and \(U_{pm3}\) from (4), we obtain the following equality:
Fig. 3a Fig. 3b
\[ \frac{U_1 Z_{p1}}{\sqrt{R_p^2+(X_p-nX_p)^2}} = \frac{K' U_1 Z_{p3}}{K''\sqrt{R_p^2+\left(3X_p-\frac{n}{3}X_p\right)^2}}, \]
or
\[ \frac{Z_{p3}}{Z_{p1}}\sqrt{R_p^2+(X_p-nX_p)^2} = \frac{K'}{K''}\sqrt{R_p^2+\left(3X_p-\frac{n}{3}X_p\right)^2}. \tag{5} \]
Let us introduce the following additional notations:
- The ratio of impedances
\[ \frac{Z_{p3}}{Z_{p1}}=m\simeq 3; \]
- The ratio of coefficients
\[ \frac{K'}{K''}=K. \]
We shall call this new coefficient \(K\) the third-harmonic amplification coefficient, and we obtain the expression for it from equality (5):
\[ K=m\sqrt{ \frac{ 1+(n^2-2n)\left(\frac{X_p}{Z_{p1}}\right)^2 }{ 1+\left(8-2n+\frac{n^2}{9}\right)\left(\frac{X_p}{Z_{p1}}\right)^2 } }. \tag{6} \]
In the numerator under the radical, unity may be neglected in comparison with the second term, and from this expression one may determine the ratio of the third-harmonic voltage at the relay terminals to the nominal voltage of the relay:
\[ K' = mK'' \frac{X_p}{Z_{p1}} \sqrt{ \frac{n^2 - 2n} {1 + \left(8 - 2n + \frac{n^2}{9}\right)\left(\frac{X_p}{Z_{p1}}\right)^2} }. \tag{7} \]
Calculation of the Protection with Allowance for the Third Harmonic
1. A metallic ground fault in the winding of the neighboring generator at a distance from the neutral point equal to \(x\), in per unit.
1. Apparent power at the relay terminals due to the first harmonic
\[ P_{\ell 1} = U_{p1}(I_{n1}-I_{v1}) + U_{p1} I_{nb1} = U_{pm1}(I_{nm1}-I_{vm1})x^2 + U_{pm1} I_{nb1}x, \]
or
\[ P_{\ell 1}=P_{\ell m1}x^2+P_{nb1}x. \tag{8} \]
where
\[ I_{nm1}=\frac{I_{zm}}{n_{t0}} \]
is the maximum current in the secondary circuit of the neutral transformer,
\[ I_{vm1}=\frac{I_{zm}}{n_{tv}} \]
is the same in the secondary circuit of the external current transformers (Fig. 1).
\(I_{zm}\) is the maximum primary ground-fault current, determined by the grounding resistance.
2. Apparent power at the relay terminals due to the third harmonic
\[ P_{\ell 3} = U_{p3}(I_{n3}-I_{v3}+I_{nb3}) = U_{pm3}(I_{nm3}-I_{vm3})(1-x)^2 + U_{pm3}I_{nb3}(1-x), \]
or
\[ P_{\ell 3}=P_{\ell m3}(1-x)^2+P_{nb3}(1-x). \tag{9} \]
The ratios of the powers in equations (8) and (9) are:
\[ \frac{P_{\ell m3}}{P_{\ell m1}} = \frac{U_{pm3}}{U_{pm1}} \cdot \frac{I_{nm3}-I_{vm3}}{I_{nm1}-I_{vm1}} = K'K'', \]
\[ \frac{P_{nb3}}{P_{nb1}} = \frac{U_{pm3}}{U_{pm1}} \cdot \frac{I_{nb3}}{I_{nb1}} = K'K_{nb}, \]
where \(K_{nb}\cong 1\) is the ratio of the third-harmonic unbalance current of the external current transformers to the first-harmonic unbalance current.
Taking this into account, expressions can be written for the total apparent power:
\[ P_{\ell} = P_{\ell 1}+K_fP_{\ell 3} = P_{\ell m1}x^2+P_{nb1}x + K_fK'K''P_{\ell m1}(1-x)^2 + K_fK'K_{nb}P_{nb1}(1-x). \]
Let us further denote \(K_1=K_fK'K''\) and \(K_2=K_fK'K_{nb}\),
where \(K_f=0.85\) is the frequency coefficient of induction relays.
Then we obtain the simplified expression for the total apparent power
\[ P_{\ell} = P_{\ell m1}\left[x^2+K_1(1-x)^2\right] + P_{nb1}\left[x+K_2(1-x)\right]. \tag{10} \]
For the selectivity of the protection operation to be observed, it is necessary that the maximum false power at the relay terminals be less than the relay pickup power:
\[ P_{pt}=K_nP_{lm}, \tag{11} \]
where \(K_n>1\) is the reliability coefficient.
To find the maximum false power, we differentiate equation (10) with respect to \(x\):
\[ \frac{dP_l}{dx}=P_{lm1}[2x-2K_1(1-x)]+P_{nb1}(1-K_2)=0. \]
The value of \(x\) corresponding to the maximum of the false power is
\[ x_m=\frac{2K_1P_{lm1}-P_{nb1}(1-K_2)}{2P_{lm1}(1+K_1)}. \tag{12} \]
Substituting the value \(x_m\) into equation (10), we obtain the following expression for the maximum false power:
\[ P_{lm}= \frac{K_1P_{lm1}^2+P_{lm1}P_{nb1}(K_1+K_2)-0.25P_{nb1}^2(1-K_2)^2} {P_{lm1}(1+K_1)}. \tag{13} \]
As will be shown below, the practical limits of the coefficients \(K_1\) and \(K_2\) are such that the last term of the numerator of expression (13), in comparison with the other terms, may be neglected.
Taking into account the selectivity condition (11), the following relation can be composed:
\[ \frac{P_{pt}}{K_n}= \frac{K_1P_{lm1}+P_{nb1}(K_1+K_2)}{1+K_1}. \tag{14} \]
From this relation one can determine the difference of the secondary currents in the protection circuit, which is expressed in the following form:
\[ I_{nm1}-I_{vm1}= \frac{P_{pt}(1+K_1)-K_nP_{nb1}(K_1+K_2)} {K_1K_nU_{pm1}}. \tag{15} \]
II. Metallic short circuit to the frame in the winding of the protected generator at a distance from the neutral point equal to \(x\).
- The power at the relay terminals from the first harmonic:
\[ P_1=U_{p1}(I_{p1}-I_{nb1})=U_{pm1}I_{nm1}x^2-U_{pm1}I_{nb1}x, \]
or
\[ P_1=P_{nm1}x^2-P_{nb1}x. \tag{16} \]
- The power at the relay terminals from the third harmonic:
\[ P_3=U_{p3}(I_{n3}-I_{nb3})=U_{pm3}I_{nm3}(1-x)^2-U_{pm3}I_{nb3}(1-x), \]
or
\[ P_3=P_{nm3}(1-x)^2-P_{nb3}(1-x). \tag{17} \]
- The total power at the relay terminals
\[ P_p=P_1+K_fP_3=P_{nm1}x^2-P_{nb1}x+K_1P_{nm1}(1-x)^2-K_2P_{nb1}(1-x); \]
or
\[ P_p=P_{nm1}[x^2+K_1(1-x)^2]-P_{nb1}[x+K_2(1-x)]. \tag{18} \]
In order that the protection operate reliably for any value of \(x\) within the limits from 0 to 1, it is necessary that the minimum power at the relay terminals be greater than the pickup power, i.e.,
\[ P_{pmin}=K_nP_{pt}, \tag{19} \]
where \(K_n>1\) is the reliability coefficient.
To find the minimum power at the relay terminals, we differentiate equation (18) with respect to \(x\)
\[ \frac{dP_p}{dx}=P_{\mathrm{nm}1}\{2x-2K_1(1-x)\}-P_{\mathrm{nb}1}(1-K_2)=0; \]
the value of \(x\) corresponding to the minimum power at the relay terminals is
\[ x_{\min}=\frac{2K_1P_{\mathrm{nm}1}+P_{\mathrm{nb}1}(1-K_2)} {2P_{\mathrm{nm}1}(1+K_1)}; \tag{20} \]
substituting the value \(x_m\) from (20) into equation (18), we obtain the following expression for the minimum power:
\[ P_{p\min}= \frac{K_1P_{\mathrm{nm}1}^{2}-P_{\mathrm{nm}1}P_{\mathrm{nb}1}(K_1+K_2)-0.25P_{\mathrm{nb}1}^{2}(1-K_2)} {P_{\mathrm{nm}1}(1+K_1)} . \tag{21} \]
Neglecting the last term of the numerator in comparison with the others in expression (21), and taking into account the sensitivity condition (19), we obtain the following relation:
\[ K_nP_{pt}= \frac{K_1P_{\mathrm{nm}1}-P_{\mathrm{nb}1}(K_1+K_2)} {1+K_1}. \tag{22} \]
From (22) one can determine the maximum current in the secondary circuit of the neutral current transformer:
\[ I_{\mathrm{nm}1}= \frac{K_n(1+K_1)P_{pt}+P_{\mathrm{nb}1}(K_1+K_2)} {K_1U_{pm1}} . \tag{23} \]
Substituting the value \(I_{\mathrm{nm}1}\) into relation (15), we obtain, for the maximum current in the secondary circuit of the external current transformers, the following expression:
\[ I_{\mathrm{bm}1}= \frac{(1+K_1)(K_n^{2}-1)P_{pt}+2K_nP_{\mathrm{nb}1}(K_1+K_2)} {K_1K_nU_{pm1}} . \tag{24} \]
The maximum primary ground-fault current due to the grounding resistance of the generator is
\[ I_{\mathrm{zm}}=n_{tv}I_{\mathrm{bm}1}. \tag{25} \]
The ratio of the transformation coefficient of the neutral current transformer to the transformation coefficient of the external current transformers is
\[ \frac{n_{to}}{n_{tv}}= \frac{(1+K_1)(K_n^{2}-1)P_{pt}+2K_nP_{\mathrm{nb}1}(K_1+K_2)} {K_n\left[K_n(1+K_1)P_{pt}+P_{\mathrm{nb}1}(K_1+K_2)\right]} . \tag{26} \]
In Fig. 4 the dependence \(I_{\mathrm{zm}}=f(n_{tv})\) is plotted for the following data:
1) relay pickup power \(P_{pt}=0.1\ \mathrm{W}\);
2) nominal relay voltage \(U_{pn}=100\ \mathrm{V}\);
3) unbalance current of the external current transformers \(I_{\mathrm{nb}1}=0.02\ \mathrm{A}\);
4) reliability coefficient \(K_n=1.25\);
5) ratio of unbalance currents \(K_{\mathrm{nb}}=1\);
6) frequency coefficient of the relay for the third harmonic (for relay IM-43) \(K_f=0.85\);
7) multiplicity of the third-harmonic voltage at the relay terminals relative to the nominal relay voltage \(K'=2\);
8) the magnitude of the third-harmonic voltage component in the generator phase voltage, in fractions of unity: \(K''=0.03\) for curve (1) and \(K''=0.05\) for curve (2).
(When measuring the percentage value of the zero-sequence current at the Kolomenskii Plant, the voltage of the third harmonic was obtained equal to 30% of the phase voltage of the generator.)
In accordance with the coefficients adopted in items 5–8, the coefficients are \(K_1 = 0.051\) and \(K_2 = 1.7\) for curve (1), and \(K_1 = 0.085\) and \(K_2 = 1.7\) for curve (2).
From Fig. 4 it is seen that the ground-fault currents obtained when this protection is used are of approximately the same order as for the protections currently in use, but the protection under consideration has no dead zone. The practical significance of this protection can be established only after testing it under operating conditions.
Fig. 4
The magnitude of the voltage between points \(a\) and \(b\) (Fig. 1) for the first harmonic can be determined from equation (3):
\[ U_1 = U_{\mathrm{ph}}\sqrt{1 + (n^2 - 2n)\left(\frac{X_p}{Z_{p1}}\right)^2}. \]
For an IM-type grounding relay the ratio
\[ \frac{X_p}{Z_{p1}} = 0.98; \]
therefore, for \(n = 12\), the voltage is
\[ U_1 = U_{\mathrm{ph}}\sqrt{1 + (144 - 24)\cdot 0.98^2} = 10.8\,U_{\mathrm{ph}}. \]
As a result, it would be necessary to use a nonstandard voltage transformer to feed the relay voltage winding, or to use the circuit of Fig. 1.