Relay Protection of Synchronous Compensators Against Voltage Drop

YU. E. NEBOLYUBOV

Associate Professor, Candidate of Technical Sciences

General Considerations

At the present time there is no substantiated choice of the settings for the pickup voltage of protection against voltage drop. In almost all cases the pickup voltage of the relay of this protection scheme is taken as equal to 60–80% of the nominal network voltage. An attempt to develop a method for a substantiated determination of the pickup voltage of minimum-voltage protection is of certain value.

The stable operation of synchronous compensators in a system is characterized by the value of the limiting voltage \((U_{\text{lim}})\).

If, during an accident, the voltage is less than this value, then the compensator will fall out of synchronism. It is this value \(U_{\text{lim}}\) that should be taken as the basis for choosing the pickup voltage of the protection under consideration.

The expressions for the limiting voltage for various load cases are given in the article by Lebedev and Zhdanov¹.

For an asynchronous load, characterized by constancy of active power with respect to voltage, the limiting voltage is found on the basis of the equivalent circuit, Fig. 1.

Fig. 1

Let us consider this derivation in somewhat greater detail. If the branches of the synchronous compensator and the external circuit are combined and an equivalent electromotive force \(E_{\text{e}}\) is introduced, then the following expressions may be written:

equivalent admittance

\[ y_{\text{e}} = y_{\text{vn}} + y_{\text{sk}}; \tag{1} \]

equivalent electromotive force

\[ E_{\text{e}}=\frac{E_{\text{sk}}y_{\text{sk}}+Uy_{\text{vn}}}{y_{\text{e}}}; \tag{2} \]

active power

\[ P=E_{\text{e}}U_{\text{n}}y_{\text{e}}\sin\delta; \tag{3} \]

reactive power

\[ Q=U_{\text{n}}^{2}b_{\text{n}}=E_{\text{e}}U_{\text{n}}y_{\text{e}}\cos\delta-U_{\text{n}}^{2}y_{\text{e}}. \tag{4} \]

In equations (3) and (4), \(\delta\) denotes the angle between the equivalent electromotive force \(E_{\mathrm{e}}\) and the electromotive force of the synchronous compensator \(E_{\mathrm{sc}}\).

The active power of the compensator is approximately equal to zero, and \(E_{\mathrm{sc}}\) coincides in phase with the nominal load voltage \((U_{\mathrm{n}})\).

Having obtained the value \(U_{\mathrm{n}}\) from equation (4) and substituting it into equation (3), we determine

\[ E_{\mathrm{e}}=\frac{A}{\sqrt{\sin\delta\cos\delta}}, \tag{5} \]

where

\[ A=\frac{\sqrt{P_{\mathrm{n}}\left(y_{\mathrm{e}}+b_{\mathrm{n}}\right)}}{y_{\mathrm{e}}}. \]

Resolving \(E_{\mathrm{e}}\) into orthogonal coordinate axes, we obtain

\[ E_{\mathrm{e}x}=E_{\mathrm{e}}\cos\delta=\frac{A}{\sqrt{\operatorname{tg}\delta}}; \]

\[ E_{\mathrm{e}y}=E_{\mathrm{e}}\sin\delta=A\sqrt{\operatorname{tg}\delta}. \]

Hence

\[ \frac{E_{\mathrm{e}x}}{A}\cdot\frac{E_{\mathrm{e}y}}{A}=1. \tag{6} \]

Equation (6) is a hyperbola referred to its asymptotes. At the same time, from equation (2) we have

\[ \frac{\dot E_{\mathrm{e}}}{A} = E_{\mathrm{sc}}\frac{y_{\mathrm{c}}}{A y_{\mathrm{e}}} + \dot U\frac{y_{\mathrm{vn}}}{A y_{\mathrm{e}}} = C E_{\mathrm{sc}}+B\dot U. \tag{7} \]

By equation (7), one can construct a circle with center coordinate \(C E_{\mathrm{sc}}\) and radius vector \(B U\) (Fig. 2).

The points of possible operation of the compensator are determined by the intersection of the circle and the hyperbola. The point corresponding to the smaller angle \(\delta\) characterizes stable operation; the point with the larger angle corresponds to unstable operation. In the event of a voltage decrease, both points approach each other and finally merge into a single point of tangency of the curves, which corresponds to the limiting conditions of stable operation. If the voltage drops still further, the curves will not intersect and normal operation becomes impossible. The conditions of tangency of the curves make it possible to derive expressions for determining the value \(B U_{\mathrm{pr}}\):

Fig. 2

\[ B U_{\mathrm{pr}}=y_1\sqrt{1+y_1^4}; \qquad C E_{\mathrm{sc}}=\frac{1-y_1^4}{y_1}, \]

where

\[ y_1=\frac{E_{\mathrm{e}y}}{A}. \]

Assigning various values to \(y_1\), one can construct the dependence of \(U_{\mathrm{lim}}B\) on \(E_{\mathrm{sk}}C\) (Fig. 3), which makes it possible to determine the limiting voltage in specific electrical systems. For this it is sufficient to draw up the equivalent circuit of the system and determine the quantities \(B\) and \(C\):

\[ B=\frac{y_{\mathrm{vn}}}{\sqrt{P_{\mathrm{n}}(b_{\mathrm{n}}+y_{\mathrm{e}})}};\quad C=\frac{y_{\mathrm{sk}}}{\sqrt{P_{\mathrm{n}}(b_{\mathrm{n}}+y_{\mathrm{e}})}} , \tag{8} \]

where \(P_{\mathrm{n}}\) is the active component of the load power; \(b_{\mathrm{n}}\) is the reactive component of the load admittance; \(y_{\mathrm{e}}\) is the equivalent admittance; \(y_{\mathrm{vn}}\) and \(y_{\mathrm{sk}}\) are the external admittance and the compensator admittance. The value of \(CE_{\mathrm{sk}}\) makes it possible, from the curve of Fig. 3, to determine \(BU_{\mathrm{lim}}\). The limiting voltage \(U_{\mathrm{lim}}\) is found by dividing \(BU_{\mathrm{lim}}\) by \(B\).

The calculation determining the limiting voltage is carried out for two cases:

1) with constancy of the e.m.f. behind the transient reactance (the transient reactance of the compensator is adopted);

2) with constancy of the no-load e.m.f. (the synchronous reactance is taken into account).

The compensator will fall out of synchronism in the very first cycle of oscillation if the emergency voltage is less than the value obtained for the first calculation case.

Fig. 3

When the emergency voltage lies between the two obtained values, the compensator will fall out of synchronism in subsequent cycles of oscillation. The oscillation time may be measured in several seconds (Fig. 4).

On the basis of the relations considered, some general considerations may be stated concerning the choice of the setting of the undervoltage protection, which must be verified by operating practice. The pickup voltage of this protection should be taken as equal to the limiting voltage obtained by calculation for the maximum system condition, taking into account the transient reactance of the compensator.

If the emergency voltage is greater than the limiting voltage, the compensator will be able to pull into synchronism without being disconnected from the network (direct synchronization).

In the event of failure of the minimum-voltage protection to operate, the compensator will fall out of synchronism and will be disconnected, after a certain time delay, by the loss-of-synchronism protection. In this way, mutual backup and selectivity of these two protections are ensured.

Fig. 4

Example of determining the pickup voltage of minimum-voltage protection for a specific system

Let us determine the pickup voltage for the protection of a compensator operating in a system whose equivalent circuit, in impedances and admittances, is shown in Fig. 5.

The distribution of powers in the system and of electromotive forces is determined by the usual electrical calculation, which is not given here.

Fig. 5

The active conductances of the machines are not taken into account in the calculation. All impedances are reduced to base conditions (total generator power \(P_b = 125\) MVA and voltage \(U_b = 110\) kV). The reduced values of the impedan-

the generator, transformer, and compensator networks are calculated by the formula

\[ Z_{\text{pu}}=\frac{Z\% \cdot P_{\text{base}}}{100\cdot P_{\text{nom}}}. \]

The impedances of the lines and load are determined by the formulas:

\[ Z_{\ell b}=\frac{P_b}{U_b^2}(r_\ell+jx_\ell) \quad \text{and} \quad Z_{nb}=\frac{U_n^2}{W_n^1}(\cos\varphi+j\sin\varphi). \]

System Constants

Generators and Synchronous Compensator

Transformers

Line

Load

a) Calculation taking into account the transient reactance of the compensator.
The equivalent circuit of the system in admittances for the asynchronous load is shown in Fig. 5. On the basis of this circuit, we determine the values of \(B\) and \(C\) by formulas (8)

Let us first calculate:

\[ y_{\mathrm{e}} = y_{\mathrm{vn}} + y_{\mathrm{sk}} = 2.04 + 0.238 = 2.278. \]

\[ b_{\mathrm{n}} = \frac{Q_{\mathrm{n}}}{U_{\mathrm{n}}^2} = \frac{0.120}{0.953^2} = \frac{0.126}{0.91} = 0.139. \]

\[ P'_{\mathrm{nb}} = \frac{P_{\mathrm{n}}}{P_{\mathrm{b}}} = \frac{114}{125} = 0.91. \]

\[ B = \frac{y_{\mathrm{vn}}}{\sqrt{P'_{\mathrm{nb}}\,(b_{\mathrm{n}}+y_{\mathrm{e}})}} = \frac{2.04}{\sqrt{0.91\,(0.139+2.278)}} = 1.38. \]

\[ C = \frac{y_{\mathrm{sk}}}{\sqrt{P'_{\mathrm{nb}}\,(b_{\mathrm{n}}+y_{\mathrm{e}})}} = \frac{0.238}{\sqrt{0.91\,(0.139+2.278)}} = 0.161. \]

\(E_{\mathrm{sk}} = 1.1\) (according to the electrical calculation).

Next we determine

\[ C E_{\mathrm{sk}} = 0.161 \cdot 1.1 = 0.177. \]

From the curve in Fig. 3 we have: \(B U'_{\mathrm{pr}} = 1.2\), whence

\[ U'_{\mathrm{pr}} = \frac{U_{\mathrm{pr}}\cdot B}{B} = \frac{1.2}{1.38} = 0.87. \]

b) Calculation taking into account the synchronous reactance of the compensator.

On the basis of transformations of the equivalent circuit of the system analogous to those in the first calculation case, we can determine the following quantities:

\[ y_{\mathrm{e}} = y_{\mathrm{vn}} + y_{\mathrm{sk}} = 2.04 + 0.063 = 2.103. \]

\[ b_{\mathrm{n}} = \frac{Q_{\mathrm{n}}}{U_{\mathrm{n}}^2} = \frac{0.126}{0.91} = 0.139. \]

\[ P'_{\mathrm{nb}} = \frac{P_{\mathrm{n}}}{P_{\mathrm{b}}} = \frac{114}{125} = 0.91. \]

\[ B = \frac{y_{\mathrm{vn}}}{\sqrt{P'_{\mathrm{nb}}\,(b_{\mathrm{n}}+y_{\mathrm{e}})}} = \frac{2.04}{\sqrt{0.91\,(0.139+2.103)}} = 1.43. \]

\[ C = \frac{y_{\mathrm{sk}}}{\sqrt{P'_{\mathrm{nb}}\,(b_{\mathrm{n}}+y_{\mathrm{e}})}} = \frac{0.063}{\sqrt{0.91\,(0.139+2.103)}} = 0.044. \]

The no-load electromotive force is \(E_{\mathrm{sk}} = 1.8\).

\[ C E_{\mathrm{sk}} = 0.044 \cdot 1.8 = 0.08 \]

From the value \(C E_{\mathrm{sk}}\) and from the curve in Fig. 3 we find \(B \cdot U_{\mathrm{pr}}\).

\[ U_{\mathrm{pr}} = \frac{U_{\mathrm{pr}} B}{B} = \frac{1.35}{1.43} = 0.94. \]

The operating voltage of the undervoltage protection is:

\[ U_{\mathrm{pr}} = U'_{\mathrm{pr}} = 0.86. \]

The numerical example considered is not entirely indicative, since it gives a significant value of \(U_{\mathrm{pr}}\).

A sharp reduction of \(U_{\mathrm{pr}}\) is always possible in systems with a large value of \(y_{\mathrm{sk}}\) and small values of external reactances \((x_{\mathrm{vn}})\).

1. Lebedev and Zhdanov. Stability of the parallel operation of synchronous compensators. VEI Bulletin No. 5, 1941. ↩