Minimum permissible residual voltages on the buses of an inverter substation during short circuits in a three-phase network
Abstract Generated abstract
This paper examines the stability limits of high-voltage direct-current inverter substations when short circuits in the receiving three-phase network reduce the alternating-current bus voltage. Using the extinction angle of mercury valves as the limiting condition, it derives formulas for minimum permissible residual voltages under symmetrical voltage reductions and under single-phase and two-phase faults, accounting for changes in commutating voltage, inverter current, counter-electromotive force, and firing angle displacement. Calculations for cable and overhead transmission variants indicate that excessive voltage reduction can cause loss of inverter control and interruption of power transfer, with two-phase short circuits found to be more severe for inverter stability than single-phase or three-phase faults. The study also compares constant rectifier voltage regulation with constant direct-current regulation, finding that the latter permits larger voltage reductions, depending on line power and parameters.
Full Text
MINIMUM PERMISSIBLE RESIDUAL VOLTAGES ON THE BUSES OF AN INVERTER SUBSTATION DURING SHORT CIRCUITS IN A THREE-PHASE NETWORK
V. K. Shcherbakov and A. T. Putilova
1. In high-power electric power transmission, conversion of direct current into alternating current is possible by means of high-voltage valves provided with a grid-control system (an inverter). In the most general case, the converted energy is fed into a three-phase network (receiving network), which has its own sources of active power—synchronous generators.
During short circuits in the receiving three-phase network, a considerable reduction of voltage on the alternating-current side of the inverter is possible; this will lead to disruption of the latter’s normal operating conditions, and, at certain limiting values of the reduction in the magnitude of the inverter counter-electromotive force, loss of inverter control will inevitably occur. In general, the following takes place here: a decrease in voltage on the alternating-current side of the inverter leads to an increase in the current delivered by the inverter to the alternating-current network; with increased inverter current, the time required for transfer of the arc from one valve to another increases; at some value of the inverter current, the arc in the valve being switched off will not be extinguished, and the commutation process in the inverter will be disturbed.
By igniting the arc of the next valve earlier, it is possible to improve the turn-off conditions of the preceding valve; however, this is associated with deterioration of the inverter power factor and is limited by a certain maximum possible advance time for ignition of the arc of the next valve.
Thus, the question arises as to the magnitude of permissible voltage reductions in the three-phase network, the question of stable operation of the direct-current-to-alternating-current converter—the inverter.
A feature of the operation of valves in the inverter mode is the presence of an advance angle \(\beta\). The grid of the valve coming into operation must be opened in advance by a certain angle relative to the point of intersection of the phase emf curves. The time determined by the magnitude of the advance angle \(\beta\) must be sufficient for commutation of the current from one valve to another (angle \(\gamma\)) and for restoration of the controlling action of the grid of the valve leaving operation (extinction angle \(\delta\)) (Fig. 1).
For stable operation of the inverter it is necessary that the extinction angle \((\delta)\) not fall below \(\delta_0 = 10\text{–}15^\circ\) for mercury valves of ordinary construction. The extinction angle is determined by the known formula:
\[ \cos \delta = \cos \beta + \frac{I_g X}{U_2}, \tag{1} \]
where \(\beta\) is the inverter advance angle,
\(X\) is the reactance of the commutation circuit (in the particular case, the reactance of the inverter transformer),
\(I_g\) is the inverted direct current,
$U_2$ is the line voltage of the three-phase network under whose action the current is commutated from one valve to another.
On the other hand, if the voltage on the buses of the direct-current source remains unchanged, then a reduction of the voltage in the three-phase network
Fig. 1. Counter-e.m.f. of a three-phase inverter
leads to a reduction of the commutating voltage and to an increase of the inverted current by the amount
\[ \Delta I_g=\frac{(1-U_g)\cos\beta}{Z_g}, \tag{2} \]
where
\[ \begin{aligned} U_g&\text{— counter-e.m.f. of the inverter,}\\ Z_g&\text{— resistance of the direct-current circuit.} \end{aligned} \]
For symmetrical voltage reductions, the permissible residual voltages of the three-phase network may be determined from the formula
\[ U_2=\frac{X+\dfrac{X}{Z_g}\cos\beta} {\cos\delta-\cos\beta+\dfrac{X}{Z_g}\cos\beta}, \tag{3} \]
which is obtained from equations (1) and (2).
The permissible residual voltages are determined on the basis of the condition that the extinction angle decreases to the minimum permissible value $\delta_0$.
In this formula and in all subsequent formulas for determining the permissible residual voltages, it is assumed that before the assumed reduction of $U_2$ a current equal to the rated current flowed through the inverter.
We shall assume that the grid control of the inverter is supplied from a separate source operating synchronously with the three-phase network, and that changes of voltage in this network do not affect the grid voltage.
- Unsymmetrical short circuits in the three-phase network lead not only to a reduction of the voltage in the network, but also to a change in the firing angles of the various valves of the inverter. As in the case of single-phase, so also in the case of two-phase short circuits, the firing angle during commutation of current from the valve of the faulty phase to the valve of a healthy phase decreases.
Consequently, the stability margin of the inverter, characterized by the magnitude of the extinction angle, under unsymmetrical voltage reductions caused by single-phase and two-phase short circuits, decreases—
is not only due to a reduction of the commutating voltage, but also as a result of a decrease in the advance angle because of the displacement of the point of intersection of the phase-voltage curves (Fig. 2).
By the permissible voltage reduction under unsymmetrical short circuits is meant such a degree of reduction at which the extinction angle of the valve that is under the most unfavorable conditions for commutation remains not less than the minimum necessary extinction angle $\delta_0$.
Fig. 2. Change in the advance angles of inverter valves during single-phase short circuits.
3. In the case of single-phase short circuits, the magnitude of the permissible residual voltages is determined as follows.
Since the valve of the damaged phase is under the most unfavorable conditions for current commutation, the permissible residual voltages are determined from the conditions of current commutation from the valve of the damaged phase to the valve of the healthy phase.
As is seen from expression (1), the extinction angle is determined by the magnitude of the inverter current and by the magnitude of the voltage $U_2$, in this case, consequently, by the line voltage of the damaged and healthy phases.
The inverted current, when the voltage in the three-phase network is reduced, increases. The value of the inverted current is determined by the voltage on the buses of the direct-current source (this voltage is assumed unchanged) and by the value of the inverter back e.m.f.
The mean value of the no-load back e.m.f. of the inverter under normal operation and under symmetrical voltage reductions caused, for example, by three-phase short circuits in the network, is determined by the known formula
\[ U_g=\frac{\sqrt{2}\,U_2 \sin \frac{\pi}{m}}{\frac{\pi}{m}} \]
or, for a double three-phase bridge circuit, $U_g = 1.35\,U_2$, where $U_2$ is the line voltage in the three-phase network supplied to the inverter.
In the case of unsymmetrical short circuits, the back e.m.f. of the inverter can no longer be determined by this formula, since the line voltages are different in magnitude and the phase-shift angles between them are also different.
In this case, the mean value of the no-load back e.m.f. of the inverter over a period is found by integrating over the curve of variation of the inverter back e.m.f., after first dividing this curve into three sections.
For single-phase short circuits, the approximate value of the back emf for a double three-phase bridge circuit is expressed by the formula:
\[ U_g=\frac{1}{3}\left(2U_{2n3}+1\right), \]
where \(U_{2n3}\) is the line voltage of the damaged and healthy phases.
This expression is substituted into formula (2) to determine the magnitude of the increase in the inverted current.
The residual voltage of the damaged phase is determined from the triangle of line voltages (Fig. 3) by the formula:
\[ U_{\phi ost}=\frac{\sqrt{3}}{2}+\operatorname{tg}\Theta-0.5, \tag{4} \]
where \(\Theta\) is the angle between the line voltage of the damaged and undamaged phases and the line voltage of the undamaged phases.
Fig. 3. Vector diagram of voltages for single-phase short circuits.
Fig. 4. Vector diagram of voltages for two-phase short circuits.
The magnitude \(\Theta\) depends on the degree of voltage reduction. The tangent of the angle \(\Theta\) is determined from an equation of the form
\[ A\operatorname{tg}^{2}\Theta+B\operatorname{tg}\Theta+e=0, \tag{5} \]
which is obtained from the simultaneous solution, with respect to \(\operatorname{tg}\Theta\), of the equations:
\[ U_{2n3}=\frac{1}{2\cos\Theta},\quad (\text{Fig. 3}), \]
and equation (1).
We find the roots of equation (5) for an extinction angle equal to \(\delta_0\), and from the two roots we take the one whose value lies within the limits \(30^\circ<\Theta<60^\circ\).
Then, by formula (4), we determine the magnitude of the permissible residual voltage for the given inverter advance angle.
The angle between the voltages of the healthy phases does not remain constant, but for remote short circuits we may take it as unchanged and equal to \(120^\circ\).
- For two-phase short circuits in a three-phase network, the mean value of the no-load back emf is also found by integrating the back-emf curve in parts. It is expressed by the formula:
\[ U_g=2U_{2n3}\frac{1+\cos\Theta}{3}, \]
where \(\Theta\) is the angle between the line voltage of the damaged and healthy phases and the line voltage of the damaged phases;
\(U_{2n3}\) is the line voltage of the damaged and healthy phases.
The shape of the counter-emf curve is strongly distorted. The average value of the counter-emf is reduced to a greater extent than in single-phase short circuits, but to a lesser extent than in three-phase short circuits.
The residual voltages are determined by the formula:
\[ U_{\phi ost}=\frac{0.5}{\sin\alpha},\quad (\text{Fig. }4), \]
where \(\alpha\) is the angle between the line voltage of the damaged phases and the phase voltage of the undamaged phase.
The angle \(\alpha\) depends on the degree of voltage reduction in the three-phase network and can be calculated as
\[ \alpha=\operatorname{arctg}\frac{1}{3}\operatorname{tg}\Theta . \]
In turn, \(\operatorname{tg}\Theta\) is found as one of the roots of a quadratic equation of the form of equation (5), obtained from the simultaneous solution of two equations: equation (1) and the equation
\[ U_{2n\beta}=\frac{1}{\dfrac{2}{\sqrt{3}}\sin\Theta},\quad (\text{Fig. }4). \]
The root of equation (5) is taken whose value lies within the limits \(60^\circ<\Theta<90^\circ\).
- On the basis of the proposed method, the permissible residual voltages in a three-phase network were calculated for various inverter advance angles for cable and overhead versions of a high-power line of length \(800\ \text{km}\), and for a cable version of a line of length \(120\ \text{km}\). The following results were obtained.
1) With parallel operation of the synchronous generators of the network and the inverter, a reduction of the voltage supplied to the inverter from the alternating three-phase current side by more than 14.7% at an advance angle of \(40^\circ\) is impermissible.
2) If short circuits occurring in the three-phase network are accompanied by a reduction of the voltage supplied to the inverter to a greater extent than is permissible, then short circuits in the three-phase network cause a short circuit of the direct-current line. The transfer of energy to the three-phase network ceases. In this case the three-phase-current network is placed in difficult conditions: the source of active power is disconnected (the active power delivered by the inverter), and an additional short circuit in this network to the inverter transformer is also possible.
3) As a comparison of different types of short circuit shows, two-phase short circuits are more dangerous for the stability of the inverter than single-phase and even than three-phase short circuits (Figs. 5, 6).
4) In the transmission of energy by high-voltage direct current over long and ultra-long distances, two methods of regulating electric power transmission are possible:
a) the voltage at the direct-current source—at the rectifier substation—is maintained constant;
b) by means of a grid-control system for the rectifiers, the current in the line is automatically maintained constant, equal to a specified value. The specified current value can be varied over wide limits as the load changes.
A comparison of these two methods of regulation shows that, with the second method of regulation, considerably greater reductions are permissible
voltage than in the first case, both for smooth static changes of voltage and for abrupt ones.
The permissible degree of voltage reduction depends on the power of the power transmission and on the parameters of the direct-current line.
From a comparison of the curves in Figs. 5 and 6 it follows that the more powerful the power-transmission line, the greater the permissible degree of reduction, and that with the air-line version of the direct-current line, under the second method of regulating the power transmission, greater voltage reductions are permissible than in the case of the cable version of this line.
Fig. 5
1—when the voltage on the buses of the rectifier substation is kept constant
2—with static changes of voltage in the receiving network at \(I_g = \mathrm{const}\).
Fig. 6
With regulation to maintain constant rectified current:
3—cable version of the line, length \(120\ \text{km}\),
4—cable version of a powerful DC transmission line, length \(800\ \text{km}\),
5—overhead version of a powerful DC transmission line, length \(800\ \text{km}\).
Permissible residual voltages on the buses of an inverter substation under single-phase, two-phase, and three-phase short circuits in the supplied network.