SOME FEATURES OF TRANSIENT PROCESSES IN A SYNCHRONOUS GENERATOR UNDER AN ACTIVE LOAD CHARACTER

Yu. I. Ryabchikov, G. A. Sipailov, A. V. Loos

Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering

A large number of works, published both in our country and abroad, have been devoted to the study of transient processes in a synchronous machine. Nevertheless, some specific phenomena occurring in transient modes of synchronous machines have still not received adequate treatment [1]. Thus, among the phenomena which, for a number of reasons, have not found sufficiently complete reflection in the theory of transient modes of synchronous machines, one may include certain processes occurring when a synchronous generator is suddenly switched on to a load of active character. This is explained in part by the fact that the main attention has been given to emergency modes, or, more precisely, to the most severe of them—the process of a sudden short circuit at the generator terminals. In this case the active resistance of the stator circuit is so small in comparison with the subtransient and transient resistance of the generator that it does not exert any appreciable influence on the magnitudes of the maximum current surges [2, 3]. The transient process is determined mainly by the generator parameters along the direct axis \(d\), while the influence of the parameters along the quadrature axis \(q\) is reduced merely to finding the double-frequency current components caused by the electromagnetic asymmetry of the inductor.

There are, however, other important cases for which the assumption that the active resistance of the armature circuits is zero cannot be accepted: for example, the sudden connection of a stationary asynchronous motor to a synchronous generator, the determination of a sudden change in voltage caused by the connection of a load resistance with any power factor, and others.

It is known that, in the presence of a finite active resistance in the armature circuit, the axis of the resultant magnetic flux in the air gap coincides neither with the direct nor with the quadrature axis of the generator rotor, but can be resolved into two components coinciding with them. A change in the current in one of the rotor circuits causes a change in both components of the magnetic flux and therefore leads to a change in the flux linkages of all circuits. Thus, despite the fact that the rotor circuits along the direct and quadrature axes are displaced relative to each other by 90 electrical degrees, they turn out to be electromagnetically coupled.

To evaluate the influence of the parameters of each circuit on the transient process, we shall use the Park–Gorev equations. In order to reduce the complexity of the mathematical derivations, we shall assume that in the direct

and transverse axes of the rotor there is only one circuit each. If necessary, the presence of two circuits in the longitudinal axis can be taken into account in accordance with [5].

For this case, the system of equations for the voltage equilibrium of the generator when operating with an active load takes the form

\[ \begin{aligned} \frac{d\Psi_d}{dt} &= -\omega \Psi_q - R i_d;\\ \frac{d\Psi_q}{dt} &= \omega \Psi_d - R i_q;\\ \frac{d\Psi_f}{dt} &= U_f - r_f i_f;\\ \frac{d\Psi_{Dq}}{dt} &= -r_{Dq} i_{Dq}, \end{aligned} \tag{1} \]

where

\[ \begin{aligned} \Psi_d &= L_d i_d + L_{ad} i_f;\\ \Psi_q &= L_q i_q + L_{aq} i_{Dq};\\ \Psi_f &= L_f i_f + L_{ad} i_d;\\ \Psi_{Dq} &= L_{Dq} i_{Dq} + L_{aq} i_q;\\ R &= r_s + R_{\mathrm{н}}. \end{aligned} \tag{2} \]

If the connection of the generator to the load was preceded by the no-load mode, the system (1) corresponds to the following initial values of the flux linkages:

\[ \Psi_{d0}=L_{ad}i_{f0}, \qquad \Psi_{f0}=L_f i_{f0}, \qquad \Psi_{q0}=\Psi_{Dq0}=0. \tag{3} \]

Since the formal solution of the system is rather cumbersome and is complicated by the need to find the roots of a fourth-degree algebraic equation, we shall give an approximate solution of the problem, consisting in neglecting the transformer e.m.f. in the voltage equilibrium equations of the stator circuits, i.e., we set

\[ \frac{d\Psi_d}{dt}=0; \qquad \frac{d\Psi_q}{dt}=0. \tag{4} \]

The error of such an assumption is mainly reduced to the fact that the solution will not take into account the aperiodic components of the phase currents of the stator and the terms caused by them [6].

Fig. 1 a, b. 1 — \(i_q\), 2 — \(i_{Dq}\), 3 — \(i_f\), 4 — \(i_{Dq}\), 5 — \(i_d\)

Figure 1 (a and b) gives the solutions of the transient process caused by the sudden connection of the generator to an active load, according to equations that, respectively, take into account (a) and do not take into account (b) the presence of transformer emf’s in the armature winding. The solutions were obtained by the method of mathematical modeling of differential equations on an EMU-10 analog computer and correspond to the following parameters:

\[ \begin{gathered} L_{ad}=L_{aq}=1;\quad L_d=L_q=1.05;\quad L_f=1.2;\\ L_{Dd}=L_{Dq}=1.03;\quad r_f=r_{Dd}=r_{Dq}=0.02;\\ R=0.1;\quad \omega=1. \end{gathered} \]

All parameters are given in relative units.

From comparison of the figures it is seen that the simplification made in the second case has not led to a significant error in the solution, which is explained by the comparatively small value of the time constant of the armature circuit under an active load.

Taking (4) into account, obtaining the analytical solution is considerably facilitated. Applying the Laplace transforms, we find the images of the currents in operator form under the initial conditions (3). We have

\[ I_d(p)=\frac{N_1(p)}{M(p)}; \tag{5} \]

\[ I_q(p)=\frac{N_2(p)}{M(p)}, \]

where

\[ N_1=-\omega^2 L_{ad}\left(pL_{Dq}L'_q+L_q r_{Dq}\right)\left(p\Psi_{f0}+U_f\right); \tag{6} \]

\[ N_2=R\omega L_{ad}\left(pL_{Dq}+r_{Dq}\right)\left(p\Psi_{f0}+U_f\right); \tag{7} \]

\[ \begin{aligned} M=p\{&p^2L_fL_{Dq}\left(R^2+\omega^2L'_dL'_q\right) +p\left[L_fr_{Dq}\left(R^2+\omega^2L'_dL_q\right)\right.\\ &\left.+L_{Dq}r_f\left(R^2+\omega^2L_dL'_q\right)\right] +r_fr_{Dq}\left(R^2+\omega^2L_dL_q\right)\}. \end{aligned} \tag{8} \]

Carrying out the transition to the originals, we obtain

\[ i_d=\sum_{k=1}^{3}\frac{N_1(p_k)}{M'(p_k)}\,e^{p_kt}; \tag{9} \]

\[ i_q=\sum_{k=1}^{3}\frac{N_2(p_k)}{M'(p_k)}\,e^{p_kt}, \tag{10} \]

where \(p\) are the roots of polynomial (8),
\(M'\) is the derivative of polynomial (8) with respect to the parameter \(p\).

With the aid of the obtained solution one can calculate the variation of currents, flux linkages, and voltages during the transient process; however, analyzing the influence of individual parameters on the nature of the transient process directly from formulas (6–10), without constructing the corresponding curves, proves to be very difficult because of the cumbersomeness of the expressions for the roots of polynomial (8). To this end, we shall first investigate the following limiting cases:

1) the windings of the rotor circuits are superconducting \((r_f=r_{Dq}=0,\ U_f=0)\);

2) the excitation winding has a finite active resistance, while the damper winding along the \(q\)-axis is open \((r_{Dq}=\infty)\);

3) the excitation winding is superconducting \((r_f=0,\ U_f=0)\), while the damper winding along the \(q\)-axis has a finite resistance. It should be noted that, by taking the rotor circuits to be superconducting, we find the initial values of the current surges, which may also be determined by substituting \(t=0\) in (9, 10). After transformations we have

\[ i_{d(0)}=-\frac{x_{ad} i_{f0} x'_q}{R^2+x'_d x'_q}; \]

\[ i_{q(0)}=\frac{R x_{ad} i_{f0}}{R^2+x'_d x'_q}. \tag{11} \]

The amplitude value of the phase current is

\[ I_{m\phi(0)}=\sqrt{i_d^2+i_q^2} =\frac{E_{m\phi0}\sqrt{(x'_q)^2+R^2}}{R^2+x'_d x'_q}, \tag{12} \]

where \(E_{m\phi0}=x_{ad} i_{f0}\) is the amplitude of the no-load phase voltage.

Studies of the influence of generator parameters on the magnitude of current surges, carried out in [7], show that the initial value of the phase-current amplitude, as a function of the parameter \(x'_q\), has a minimum at \(x'_q=x'_d\); an increase or decrease of \(x'_q\), with the remaining parameters unchanged, leads to an increase in \(I_{m\phi(0)}\). Let us additionally investigate the abrupt change in the magnitude of the magnetic flux in the gap and the change in the voltage at the generator terminals after switching on an active load. The expression for determining the magnitude of the magnetic flux in the gap, obtained on the basis of equations (2, 3) with (12) taken into account, and the expression for determining the initial value of the phase-voltage amplitude have the form

\[ \Phi_\delta= \frac{E_{m\phi0}\sqrt{R^2+(x'_q)^2}}{\omega\left(R^2+x'_d x'_q\right)} \sqrt{R^2+x_s^2}, \tag{13} \]

\[ U_{m\phi(0)}= \frac{E_{m\phi0}\sqrt{R^2+(x'_q)^2}}{R^2+x'_d x'_q}\,R. \tag{14} \]

It is of interest to investigate (13) and (14) for an extremum with respect to the parameter \(R\). Having done this, we find that both functions have one maximum each, respectively at

\[ R_1=\sqrt{ \frac{x'_q\left[x'_d(x'_q)^2-2x'_q x_s^2+x'_d x_s^2\right]} {(x'_q)^2-2x'_d x'_q+x_s^2} }; \tag{15} \]

\[ R_2=\sqrt{\frac{x'_d(x'_q)^2}{x'_q-2x'_d}}. \tag{16} \]

Consequently, if the generator parameters are such that expressions (15, 16) give real values, then there exists a certain interval of variation in the magnitude of the active load, switching onto which should lead to an increase in the magnetic flux in the air gap and in the voltage at the generator terminals in comparison with no-load operation. It should be noted that this is possible only in the case of electromagnetic asymmetry of the rotor; otherwise the quantities determined by formulas (15, 16) are expressed by complex numbers (it is not difficult to see, for example, that in order for the quantity determined by (16) to have a real value, the condition \(x'_q>2x'_d\) must be satisfied). As studies show, the greatest increase in the flux in the gap, as well as in the amplitude of the generator voltage, will occur under limiting asymmetry, for example, the complete absence of a damper winding along the transverse axis of a laminated rotor.

The noted effects can easily be explained on the basis of the generally accepted theory of two reactions [4]. Indeed, after switching on an active load, the magnitude of the magnetic flux in the air gap can be determined from its components \(\Phi_{ad}\) and \(\Phi_{aq}\):

\[ \Phi_\delta=\sqrt{\Phi_{ad}^{2}+\Phi_{aq}^{2}}. \tag{17} \]

For simplicity, let us assume that the rotor windings are superconducting and do not produce leakage fluxes. Then, in the case of electromagnetic symmetry of the rotor, the magnitude of the magnetic flux will not change in comparison with the no-load flux \((\Phi_{ad}=\Phi_{ad0}, \Phi_{aq}=0)\). If, however, there is no damper winding along the transverse axis, the component \(\Phi_{ad}\) will remain unchanged, but the component \(\Phi_{aq}\), in the case of an active load character, will no longer be equal to zero. As a result, the magnitude of the flux determined by (17) will turn out to be greater than before the load was connected. The increase in the magnetic flux in the air gap, in turn, causes an increase in the rotational e.m.f. in the armature winding, which may lead to an increase in the voltage at the generator terminals.

As was noted, changes in the magnitudes of currents, flux linkages, and the associated rotational e.m.f. values can occur in a real machine only over a finite time. The formulas given, however, do not reflect this, which is a consequence of neglecting transformer e.m.f.s in their derivation. In addition, since the rotor windings of a real generator possess finite active resistances, two processes occur simultaneously: the process of change of currents, flux linkages, and associated quantities, in accordance with the time constant of the armature circuit, and the process of decay of these quantities, determined by the time constants of the rotor circuits in the transient regime. As a result, the existence of increased voltage and flux in the generator air gap after its connection to the load can be only short-lived, while formulas (13), (14) give somewhat overestimated values.

Fig. 2

Figure 2 presents an experimental oscillogram of the line voltage during sudden connection of the generator to a symmetrical active load for the case when the damper winding along the transverse axis is open-circuited. The generator is made in the frame size of an AK-51-4 induction motor. The parameters of the generator and load are: \(x_{ad}=x_{aq}=1.1\); \(x'_d=0.15\); \(R=0.4\). The magnetic circuit of the generator is unsaturated. The oscillogram clearly shows the effect of an increase in voltage immediately after the generator is connected to the load, which confirms the correctness of the theoretical conclusions made. Let us note one more feature of the transient process in a synchronous generator during its sudden connection to an active load. It is generally accepted that the damping of transient currents in the rotor circuits leads to a gradual decrease in the amplitude of the phase-current magnitude. In this connection, it is of interest to consider the case when the damper winding along the \(q\) axis has a finite active resistance, while the resistance of the excitation winding is zero. Under this condition, the expressions for the currents obtained on the basis of (6–10) can be written in the form

\[ \begin{aligned} i_d &= i_{d2} - (i_{d2}-i_{d1})e^{pt};\\ i_q &= i_{q2} - (i_{q2}-i_{q1})e^{pt}, \end{aligned} \tag{18} \]

where

\[ p=-\frac{r_{Dq}}{L_{Dq}}\frac{R^2+x'_d x_q}{R^2+x'_d x'_q}; \tag{19} \]

\(i_{d1}, i_{q1}\) are the initial surges of the corresponding currents, determined by (11) for the case of a closed damper winding along the \(q\)-axis;

\(i_{d2}, i_{q2}\) are, respectively, for the case of an open damper winding along the \(q\)-axis.

Formulas (18—19) express the exponential law of variation with time of the currents \(i_d, i_q\), from values equal to the initial surges of the corresponding currents with the damper winding along the \(q\)-axis closed, to values equal to the initial surges of the currents if the damper winding along the transverse axis were open. Since the initial value of the phase-current amplitude, determined by (12), proves in the second case to be larger [7], it may be concluded that the decay of the transient current in the damper winding along the \(q\)-axis, with superconducting circuits in the longitudinal axis of the rotor, leads to a gradual increase of the periodic stator current. The same applies to the change in the magnitude of the flux in the air gap and in the amplitude of the voltage at the generator terminals.

The case considered, as well as the case of superconducting rotor circuits, is idealized. Nevertheless, if the time constant of the rotor winding along the longitudinal axis is much greater than the time constant of the damper winding along the transverse axis, then for some time after such a generator is switched onto an active load, an increase in the amplitude of the current and voltage may be observed; after it there occurs a decrease of these quantities to their steady-state values.

Thus, the studies carried out show that, in comparison with the process of a sudden short circuit or of switching onto an inductive load, the transient process when a synchronous generator is suddenly switched onto a symmetrical active load has a number of essential differences. The latter are manifested in the special influence of the electromagnetic asymmetry of the rotor on the magnitudes of the initial current surges, on the sudden change of the magnetic flux in the air gap, on the change of the voltage at the generator terminals, and on the character of the process associated with the decay of transient currents in the rotor circuits.

LITERATURE

1. E. Ya. Kazovskii. Transient Processes in Alternating-Current Electrical Machines. Moscow—Leningrad, Publishing House of the Academy of Sciences of the USSR, 1962.

2. A. I. Vazhnov. Electrical Machines. Leningrad, “Energiya,” 1969.

3. K. P. Kovach, I. Rats. Transient Processes in Alternating-Current Machines. Moscow—Leningrad, Gosenergoizdat, 1963.

4. M. P. Kostenko, L. M. Piotrovskii. Electrical Machines. Part 2, Leningrad, “Energiya,” 1973.

5. L. G. Mamikonyants. On transient processes in synchronous machines with damping circuits on the rotor. “Elektrichestvo,” 1954, No. 7.

6. M. I. Alyab’ev. General Theory of Ship Electrical Machines. Leningrad, Publishing House “Sudostroenie,” 1965.

7. Yu. I. Ryabchikov, G. A. Sipailov, A. V. Loos. On the influence of the electromagnetic asymmetry of the rotor on the load current magnitude of a pulse machine-converter generator. Izvestiya TPI, vol. 284, 1973.