On the Issue of Balancing the Arms of Differential Protection of Power Transformers
Abstract Generated abstract
This paper examines how the ratio of secondary circuit arm parameters affects unbalance current in transformer differential protection during external short circuits. It derives steady state balance conditions for several protection arrangements, including circuits with autotransformers, saturating transformers, different transformer winding connections, and VTN-561 with ET-561 equipment, and proposes a calculation method based on current transformer no-load characteristics to select optimal connecting lead resistances. Experimental tests with varied current transformer cores, auxiliary devices, and added arm resistances supported the method, showing that suitable balancing of the protection arms can substantially reduce steady state unbalance current and improve selectivity.
Full Text
ON THE ISSUE OF BALANCING THE ARMS OF DIFFERENTIAL PROTECTION OF POWER TRANSFORMERS
V. S. Kolesnikov, I. D. Kutyavin
In recent years, extensive research work has been carried out in the Soviet Union to improve the differential protection of power transformers. The introduction into practice of saturating auxiliary transformers and autotransformers (BNT, VTN) has significantly increased the selectivity of the protection. Even more effective appears to be the use of transformer relays with braking and with saturating elements (RNT, MZT, DZT), which were recently proposed by A. D. Drozdov.
It is noteworthy that almost all studies concerning the differential protection of transformers do not at all touch upon the question of balancing the arms of the protection. At the same time, it has long been evident that the ratio of the parameters of the arms of transformer differential protection has a substantial influence on the magnitude of the unbalance current and, consequently, on the selectivity of its operation.
At present, at the Department of Electric Power Stations, Networks, and Systems of the Tomsk Polytechnic Institute, theoretical and experimental investigations are being conducted into the effectiveness of balancing the arms of transformer differential protection. The work is divided into a number of stages. The present article gives the results of investigations of the steady-state regime.
Since at present not only new types of differential protection of transformers are being used in operation, but also protections without the use of saturating elements, it is of interest to analyze the circuits of both old and new types of protection.
Derivation of the equilibrium condition for the arms of transformer differential protection in the steady-state regime
Let us consider the schematic diagram of the differential protection of a power transformer with a \(Y/Y-12\) winding connection (Fig. 1).
\(I_1\) and \(I_2\)—primary and secondary currents of the power transformer;
\(i_1\) and \(i_2\)—secondary currents of the current transformers;
\(i_{01}\) and \(i_{02}\)—magnetizing currents of the current transformers;
\(i_a\) and \(i_{0a}\)—secondary current and magnetizing current of the autotransformer;
\(N; n_1; n_2; n_a\)—transformation ratios of the power transformer,
current transformers and the autotransformer, with:
\[ n_a = N \frac{n_2}{n_1}. \]
The magnetizing current of the power transformer may be neglected; in this connection the equality holds
\[ I_1 = N I_2. \]
The secondary circuit of the differential protection shown in Fig. 1 may be represented in the form of the equivalent circuit shown in Fig. 2.
Since the autotransformer operates under short-circuit conditions, it may be regarded as an ordinary transformer with the resistances of the primary and secondary windings reduced by
\[ \left(1 - \frac{1}{n_a}\right) \]
times [1].
Fig. 1.
Fig. 2.
\(z_{T1}\) and \(z_{T2}\)—resistances of the secondary windings of the current transformers;
\(z_{pr1}\) and \(z_{pr2}\)—resistance of the connecting leads from the current transformer to the autotransformer or to the relay tap;
\(z_{a1}\) and \(z_{a2}\)—resistances of the windings of the transformer equivalent to the autotransformer;
\(z_{pra}\)—resistance of the connecting leads from the relay tap to the autotransformer;
\(z_{01}\) and \(z_{02}\)—resistances of the magnetizing branches of the current transformers;
\(z_{0a}\)—resistance of the magnetizing branch of the autotransformer.
Let us denote:
\[ z_{T1} + z_{pr1} + z_{a1} = z_1, \tag{1} \]
\[ z_{T2} + z_{pr2} = z_2, \tag{2} \]
\[ z_{a2} + z_{pra} = z_3. \tag{3} \]
In view of the fact that in the steady-state regime \(z_{oa} \gg z_3\), we take \(z_{oa} = \infty\). Taking this assumption into account, we write the equation according to Kirchhoff’s second law for the secondary-circuit loop of the first current transformer:
\[ E_{T1}=i_{01}\cdot z_{01}=i_1\cdot z_1+i_a\cdot z_3\frac{1}{n_a}, \tag{4} \]
but
\[ i_1=(i_a+i_{oa})\cdot n_a. \tag{5} \]
Substituting (5) into (4), we obtain:
\[ i_{01}z_{01}n_a=(i_a+i_{oa})z_1n_a^2+i_az_3. \tag{6} \]
Solving the last equation with respect to \(i_a\),
\[ i_a=\frac{n_a\left(i_{01}z_{01}-i_{oa}n_az_1\right)}{n_a^2z_1+z_3}. \tag{7} \]
Considering in an analogous manner the arm circuit of the second current transformer, we find
\[ i_2=\frac{i_{02}z_{02}}{z_2}. \tag{8} \]
The current in the differential-protection relay (the unbalance current) is
\[ i_p=i_a-i_2. \tag{9} \]
Setting the current in the relay equal to zero and using expressions (7), (8), and (9), we obtain an equation expressing the condition for equilibrium of the arms of the differential protection:
\[ \frac{n_a\left(i_{01}z_{01}-i_{oa}n_az_1\right)}{n_a^2z_1+z_3} = \frac{i_{02}z_{02}}{z_2}. \tag{10} \]
When high-class autotransformers are used, for example VU-25, the magnetizing current of the autotransformer may be neglected. In this case: \(i_{oa}n_zz_1=0\), and
\[ \frac{i_{01}}{n_a}=i_{02}. \tag{11} \]
As a result, equality (10) is simplified:
\[ \frac{n_a^2z_{01}}{n_a^2z_1+z_3}=\frac{z_{02}}{z_2}. \tag{12} \]
In the absence of an autotransformer, \(n_a=1;\ z_3=0\), and accordingly we obtain:
\[ \frac{z_{01}}{z_1}=\frac{z_{02}}{z_2}. \tag{13} \]
When the windings of the power transformer are connected according to the $\Delta/Y\text{—}11$ scheme, and the windings of the current transformers are connected on the delta side in star and on the star side in delta, the resistance of the protection arm of one phase on the side of the current transformers connected in $\Delta$ will be equal to:
\[ z_2 = z_{T2} + 3z_{\mathrm{pr}} . \tag{14} \]
In this case the transformation ratio of the autotransformer is determined by the following expression:
\[ n_a = \frac{N n_2}{\sqrt{3}\, n_1}, \]
and the relation between the magnetizing currents of the current transformers is:
\[ \frac{i_{01}}{n_a} = \frac{i_{02}}{\sqrt{3}} . \tag{15} \]
Method for Calculating the Optimal Ratio of the Arm Resistances of Transformer Differential Protection
Balancing the arms of the differential protection of power transformers can be accomplished by the corresponding selection of the cross sections of the connecting wires of the differential-phase circulation circuit. For this purpose it is necessary to have typical no-load characteristics for the individual current transformers $E(I_0)$.
Having specified the minimum permissible cross section of the connecting wires in the circuit of the more powerful current transformer and having determined the total resistance of this arm, one can calculate the value of the e.m.f. of the secondary winding of the current transformer. With reference to the scheme of Fig. 1, the e.m.f. of the secondary winding of the first current transformer is determined by the formula
\[ E_1 = \frac{I_{\mathrm{nom}} k}{n_1} \left( z_1 + \frac{z_3}{n_a^2} \right), \tag{16} \]
where $I_{\mathrm{nom}}$ is the rated current of the primary winding of the power transformer,
$k$ is the multiplicity of the external short-circuit current.
The quantities $z_1$ and $z_3$ are calculated by formulas (1) and (3).
Having determined from the curve $E(I_0)$ the value of the magnetizing current $(i_{01})$ of the first current transformer, we calculate, using equality (11), the value of the magnetizing current $(i_{02})$ of the current transformer installed in the opposite arm.
From the curve $E(I_0)$ we find the value of the e.m.f. of the second current transformer, $E_2$, corresponding to the value $i_{02}$, and we calculate, using equation (12), the optimal value of the arm resistance of the second current transformer:
\[ z_2 = \frac{E_2}{E_1}\,\frac{n_a^2 z_1 + z_3}{n_a}. \tag{17} \]
When the windings of the power transformer are connected according to the $\Delta/Y\text{—}11$ scheme and the current transformers are connected on the delta side in star (index 1), and on the star side in delta (index 2), the calculation procedure does not change. However, the relation between the magnetizing currents is determined by expression (15). The arm resistance of the second current transformer, as in the preceding case, is found from expression (17).
Equilibrium Condition for the Arms of Differential Protection Implemented Using VTN-561 with ET-561
Obviously, in the case when the equalizing winding of the VTN is not used (when \(n_1 = N n_2\)), the condition for the minimum unbalance current will be determined by expression (13). The procedure and sequence of calculation will be similar to that set forth above.
Consider the circuit of the differential protection of a power transformer for the case of a \(Y/Y\text{—}12\) winding connection and using the VTN windings both as differential and equalizing windings (Fig. 3).
Fig. 3.
The designations are analogous to those adopted for Fig. 1; in addition:
\(w_d\)—number of turns of the differential winding of the VTN,
\(w_e\)—number of turns of the equalizing winding of the VTN, \(i_1 > i_2\).
As is known, the number of turns of the equalizing winding is determined from the equality:
\[ w_d(i_1-i_2)=i_2w_e. \tag{18} \]
Moreover, the number of turns of the equalizing winding obtained by calculation is rounded to an integer.
In an external short circuit, the current in the ET-561 relay will be equal to zero in the case when the magnetic flux in the VTN core is equal to zero. Taking the foregoing into account, for the case of arm equilibrium (with \(I_{nb}=0\)), the equivalent circuit of the secondary circuit of the differential protection shown in Fig. 4 will be valid.
Fig. 4.
The designations correspond to those adopted for Fig. 2, taking into account formulas (1) and (2); in addition:
\(z_{wd}\)—resistance of the differential winding of the VTN;
\(z_{we}\)—resistance of the equalizing winding of the VTN.
We neglect the magnetizing current of the power transformer.
Applying Kirchhoff’s first law to nodes 1 and 2, we find:
\[ i_1=I'_1-i_{01}=\frac{I_1}{n_1}-i_{01}, \tag{19} \]
\[ i_2=I'_2-i_{02}=\frac{I_1}{N n_2}-i_{02}. \tag{20} \]
Substituting expressions (19) and (20) into equality (18), after a number of simple transformations we find the relation between the magnetizing currents of the current transformers that determines the minimum unbalance current in the ET-561 relay:
\[ i_{01}=i_{02}\frac{w_d+w_v}{w_d} +I_{\mathrm{nom}}\,k\left[ \frac{1}{n_1} -\frac{1}{N n_2}\cdot\frac{w_d+w_v}{w_v} \right]; \tag{21} \]
\[ i_{02}=i_{01}\frac{w_d}{w_d+w_v} +I_{\mathrm{nom}}\,k\left[ \frac{1}{N n_2} -\frac{1}{n_1}\frac{w_d}{w_d+w_v} \right]. \tag{22} \]
The presence of a voltage drop across the differential winding of the CTN, which arises in connection with the flow through it of the current \((i_1-i_2)\), must be taken into account when determining the load on the current transformers.
In particular, for \(i_1>i_2\), the presence of a voltage drop across the differential winding is equivalent to an increase in the load on the first current transformer by the amount
\[ \Delta z_1=z_{w d}\left(1-\frac{n_1}{N n_2}\right) \tag{23} \]
and to a decrease in the load on the second current transformer by the amount
\[ \Delta z_2=z_{w d}\left(\frac{N n_2}{n_1}-1\right). \tag{24} \]
The method and sequence for calculating the optimal ratio of the arm resistances will be completely analogous to that set forth earlier; however, the calculation formulas are somewhat modified. Thus, the e.m.f. of the secondary winding of the first current transformer is determined by the formula:
\[ E_1=\frac{I_{\mathrm{nom}}\,k}{n_1}z_1, \tag{25} \]
where
\[ z_1=z_{T1}+z_{pr1}+\Delta z_1. \tag{26} \]
The magnitude of the magnetizing current of the second current transformer should be determined from expression (22). The total arm resistance of the second current transformer will be equal to
\[ z_2=\frac{E_2}{E_1}\frac{N n_2}{n_1}z_1, \tag{27} \]
where:
\[ z_2=z_{T2}+z_{pr2}+z_{w v}-\Delta z_2. \tag{28} \]
When the windings of the power transformer are connected according to the \(\Delta, Y\)—11 scheme, and provided that the first current transformer is installed on the delta side, the magnitude of the magnetizing current of the second transfor-
the magnetizer and the resistance of its arm can be determined from the following expressions:
\[ i_{02}=i_{01}\frac{w_{\mathrm{д}}}{w_{\mathrm{д}}+w_{\mathrm{в}}} +I_{1\mathrm{nom}}\,k\left[ \frac{\sqrt{3}}{N n_{2}} -\frac{1}{n_{1}}\frac{w_{\mathrm{д}}}{w_{\mathrm{д}}+w_{\mathrm{в}}} \right], \tag{30} \]
\[ z_{2}=\frac{E_{2}N n_{2}}{E_{1}\sqrt{3}\,n_{1}}\cdot z_{1}. \tag{31} \]
Experimental verification of the calculation method set forth above gave sufficiently good results. For the investigation, transformers were made from different grades of steel (the curves of the dependence \(E(I_{0})\) for one of them are presented in Fig. 5), with different numbers of turns of the secondary windings, and with different cross-sectional areas and lengths of the mean lines of the cores. To equalize the secondary currents of the current transformers, a VU-25 autotransformer and a VTN-561 saturating transformer were used.
Fig. 5.
Additional resistances of various magnitudes were connected in series into the arms of the circulating circuit. The value of the current passing through the primary windings of the current transformers was selected so as to create in the cores of these transformers inductions corresponding to the passage of a 6–8-fold through short-circuit current in the actual circuit of the differential protection of a power transformer. Figure 6 presents the curve of the dependence of the magnitude of the steady-state unbalance current on the ratio of the arm resistances for one of the pairs of current transformers. Figure 7 gives the curve of the dependence of the magnitude of the steady-state unbalance current in the winding of the ET-561 relay on the ratio of the arm resistances. The differential winding of the VTN-561 had 20 turns in this case.
Figure 6.
Visible graph labels: \(I_{\text{[[unclear]]}}\); \(\dfrac{R_1}{R_2}\).
Figure 7.
Visible graph labels: \(I_{\text{[[unclear]]}}\); \(\dfrac{R_1}{R_2}\).
The theoretical and experimental studies carried out make it possible to draw the following conclusions:
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Balancing the arms of differential protection by selecting the optimal ratio of the arm resistances leads to a substantial reduction in the magnitude of the unbalance current during external short circuits.
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If the dependence curves \(E(I_0)\) are available for the current transformers used in differential protections, the calculation of the optimal ratios of the arm resistances can be carried out quite simply by the proposed method.
REFERENCES
- Piotrovsky L. M. Electrical Machines. GEI, 1949.