ACCOUNTING FOR GENERATOR PARAMETERS IN DETERMINING THE COMMUTATION ANGLE

V. A. Lukutin

(Presented by Professor, Doctor of Technical Sciences V. K. Shcherbakov)

Owing to the anode reactances in an operating converter, when the next anode is ignited the current is transferred to it not immediately, but gradually; moreover, at the moment of this transfer the extinguishing anode and the newly ignited anode conduct simultaneously, which is customarily called commutation. The fraction of the period of the fundamental frequency, in angular units, during which simultaneous conduction of the anodes occurs is called the commutation angle.

From the general theory of converters it is known that the commutation angle depends on such factors as the magnitude of the operating current, the reactance of the anode circuits, and the voltage at the input and output of the converter. If the power of the converter is small in comparison with the power of the supplying power system, then the resistance of the supply transformer is taken as the anode reactance, and the voltage at the converter input is taken to be the counter-e.m.f.

At the present time converter installations are being built for very large powers, and sometimes they constitute the main load of the power system. In such cases the parameters of the electrical system cannot be neglected, and as the anode reactance one should take the equivalent resistance of the system, just as, instead of the voltage at the converter input, one should take the equivalent e.m.f. of the system. The equivalent e.m.f. is determined by the e.m.f. of the generators of the system. But it is known that, in calculation circuits, generators may be replaced by synchronous, transient, or subtransient parameters, depending on the operating condition of these generators.

In determining the commutation angle, the parameters of the elements of the electrical system can be taken into account most fully if the overlap of the anodes is regarded as a two-phase short circuit behind the supply transformer of the converter.

If the direct current of the converter contains no pulsations \((I_d = \mathrm{const})\), then the overlap will continue until the commutation current in the newly ignited anode reaches the value of the current \(I_d\). Consequently, the commutation angle will depend on the magnitude of the two-phase short-circuit current, and the latter depends on time.

The commutation process lasts for a very short time (about \(0.001\) sec), and therefore there is an opinion that in calculation circuits generators should be replaced by subtransient parameters. However, this does not correspond to reality and may lead to errors.

It is known that the nature of the variation of the stator current of synchronous machines during short circuits can be traced from the processes of variation of their magnetic fluxes, among which the rotor flux is of decisive importance. Using this method, let us analyze the time variation of the amplitude value of the stator current under uniformly alternating short circuits. For simplicity

we shall assume that alternating short circuits occur in a generator operating at no load.

In the no-load mode of the generator, a current flows in its excitation winding, by virtue of which the rotor magnetic flux \(\Phi_{f0}\) is produced. At the first closing of the circuit (the rectifier begins to operate), a short-circuit current arises in the stator winding, whose amplitude reaches the value \(I_{mk0}\).

At the same time, two new magnetic fluxes appear in the rotor: one is a consequence of the stator current—the stator-reaction flux \(\Phi_{ad}\); the other is due to the free currents of the rotor circuits, \(\Delta\Phi_p\).

If the leakage fluxes of the rotor are not taken into account, then at the initial instant \(\Phi_{ad}\) and \(\Delta\Phi_p\) completely balance each other. If the generator has damper windings or circuits equivalent to them, the free flux \(\Delta\Phi_p\) will consist of two components. One component, \(\Delta\Phi'_p\), is due to the free current of the excitation winding, while the other, \(\Delta\Phi_d\), appears under the action of the free currents in the damper circuits.

Both components of the free flux, immediately after their appearance, begin to decay, since the currents that caused them have no supporting emf. However, they decay with different time constants. Thus, the component \(\Delta\Phi_d\) decays very rapidly because of the small inductance of the damper circuits and disappears in less than \(0.5\) sec, whereas the flux \(\Delta\Phi'_p\) decays much more slowly, over several seconds.

The resultant magnetic flux \(\Phi_f\), linked with the stator winding and equal to the sum

\[ \Phi_f=\Phi_{f0}+\Delta\Phi'_p+\Delta\Phi_d-\Phi_{ad}, \]

will also decay.

If the short circuit were to continue for an indefinitely long time, the flux \(\Phi_f\) would decrease to a certain value corresponding to the steady-state short-circuit current, and the free flux \(\Delta\Phi_p\) would disappear completely.

However, in our case, after a short interval of time (for a bridge circuit of the order of \(t=0.001\) sec), the short circuit—or, in other words, commutation—ceases. The stator-reaction flux also disappears, and the resultant flux \(\Phi_f\) proves to be less than its forced value \(\Phi_{f0}\).

At the first instant after the elimination of the short circuit, in accordance with the principle of inertia, the resultant magnetic flux retains its value at the expense of free currents in the rotor circuits. These currents arise both in the excitation winding and in the damper circuits and will again decay with the same time constants as the free currents during the short circuit. The only difference will be that the direction of these currents will be opposite to the forced current in the excitation winding, whereas during overlap of the anodes the free currents were directed in the same way as the forced current.

The relation between the free fluxes of the rotor \(\Delta\Phi'_p\) and of the damper winding \(\Delta\Phi_d\) will be the same when the reaction flux \(\Phi_{ad}\) arises and when it disappears, as is seen in Fig. 1.

This relation is determined by the parameters of the windings.

In Fig. 1 the following are denoted:

\(\Phi''_{f0}\)—the rotor flux corresponding to the subtransient short-circuit current;

\(\Phi'_{f0}\) — the rotor flux corresponding to the transient short-circuit current;

\(\Phi_k\) — the rotor flux corresponding to the steady-state value of the short-circuit current.

Over the short time intervals corresponding to the overlap of the anodes and to their separate conduction, the free component of the magnetic flux of the rotor winding does not have time to change appreciably, since the time constant of this winding is very large. The flux of the damper winding, however, decays appreciably over these time intervals, as is shown in Fig. 1. The total magnetic flux piercing the air gap of the machine will at any instant be equal to the sum of all the fluxes:

\[ \Phi_f=\Phi''_{f0}-\Phi_{ad}+\Delta\Phi'_p+\Delta\Phi_{\mathrm{e}}. \tag{1} \]

The curve of variation of this flux consists of segments of exponential functions. By the instant of repeated overlap of the anodes, the rotor flux \(\Phi_f\) will be smaller than at the first overlap. Thereafter the pattern will repeat, and in each new cycle the resultant flux will become smaller and smaller, tending toward a certain value determined by the given ratio of the durations of the commutation and non-commutation periods.

Fig. 1.

Fig. 2.

In the first commutation sections the resultant magnetic flux repeats the form of the flux of the damper winding, since the flux of the rotor winding does not have time to change during this time. It therefore becomes possible to consider separately the processes due to the damper and rotor windings.

We shall assume that the transient process due to the flux of the damper winding assumes a certain steady value, while in the rotor winding the magnetic flux \(\Delta\Phi'_p\) has not yet had time to change. Then the resultant flux will vary along the curve shown in Fig. 2.

Let us find the steady-state value of the rotor magnetic flux \(\Phi_f\) for a given ratio of the time of the commutation interval \((1-b)T\) and the non-commutation interval \(bT\).

On the first commutation interval the flux varies according to the law:

\[ \Phi''_f=\Phi'_{f0}+\left(\Phi''_{f0}-\Phi'_{f0}\right)e^{-\frac{t}{\tau}}. \]

By the end of the overlap, at \(t=(1-b)T\), its value will be:

\[ \Phi''_f=\Phi'_{f0}+\left(\Phi''_{f0}-\Phi'_{f0}\right)e^{-\frac{(1-b)T}{\tau}}. \]

By analogy, in the first non-commutation interval the flux will vary according to the following law:

\[ \Phi_f''=\Phi_{f0}''-(\Phi_{f0}''-\Phi_f')e^{-\frac{t}{\tau}}, \]

or

\[ \Phi_f''=\Phi_{f0}''-\Phi_{f0}''e^{-\frac{t}{\tau}}+\Phi_{f0}'e^{-\frac{t}{\tau}}+(\Phi_{f0}''-\Phi_{f0}')e^{-\frac{(1-b)T+t}{\tau}}. \]

After a time interval \(T\), i.e., by the end of the first cycle, the flux assumes the value:

\[ \Phi_{f1}''=\Phi_{f0}''-(\Phi_{f0}''-\Phi_{f0}')e^{-\frac{bT}{\tau}}+(\Phi_{f0}''-\Phi_{f0}')e^{-\frac{T}{\tau}}. \tag{2} \]

The quantity \(\Phi_{f1}''\) determines the initial value of the flux at the beginning of the repeated overlap. By matching the subsequent intervals in this way, one can find the initial value of the rotor magnetic flux during overlap in the \((n+1)\)-th cycle:

\[ \Phi_{fn}''=\Phi_{f0}''-(\Phi_{f0}''-\Phi_{f0}')\left[ e^{-\frac{bT}{\tau}}-e^{-\frac{T}{\tau}}+e^{-\frac{T+bT}{\tau}}-e^{-\frac{2T}{\tau}}+ \right. \]

\[ \left. +\ldots+e^{-\frac{(n-1)T+bT}{\tau}}-e^{-\frac{nT}{\tau}} \right]. \tag{3} \]

It is not difficult to verify that in formula (3) the second bracket becomes unity when \(b=0\), and becomes zero when \(b=1\).

If the sum of the infinite alternating series that stands in this bracket is denoted by \(S_\infty\), then the following obvious inequality may be written:

\[ 0\leq S_\infty \leq 1. \tag{4} \]

The alternating series mentioned can easily be represented graphically, as shown in Fig. 3.

According to this figure, for sufficiently small \(T\) the sum of the infinite alternating series will be equal to:

\[ S_\infty=(1-b), \tag{5} \]

where \(b\) is the fraction of the non-commutation time interval in the sum of the times of overlap and separate conduction of the anodes.

Fig. 3.

Taking into account relation (5), one can find the limiting value of the magnetic flux at the beginning of infinite overlap:

\[ \Phi_{f\infty}''=\Phi_{f0}''-(\Phi_{f0}''-\Phi_{f0}')(1-b), \]

or, in another form,

\[ \Phi_{f\infty}''=\Phi_{f\infty}'+(\Phi_{f0}''-\Phi_{f0}')b. \tag{6} \]

In this last formula \(\Phi_{f\infty}'\) denotes the established value of the rotor magnetic flux in an infinite short circuit. It is not difficult to note that this value of the magnetic flux can be found in a completely analogous manner:

\[ \Phi_{f\infty}'=\Phi_k+(\Phi_{f0}'-\Phi_k)b. \tag{7} \]

Substituting (7) into (6), one can obtain the following equality:

\[ \Phi''_{f\infty}=\Phi_k+(\Phi''_{f0}-\Phi_k)b. \tag{8} \]

If the generator has no damper circuits, then instead of formula (8) one should use formula (7).

If one takes into account that the magnetic flux \(\Phi_k\) corresponds to the reactance \(x_d\)—the synchronous reactance of the generator, the flux \(\Phi'_{f0}\) corresponds to the transient reactance \(x'_d\), and the flux \(\Phi''_{f0}\) to the subtransient reactance \(x''_d\), then formulas (7) and (8) may be written in the following form:

\[ \begin{aligned} x''_2 &= x_d+(x''_d-x_d)b;\\ x'_2 &= x_d+(x'_d-x_d)b. \end{aligned} \tag{9} \]

From these formulas it is evident that, for different durations of short circuits, the generator should be introduced into the calculation with different reactances. Thus, for example, if periodic short circuits last for a very short time (which is true for \(\gamma \to 0\)), then \(b \to 1\), and the generator should be represented, respectively, by \(x''_d\) and \(x'_d\). If, however, the short-circuit process is prolonged, which corresponds to \(b \to 0\), then for both types of generators one should use \(x_d\).

If the commutation process is regarded as an alternating short circuit beyond the transformer of the converter, then the conclusions given above may also be extended to the case of generator operation feeding a converter. In this case the generator should be introduced into the calculations with reactances computed by formula (9). In addition, it should be borne in mind that the overlap angle of the converter, after the latter begins operation, will change for some time until it reaches a certain value for the given circuit parameters.