INFLUENCE OF STATION LOCAL LOAD ON SYSTEM STABILITY DURING THREE-PHASE AUTOMATIC RECLOSING OF SINGLE-CIRCUIT TRANSMISSION LINES

G. F. PUKHOVA

In our article [2] it was shown that when a station having no significant local load operates through a single-circuit transmission line onto the busbars of a powerful interconnection, there exists a certain range of permissible separation time from the standpoint of maintaining system stability. This time depends on the type of station (hydrogenerator, turbogenerator), on the reactance of the power transmission, on the duration and severity of the fault, etc. The calculations carried out showed that the limit of permissible separation time for stations with large inertia constants (16 sec), operating before automatic reclosing with rated load, even under extremely light faults (single-pole short circuit in networks with isolated neutral), is 0.55 sec. In all cases of practical importance, however, it is considerably smaller.

In reality, however, a station operating onto the busbars of a powerful interconnection almost always has some local load. It may be expected that this load, by changing the surplus power on the station shaft during the period of separated operation, should exert a substantial influence on the stability of the entire system and, consequently, change the permissible time of interruption of synchronous connection. Clarifying the character of this influence in a first approximation is the aim of the present work.

Fig. 1. Diagram of the considered section of the system

The diagram of the section of the system that interests us in this case will have the form shown in Fig. 1. Let \(P_{\text{load}}\) be the load power connected to the station busbars under the normal operating condition preceding the fault. In approximate calculations we consider it possible to replace the load

by an equivalent shunt with impedance \(Z_{\text{load}}\). After such a replacement, the equivalent circuit of the given section of the system can easily be reduced to the form shown in Fig. 2.

Using the superposition principle, we obtain that the active power delivered by the station is determined by the formula

\[ P_{\text{el}}=\frac{E_1^2}{Z_{11}}\sin\alpha_{11}+\frac{E_1E_2}{Z_{12}}\sin(\delta-\alpha_{12}). \tag{1} \]

The notation is given in the appendix.

It is obvious that equations of this form will correspond to the power delivered by the station during the periods of normal operation and of the fault. In the phase of isolated operation, owing to the complete interruption of the connection with the system, the entire load will fall only on the given station. Carrying out the whole calculation in a first approximation, we consider it possible to regard the power of the load connected to the station buses as constant and equal to \(P_{\text{load}}\) throughout the entire interval of isolated operation. Thus, the equations of motion of the rotor in the various phases of the automatic reclosing cycle will have the form:

Fig. 2. Equivalent circuit of a section of the system

Phase 1. Fault

\[ K\frac{d^2\delta}{dt^2} = P_{\text{mech I}} - \frac{E_1^2}{Z_{11}}\sin\alpha_{11\text{I}} - \frac{E_1E_2}{Z_{12\text{I}}}\sin(\delta-\alpha_{12\text{I}}). \tag{2} \]

Phase 2. Isolated operation

\[ K\frac{d^2\delta}{dt^2}=P_{\text{mech II}}-P_{\text{load}}. \tag{3} \]

Phase 3. Parallel operation after automatic reclosing

\[ K\frac{d^2\delta}{dt^2} = P_{\text{mech III}} - \frac{E_1^2}{Z_{11\text{III}}}\sin\alpha_{11\text{III}} - \frac{E_1E_2}{Z_{12\text{III}}}\sin(\delta-\alpha_{12\text{III}}). \tag{4} \]

Here the indices I, II, III indicate belonging to the corresponding phase of operation.

In order that the AR operation not cause a violation of stability, the following condition must be satisfied:

\[ \int_{\delta_0}^{\delta_1} \left[ P_{\mathrm{mech} I} - \frac{E_1^2}{Z_{11 I}}\sin \alpha_{11 I} - \frac{E_1E_2}{Z_{12 I}}\sin(\delta-\alpha_{12 I}) \right]\,d\delta + \int_{\delta_1}^{\delta_{II}} (P_{\mathrm{mech} II}-P_{\mathrm{load}})\,d\delta + \]

\[ + \int_{\delta_{II}}^{\delta_l} \left[ P_{\mathrm{mech} III} - \frac{E_1^2}{Z_{11 III}}\,\alpha_{11 III} - \frac{E_1E_2}{Z_{12 III}}\sin(\delta-\alpha_{12 III}) \right]\,d\delta \leq 0 . \tag{5} \]

Denoting

\[ P_{\mathrm{mech} I}-\frac{E_1^2}{Z_{11 I}}\sin \alpha_{11 I} \quad \text{by } P_I,\qquad P_{\mathrm{mech} III}-\frac{E_1^2}{Z_{11 III}}\sin \alpha_{11 III} \quad \text{by } P_{III}, \]

\[ \frac{E_1E_2}{Z_{12 I}} \quad \text{by } q_I; \qquad \frac{E_1E_2}{Z_{12 III}} \quad \text{by } q_{III} \]

and carrying out the integration of inequality (5), we obtain the stability condition for the transition under consideration in the form:

\[ P_I(\delta_1-\delta_0) + q_I\left[\cos(\delta_1-\alpha_{12 I})-\cos(\delta_0-\alpha_{12 I})\right] + (P_{\mathrm{mech} II}-P_{\mathrm{load}})(\delta_{II}-\delta_1) + \]

\[ + P_{III}(\delta_l-\delta_{II}) + q_{III}\left[\cos(\delta_l-\alpha_{12 III})-\cos(\delta_{II}-\alpha_{12 III})\right] \leq 0 . \tag{6} \]

Denoting

\[ \frac{P_{\mathrm{mech} II}-P_{\mathrm{load}}-P_{III}}{q_{III}} \quad \text{by } a, \]

\[ \frac{ P_I(\delta_1-\delta_0) + q_I\left[\cos(\delta_1-\alpha_{12 I})-\cos(\delta_0-\alpha_{12 I})\right] - (P_{\mathrm{mech} II}-P_{\mathrm{load}})\delta_1 + P_{III}\delta_l }{q_{III}} + \]

\[ + \frac{q_{III}\cos(\delta_l-\alpha_{12 III})}{q_{III}} \quad \text{by } A, \]

we obtain the following expression for determining the limiting permissible value of the rotor run-out angle in the phase of separated operation \((\delta_{II,\max})\):

\[ A+a\delta_{II,\max}=\cos(\delta_{II,\max}-\alpha_{12 III}) . \tag{7} \]

Since, in the phase of separated operation, the rotor equation of motion has the form:

\[ K\frac{d^2\delta}{dt^2}=P_{\mathrm{mech} II}-P_{\mathrm{load}}, \]

the limiting duration of this period is determined by the equation

\[ t_{II,\max} = - \frac{ \left(\dfrac{d\delta}{dt}\right)_{\delta_1}\cdot K }{ P_{\mathrm{mech} II}-P_{\mathrm{load}} } \pm \sqrt{ \frac{ \left(\dfrac{d\delta}{dt}\right)_{\delta_1}^{2}K^{2} }{ (P_{\mathrm{mech} II}-P_{\mathrm{load}})^2 } + \frac{ (\delta_{II,\max}-\delta_1)2K }{ P_{\mathrm{mech} II}-P_{\mathrm{load}} } }. \tag{8} \]

For the choice of the sign before the radical in equation (8), see the Appendix.

It should be noted that, in the case under consideration, the steady-state conditions before and after reclosing will, strictly speaking, no longer be identical. Indeed, the possibility is not excluded that a number of electric-power consumers which do not permit a short-term reduction in voltage will be disconnected from the network. In our subsequent calculations,

However, we consider it possible not to take this factor into account, since in an ordinary industrial load the share of such receivers is small.

The formulas given make it possible to carry out, comparatively easily, calculations of the permissible time of separated operation for cases when

$P_{\text{load}} \ll P_{\text{mech II}}$.

The question is somewhat more complicated when $P_{\text{load}} > P_{\text{mech II}}$, and for this reason we consider it expedient to determine the nature of the course of the automatic reclosing operation in this case. It is evident that all possible cases of automatic reclosing for $P_{\text{load}} > P_{\text{mech II}}$ may conventionally be divided into two categories:

1. When automatic reclosing with a duration of separated operation $(t_{\text{II}})$ equal to zero is accompanied by preservation of stability.
2. When automatic reclosing at $t_{\text{II}} = 0$ causes a loss of stability.

According to the rule for choosing signs given in the Appendix, cases falling into the first category will correspond to the sign $+$ before the root in equation (8). The time of separated operation calculated from this equation will be the maximum permissible one, and decreasing it must be accompanied by an improvement in stability conditions.

Cases assigned to the second category correspond in equation (8) to the sign $\pm$ before the root. This means that the phase of separated operation here is bounded by a two-sided limit: it cannot last less than the time $t'_{\text{II max}}$, calculated with the sign $-$, and, on the other hand, its duration must not exceed $t''_{\text{II max}}$, calculated with the sign $+$. The time of separated operation that does not lie in the interval $t'_{\text{II max}} \div t''_{\text{II max}}$ must, consequently, lead to a loss of synchronism. This circumstance makes the performance of the automatic reclosing operation extremely complicated and compels us to regard it here as practically difficult to carry out. In what follows this case is not considered by us.

Putting in inequality (6) the quantity $\delta_{\text{II}} - \delta_{\text{I}}$ equal to zero, we obtain that, in the cases assigned by us to the first category, the following condition must be satisfied:

\[ q_{\text{III}} \ge \frac{ P_1(\delta_1-\delta_0)+q_1[\cos(\delta_1-\alpha_{12\text{I}})-\cos(\delta_1-\alpha_{12\text{I}})]+P_{\text{III}}(\delta_1-\delta_{\text{I}}) }{ \cos(\delta_1-\alpha_{12\text{III}})-\cos(\delta_{\text{I}}-\alpha_{12\text{III}}) } \]

For a visual representation of the nature of the influence of the station local load on the time of separated operation permissible during automatic reclosing, we carried out a series of calculations of this time for a station with $K = 0.03$, connected by a single line 100 km long to the buses of a powerful interconnected system. The results of these calculations are presented in the form of curves $t_{\text{II max}} = f(P_{\text{load}})$ in Fig. 3.

Analysis of the curves presented makes it possible to draw the following conclusions:

1. Under extremely light faults, characterized by $X_k = \infty$, the load connected to the station buses, in the range considered $(0 < P_{\text{load}} < 2)$, increases the permissible time of separated operation, thereby facilitating the automatic reclosing operation. In the section $P_{\text{load}} = 0 \div 1$ the curve $t_{\text{II max}} = f(P_{\text{load}})$ has an ascending character, tending to infinity at $P_{\text{load}} = 1$. Thus, an increase of load here is accompanied by an increase in the permissible time of separated operation, and at $P_{\text{load}} = 1$ it may theoretically be taken as arbitrarily large. The physical meaning of the latter conclusion is obvious. Indeed, disconnection of the line cannot disturb the steady motion of the rotor of a station having on its buses a load equal to the power of its prime mover.

A further increase of load is accompanied by a smooth decrease in the permissible time of separated operation, and even at the boundary of the considere-

of the interval, i.e., at \(P_{\text{load}}=2\), its value \((t_{II\,\max}=0.32)\) fully permits carrying out the automatic-reclosing operation.

1. In severe disturbances characterized by \(X_k=0\), the influence of the load on the permissible time of separate operation is also positive, but the very character of the curve \(t_{II\,\max}=f(P_{\text{load}})\) differs significantly from that considered in item 1. Here, in the interval \(P_{\text{load}}=0 \div 1\), the curve \(t_{II\,\max}=f(P_{\text{load}})\) is almost rectilinear, and at \(P_{\text{load}}=1\) \(t_{II\,\max}\) has a quite definite finite value. With a further increase in \(P_{\text{load}}\), the permissible time of separate operation grows more intensively and, finally, at some critical value \(P_{\text{load}}=P_{\text{load,cr}}\), the curve \(t_{II\,\max}=f(P_{\text{load}})\) undergoes a discontinuity, accompanied by a sharp increase in the value of \(t_{II\,\max}\). It is not difficult to understand the physical meaning of the indicated discontinuity of the curve at \(P_{\text{load}}=P_{\text{load,cr}}\) from the considerations set forth below. First of all, attention must be paid

Fig. 3. Dependence of the time of separate operation permissible under automatic reclosing on the load connected to the station buses

to the fact that, at \(P_{\text{load}}>P_{\text{mech}}\), equation (7) may have several solutions with respect to the angle \(\delta_{II\,\max}\). This circumstance is most easily traced in the following example: suppose that the station rotor, having received as a result of the disturbance a certain reserve of kinetic energy, during the period of separate operation continues to move in the initial direction, experiencing a certain braking associated with a power deficit equal to \(P_{\text{load}}-P_{\text{mech}}\). Let us choose the value of \(P_{\text{load}}\) so that the stopping of the relative motion of the rotor occurs at the angle \(\delta_{II0}=\delta_l\). Then equation (7) can give us the following three solutions with respect to the angle \(\delta_{II\,\max}\): 1) the angle \(\delta^{a}_{II\,\max}\), lying in the direction of the initial motion of the rotor and somewhat smaller in magnitude than the angle \(\delta_l\); 2) the angle \(\delta^{v}_{II\,\max}\), equal to the angle \(\delta_l\),

3) the angle \(\delta^{c}_{II_{\max}}\), lying on the side opposite to the initial deviation of the rotor, i.e., smaller in magnitude than the angle \(\delta_l\).

The question arises: which of the three indicated solutions should be taken into account in the calculations?

It is obvious that, in time, the rotor will first of all reach the angle \(\delta^{a}_{II_{\max}}\), at which

\[ \int_{\delta^{a}_{II_{\max}}}^{\delta_l} \left[P_{\text{mech}_{III}}-P_{\text{el}_{III}}\right]\,d\delta = \int_{\delta^{a}_{II_{\max}}}^{\delta_l} \left(P_{\text{mech}_{II}}-P_{\text{load}}\right)\,d\delta \]

and at any point of the interval \(\delta_l \div \delta^{a}_{II_{\max}}\) reclosing will not cause a loss of synchronism. If, however, we perform this operation not in the indicated interval, but at \(\delta^{b}_{II_{\max}}=\delta_l\), then synchronism will also be preserved, but here an arbitrarily small reduction of the duration of separate operation relative to the time corresponding to the angle \(\delta^{b}_{II_{\max}}=\delta_l\) is sufficient for the system to lose stability. Indeed, in real cases one can always find such a value \(\Delta\delta\) for which

\[ \int_{\delta_l-\Delta\delta}^{\delta_l} \left(P_{\text{mech}_{III}}-P_{\text{el}_{III}}\right)\,d\delta < \int_{\delta_l-\Delta\delta}^{\delta_l} \left(P_{\text{mech}_{II}}-P_{\text{load}}\right)\,d\delta \]

and inequality (5) is not satisfied.

Thus, we must reckon with the fact that, for angles \(\delta_{II}\) exceeding \(\delta^{a}_{II_{\max}}\), a band of unstable transients is possible and, in order to avoid unsuccessful reclosures, we consider it necessary in the calculations to take into account only that angle \(\delta_{II_{\max}}\) which the rotor reaches earlier than the others in time.

We have just examined the case when the cessation of relative rotor motion occurs at the angle \(\delta_{II_o}=\delta_l\). Suppose that we slightly increase \(P_{\text{load}}\). The cessation of relative rotor motion will now occur at the angle \(\delta_{II_o}<\delta_l\). It is quite obvious that reclosing performed at the angle \(\delta_{II_o}\) will in this case be accompanied by preservation of synchronism, but it is not difficult to prove that here the second solution of equation (7) must give a value of the angle \(\delta^{b}_{II_{\max}}\) smaller than the angle \(\delta_{II_o}\). Indeed, one can always find such a value \(\Delta\delta\) for which the condition would be satisfied:

\[ \int_{\delta_{II_o}-\Delta\delta}^{\delta_l} \left(P_{\text{mech}_{III}}-P_{\text{el}_{III}}\right)\,d\delta > \int_{\delta_{II_o}-\Delta\delta}^{\delta_{II_o}} \left(P_{\text{mech}_{II}}-P_{\text{load}}\right)\,d\delta \]

and, consequently, stable AR operations will be those carried out not only at the angle \(\delta_{II_o}\), but also in a certain region of smaller angles adjoining this point. On the other hand, an arbitrarily small increase in the value of \(P_{\text{load}}\) relative to the load power value corresponding to \(\delta_{II_o}=\delta_l\) must lead to the first solution of equation (7) giving

the value of the angle \(\delta_{II\,\max}^{a}\), increased in comparison with that obtained for \(\delta_{II_0}=\delta_l\), since the equality

\[ \int_{\delta_{II\,\max}^{a}}^{\delta_l} \left(P_{\mathrm{mech}\,II}-P_{\mathrm{el}\,II}\right)d\delta = \int_{\delta_{II\,\max}^{a}}^{\delta_{II_0}} \left(P_{\mathrm{mech}\,II}-P_{\mathrm{load}}\right)d\delta \]

with increasing \(P_{\mathrm{load}}\) and with decreasing \(\delta_{II_0}\) will be satisfied already at larger values of \(\delta_{II\,\max}\). Thus, we must draw an important conclusion: with increasing \(P_{\mathrm{load}}\), the instability band lying between the angles \(\delta_{II\,\max}^{a}\) and \(\delta_{II\,\max}^{b}\), corresponding to the first and second solutions of equality (7), narrows. The break of curve 3–4 in Fig. 3 that is of interest to us corresponds to the case of the complete disappearance of this band. In this case the angles \(\delta_{II\,\max}^{a}\) and \(\delta_{II\,\max}^{b}\) merge, and AR becomes possible over the entire course of rotor motion from its position at the end of the fault period until it reaches the angle \(\delta_{II\,\max}^{c}\). The permissible duration of separate operation in this case increases sharply, since it now includes the time of the rotor’s double traverse of the interval \(\delta_l \div \delta_{II_0}\) plus the time of traverse of the interval \(\delta_l \div \delta_{II\,\max}^{c}\). With further growth of the load it is necessary to limit only the return motion of the rotor, and equality (7) will give one solution—the angle \(\delta_{II\,\max}^{c}\), lying on the side opposite to the initial deviation. Subsequently, i.e., in the interval \(P_{\mathrm{load\,cr}}\div 2\), the curve \(t_{II\,\max}=f(P_{\mathrm{load}})\) decreases smoothly, giving, however, a value \(t_{\max}\) fully allowing the AR operation to be carried out without difficulty.

1. Comparison of branches 2 and 4 of the curves shown in Fig. 3 makes it possible to conclude that, in cases where \(P_{\mathrm{load}}>P_{\mathrm{load\,cr}}\), the permissible duration of the period of separate operation under severe faults at the station buses is greater than under light ones. AR must proceed more severely in the interval \(P_{\mathrm{load\,cr}}\div 2\) for short circuits at the opposite end of the line, since here the inflow of power from the system is interrupted, while the load on the station buses remains of considerable magnitude. It is not difficult to see that in the limiting case, when during a fault the connection with the system is completely absent and the entire load falls on the station, the permissible times of separate operation will be less than those found from curve 2 (Fig. 3) by the magnitude of the fault-clearing time, i.e., by \(t_l=0.15\) sec. In Fig. 3 this case corresponds to curve 5; from comparison of it with branch 4 it follows that, for \(P_{\mathrm{load}}=P_{\mathrm{load\,cr}}\div 2\), AR with a duration of the period of separate operation of 0.2 sec is possible for any types of faults.

2. The totality of all the curves shown in Fig. 3 makes it possible to assume that, as the faults are made lighter, the values \(P_{\mathrm{load\,cr}}\) will decrease, approaching in the limit \(P_{\mathrm{load\,cr}}=P_{\mathrm{mech}\,II}\), while the corresponding times \(\left(t_{II\,\max},\,t_{II\,\max}\right)P_{\mathrm{load\,cr}}\) must increase.

In accordance with this proposition and taking into account the conclusions of item 3, one may conclude that the most severe cases for carrying out AR operations should be considered:

1) for \(P_{\mathrm{load}}<P_{\mathrm{mech}\,II}\)—a three-phase short circuit at the station buses;

2) for \(P_{\mathrm{load}}>P_{\mathrm{load\,cr}}\)—a three-phase short circuit at the oppo-

\[ (3) \]

positive end of the line (here \(P_{\text{load}\,cr}\) is the value of \(P_{\text{load}\,cr}\) for a three-phase short circuit at the station buses).

3) For \(P_{\text{mech}\,n} < P_{\text{load}} < P_{\text{load}\,cr(3)}\)—a three-phase short circuit either at the station buses or at the interconnection buses. Obviously, the first case will correspond to values of \(P_{\text{load}}\) lying to the left of the projection of the point of intersection of curves 3 and 5 (Fig. 3); the second case occurs for \(P_{\text{load}}\) lying to the right of this projection.

1. The considerations set forth in items 1, 2, 3, 4 lead us to the conclusion that, in the interval considered, \(P_{\text{load}} = 0 \div 2\), the load connected to the station buses considerably facilitates the performance of the automatic reclosing operation; moreover, its favorable influence is most pronounced at values of \(P_{\text{load}}\) equal to or somewhat exceeding the value \(P_{\text{load}\,cr}\).

Appendix

Determination of the Sign Before the Root in Equation (8)

In all cases where the acceleration

\[ \frac{P_{\text{mech}\,n} - P_{\text{load}}}{K} \]

and the initial velocity

\[ \left(\frac{d\delta}{dt}\right)_{\delta_1} \]

have the same signs, a plus sign must be placed before the root in equation (8).

In those cases where the indicated acceleration and velocity have different signs, oscillation of the rotor is possible during the separated-operation phase. The question is most simply resolved in this case if the angle \(\delta_{I\max}\) lies on the side opposite to the initial deviation, since here, obviously, a plus sign must also be placed before the root. If, however, this angle is located on the side of the initial deviation and the rotor velocity during the period of separated operation does not change sign, it is necessary to find the value of the sum:

\[ \int_{\delta_0}^{\delta_I} [P_I - q_I \sin(\delta - \alpha_{12I})]\,d\delta + \int_{\delta_I}^{\delta_l} [P_{\text{ш}} - q_{\text{ш}} \sin(\delta - \alpha_{12\text{ш}})]\,d\delta . \]

In the case where the indicated sum is less than zero, a minus sign must be taken before the root in equation (8). In the case where the indicated sum is greater than zero, both signs \(\pm\) must be placed before the root, and the times \(t'_{II\max}\) and \(t''_{II\max}\), corresponding to each of these signs, must be found. Preservation of stability under automatic reclosing will here be possible only under the condition that the actual time of separated operation \(t_{II}\) satisfies the condition:

\[ t'_{II\max} \leqslant t_{II} \leqslant t''_{II\max}, \]

where \(t'_{II\max}\) is the value of \(t_{II\max}\) calculated under the condition that in equation (8) a minus sign stands before the root; \(t''_{II\max}\) is the same, but when a plus sign stands before the root.

Notation Used

$E_1$ — e.m.f. of the station behind the longitudinal transient reactance.
$E'_2$ — voltage of the infinite-power busbars.
$Z_{11}=R_{11}+jX_{11}$ — self-impedance of the station.

\[ \alpha_{11}=90-\psi_{1};\quad \psi_{11}=\operatorname{arc tg}\frac{X_{11}}{R_{11}}. \]

$Z_{12}=R_{12}+jX_{12}$ — mutual impedance of the station and the interconnection.

\[ \alpha_{12}=90-\psi_{12};\quad \psi_{12}=\operatorname{arc tg}\frac{X_{12}}{R_{13}}. \]

$K$ — inertia coefficient of the station.
$C$ — inertia constant of the station.
$\omega$ — angular velocity of rotation of the synchronous axis of the system.
$\delta$ — angle between the transverse axis of the rotor and the synchronous axis of the system.
$t$ — time.
$P_{\text{el}}$ — electrical power delivered by the station.
$P_{\text{mech}}$ — power of the station prime mover.
$P_{\text{load}}$ — power of the load connected to the station busbars before the fault.

$\delta_0$ — rotor angle in the operating condition preceding the fault.
$\delta_1$ — value of the angle $\delta$ at the end of the fault.
$\delta_{\mathrm{II}}$ — the same at the end of the phase of separate operation.
$\delta_l$ — value of the angle $\delta$ in the established but unstable motion of the rotors in the third phase of the automatic reclosing operation, measured in the direction of motion at the beginning of this phase.
$X_K$ — reactance of the shunt at the place of the short circuit.

REFERENCES

1. Gorev, A. A. — Introduction to the Theory of Stability of Parallel Operation of Electric Stations, Kubuch, 1936.
2. Pukhova, G. F. — Stability of a system under three-phase automatic reclosing of single transmission lines, Scientific Notes of the Lviv Polytechnic Institute, Issue II, 1948.