CALCULATION OF THE EFFECT OF POWER SWINGS IN SYSTEMS ON THE OPERATION OF RELAY PROTECTION

I. D. Kutyavin
Associate Professor, Candidate of Technical Sciences

During power swings in a system, individual electrical quantities may assume values similar to those obtained during short circuits, as a result of which incorrect operation of protection is possible. To clarify the behavior of protection during power swings, it is necessary to determine the values of the electrical quantities to which the protection responds, as functions of the angle of divergence of the emf vectors of the stations of the system. The “Guidelines for Relay Protection of Electrical Networks” recommend for this purpose a graphical calculation method proposed by Eng. A. B. Barzam.

This method assumes a whole series of simplifications, namely: 1) equality of the emfs of the generators of the system; 2) absence of damage in the system during swings; 3) absence of load in the system.

These assumptions can cause significant distortions of the final result and lead to an incorrect conclusion.

The fundamentals of the proposed method for calculating the effect of swings on the operation of protection were given by N. Shchedrin in his book “Short-Circuit Currents of Electrical Systems.”

Fig. 1

In Fig. 1a the simplest system is shown, with two generating stations and with load concentrated on the high-voltage buses; in Fig. 1b the equivalent circuit of the indicated system is given, on which the voltages, currents, and resistances corresponding to ...

normal operating condition of the system. To set up the basic equations during swings, one may use the normal condition (Fig. 1b)

\[ \dot I_{n1}\dot Z_{n1}=\dot I_c \dot Z_c+\dot I_{n2}\dot Z_2; \tag{1} \]

\[ \dot E_1=\dot I_1\dot Z_1+\dot I_{n1}\dot Z_{n1}; \tag{2} \]

\[ \dot E_1-\dot E_2=\dot I_1\dot Z_1+\dot I_c\dot Z_c-\dot I_2\dot Z_2; \tag{3} \]

\[ \dot I_1=\dot I_{n1}+\dot I_c; \tag{4} \]

\[ \dot I_{n2}=\dot I_c+\dot I_2. \tag{5} \]

The simultaneous solution of these five equations gives the following expressions for the currents:

\[ \dot I_1=\dot E_1 \left[ \frac{ \dot Z_{n1}\dot Z_{n2}+Z_{n1}\dot Z_2+\dot Z_{n2}\dot Z_2+\dot Z_{n2}\dot Z_c+Z_2Z_c }{\dot N} \right] -\dot E_2\frac{Z_{n1}\dot Z_{n2}}{\dot N} =\dot I'_1+\dot I''_1; \tag{6} \]

\[ \dot I_2=-\dot E_1\frac{\dot Z_{n1}\dot Z_{n2}}{\dot N} +\dot E_2 \left[ \frac{ \dot Z_{n1}\dot Z_{n2}+\dot Z_{n1}\dot Z_1+\dot Z_{n1}\dot Z_c+\dot Z_1\dot Z_{n2}+\dot Z_1\dot Z_c }{\dot N} \right] =\dot I'_2+\dot I''_2; \tag{7} \]

\[ \dot I_c=\dot E_1\frac{\dot Z_{n1}(\dot Z_{n2}+\dot Z_2)}{\dot N} -\dot E_2\frac{\dot Z_{n2}(\dot Z_{n1}+\dot Z_1)}{\dot N} =\dot I'_c+\dot I''_c. \tag{8} \]

Expressions (6), (7), and (8) give the values of the currents \(\dot I_1\), \(\dot I_2\), and \(\dot I_c\) for the normal operating condition; moreover, the currents \(\dot I'\) and \(\dot I''\) for each expression may be obtained when calculating the currents \(\dot I_1\), \(\dot I_2\), and \(\dot I_c\) by the method of superposition.

During swings in the system, the vectors of the e.m.f. \(\dot E_1\) and \(\dot E_2\) will separate by an angle \(\delta\), as a result of which the currents \(\dot I'\) and \(\dot I''\) in expressions (6), (7), and (8) will also separate by this angle; then the expressions for the currents during swings will have the form:

\[ \dot i_1=\dot i'_1 e^{j\delta}+\dot i''_1; \]

\[ \dot i_2=\dot i'_2 e^{j\delta}+\dot i''_2; \]

\[ \dot i_c=\dot i'_c e^{j\delta}+\dot i''_c. \tag{9} \]

Expressions (9) for the currents \(\dot i_1\), \(\dot i_2\), and \(\dot i_c\) are equations of a circle with center coordinates \(\dot i''\) and radius \(\dot i'\).

To construct circular diagrams of the currents for individual points of the system in which protection is installed, it is necessary to know the distribution of currents and voltages in the system under the normal condition. This problem can be solved by already known methods. On the basis of the data for the normal condition, a vector diagram of voltages is constructed for the given point (Fig. 2), with the e.m.f. \(\dot E_1\), taken as rotating, placed along the real axis. In Fig. 2, circular diagrams are constructed for the current \(\dot I_1\) and the voltage \(\dot U_1\) for the system under consideration in Fig. 1, point (1).

In addition, in order to construct the circular diagram of the current, it is necessary to determine the vector \(\dot I''\) for \(\dot E_1=0\). Laying off this vector \(\dot I''\) on the current scale on the vector diagram, we obtain the center of the circle for the circular diagram, \(O_1\), whose radius is equal to the segment \(O_1a\), connecting the center of the circle \(O_1\) with the end of the current vector corresponding to the normal operating condition.

It is necessary to indicate that the vector of the center \(\dot I_1''\), according to expressions (6), (7), and (8), is equal to the current at \(\dot E_1=0\) with a positive sign if this current coincides in direction with the current of the normal regime, and with a negative sign if the current at \(\dot E_1=0\) is opposite to the current of the normal regime.

What has been said is illustrated by the arrangement of the arrows in Fig. 1b, where

Fig. 2

the arrows indicating the direction of the current of the normal regime are drawn on the circuit contour, while the arrows of the current at \(\dot E_1=0\) are parallel to the contour.

The initial position of the current vector during swings, from which the angle \(\delta\) is reckoned, coincides with the position of the current on the vector diagram in the normal regime (Fig. 2).

On the same diagram the circle diagram of the voltage is constructed for the given point of the system; the expression for the voltage \(U_1\) can be obtained as follows:

\[ \dot U_1=\dot E_1 e^{j\delta}-\dot I_1 \dot Z_1 . \tag{10} \]

Substituting into this equation the expression for the current \(\dot I_1\) from (9), we obtain the expression for the voltage

\[ \dot U_1=(\dot E_1-\dot I_1' \dot Z_1)e^{j\delta}-\dot I_1'' \dot Z_1 =\dot U_1' e^{j\delta}+\dot U_1'' . \tag{11} \]

This expression is similar to the expressions for the currents (9) and, for \(\delta=0\), gives the voltage in the normal regime at the given point of the system, i.e.,

\[ \dot U_1=\dot U_1'+\dot U_1''=\dot U_{1\text{norm}}=\overline{OB}. \]

For constructing the circle diagram of the voltage it is necessary only to determine the vector of the center

\[ \dot U_1'=-\dot I_1'' \dot Z_1=\overline{OO_2} \tag{12} \]

and lay it off on the vector diagram in Fig. 2, and then, having obtained the center of the circle \(O_2\), draw a circle with radius \(\overline{O_2B}=U_1'\). In this case the initial position of the voltage vector during swings coincides with its position under normal conditions—the vector \(\overline{OB}\) (Fig. 2).

Using the diagram constructed, one can determine the behavior of all electrical quantities as a function of the angle \(\delta\). Unfortunately, the powers and impedances measured by the relay during swings cannot be represented by a circular diagram; therefore their graphical determination

Fig. 3

is difficult, as a result of which it is simpler to calculate the power from the equation

Fig. 4a

from the equation

\[ \dot{W}=\dot{U}\dot{I}, \]

the impedance

\[ Z=\frac{U}{I} \]

and the reactance

\[ X=\frac{Q}{I^2}, \]

where \(U\) is the voltage, \(I\) the current, and \(Q\) the reactive power for the given angle \(\delta\). In Fig. 3 the curves of variation of \(U_1\) and \(I_1\) as functions of \(\delta\), constructed from the circle diagrams of Fig. 2, are given; and in Figs. 4a and 4b the curves of variation of \(P_1\), \(Q_1\), \(Z_1\), and \(X\), measured by relay 1 during swings in the system, are given.

Fig. 4b

Further, in Fig. 5 circle diagrams are plotted for \(U_1\) and \(I_c\), and in Figs. 6a and 6b the curves of variation of \(U_1\), \(I_c\), \(P_c\), \(Q_c\), \(Z_c\), and \(X_c\), measured by relay 2 during swings, are given.

For the numerical example the following was assumed:

1. The capacity of each station (Fig. 1) is \(3 \times 50\) MW.
2. The load on the buses of the first station is 100 MW, and of the second station 200 MW.

Fig. 5

1. The capacity of the step-up transformers at each station is \(3 \times 60\) MVA.
2. Under normal operating conditions, 50 MW is transmitted over the transmission line.

1. The line impedance in relative units is \(0.03 + j0.07\).

From a comparison of the results obtained for points 1 and 2 it is evident that neglecting the load may cause significant deviations of the calculated quantities from the actual ones. In particular, in the present example the current

Fig. 6a

will be the same at points 1, 2, 3, and 4 if the load is neglected; in reality this current may differ not only in magnitude but also in direction, depending on the magnitude of the transmission-line impedance. The difference becomes especially striking when the results obtained are compared with the calculation results given in the “Guiding Instructions

Fig. 6b

on relay protection of networks” (Section O-2). These results were obtained without taking the load into account.

The above considerations concerning the calculation of the behavior of electrical quantities during swings in a system with two generating stations can also be extended to the case with the number of generating stations equal to \(n\). In this case the expressions for the currents will have the form:

\[ J_k = \dot{I}'_k e^{j\delta} + \dot{I}''_{k1} + \dot{I}''_{k2} + \dot{I}''_{k3} + \cdots + \dot{I}''_{k(n-1)}, \]

where \(\dot I_{k1}^{\prime\prime}, \dot I_{k2}^{\prime\prime}, \ldots, \dot I_{k(n-1)}^{\prime\prime}\) are the currents caused at the point of the system under consideration by the individual emfs; moreover, the center vector for the circular diagram of the current will be equal to the sum of these currents, i.e.

\[ \dot I_k'=\dot I_{k1}^{\prime\prime}+\dot I_{k2}^{\prime\prime}+\ldots+\dot I_{k(n-1)}^{\prime\prime}; \]

and the center vector for the circular voltage diagram can in this case also be determined from equation (10).

Above, the question of the behavior of electrical quantities during swings not accompanied by a fault in the system was considered. However, all the considerations set forth above are also valid in the case when swings in the system are accompanied by a short circuit. In this case, the impedance up to the point of the short circuit may be included in the load impedance if the short circuit occurred on a branch, and included separately if the short circuit occurred on the tie line. To construct circular diagrams in this case it is necessary to determine the distribution of currents and voltages in the system: 1) with the short circuit accompanying the swings, and 2) with \(\dot E_1=0\), taking the indicated short circuit into account.

Fig. 7

The data obtained should be used in constructing the diagrams in the same way as for a normal operating condition accompanied by swings.

In conclusion, it should be noted that this method of calculation, not differing in complexity in comparison with other methods of calculation, undoubtedly gives a more correct result. In addition, this method makes it possible easily to clarify the physical essence of the phenomena occurring during swings in systems.

Noted Misprints