CALCULATION OF DYNAMIC STABILITY OF ELECTRIC POWER SYSTEMS WITH POWERFUL INVERTER INSTALLATIONS

V. A. Lukutin

(Presented by Professor, Doctor of Technical Sciences
I. D. Kutyavin)

Any disturbance of the operating regime of an electric power system consisting of transmission lines connecting generating stations and loads causes a redistribution of power among the individual elements of this system. When generators, transformers, or transmission lines are disconnected, and also during short circuits, the powers delivered by electric stations change abruptly. The prime movers of generators are equipped with power regulators, but these regulators have great inertia and respond to changes in power with a considerable delay. As a result, the balance between the electric power delivered by the generators to the network and the power developed by the prime movers is disturbed. Excess torques are produced on the shafts of the machines, under the action of which the rotational speeds of the rotors and their relative angles change. In the course of the oscillations that arise, the distribution of power among the generators changes continuously and depends on many factors: on the inertia of the rotating masses, on the magnitude and phase of the generators’ e.m.f., on voltage regulators, etc.

The method for calculating the swinging process of synchronous machines has been very well developed in special courses [3]. But in recent years powerful converter installations have come into wide use for industrial purposes, as well as for transmitting energy by direct current. The presence in electric systems of powerful converter installations will in a definite way affect their operation, especially under nonstationary regimes in the system. Calculations of power-system operating regimes, the stability of parallel operation of electric stations, and short-circuit currents acquire a distinctive character [7].

The inverter is a new element, and moreover a very distinctive one, responding rapidly to any change in the regime of the power system supplied by it. Short circuits, especially three-phase ones, usually lead to a reduction of voltage at many points of the power system, including on the buses of the inverter substation. As a result of such voltage changes, the inverter begins to deliver to the system changed active and reactive power, which in turn affects the magnitude of the excess torques on the shafts of synchronous machines. But it is not only voltage that affects the magnitude of the power delivered by the inverter. Here an important role is played by the change in system reactance that occurs as a result of short circuits.

The active and reactive powers of the inverter are functions of the control angle, the overlap angle, and the direct-current voltage at the inverter input [1, 4, 6, 8]; moreover, the control angle itself may be a function of current or power. To take into account analytically the entire complex

these factors is extremely difficult, especially since they are poorly covered in the literature. The problem consists in finding sufficiently simple methods for taking into account the influence of the inverter on the oscillation process of synchronous machines. Specifically, this can be reduced to determining the fundamental harmonic of the voltage on the buses of the inverter substation; knowing this, it is easy to determine the value of the active and reactive power delivered by the synchronous generators. Finding the electrical power of the generators at any instant of time is the most labor-intensive operation in calculations of the dynamic stability of the system.

In the present article only symmetrical short circuits will be considered in an electric system into which an inverter, operating according to a double three-phase bridge circuit, delivers its power. We shall neglect the higher harmonics of the inverter current, taking into account in the calculations only the fundamental-frequency current. Besides simplifying the calculations, this is also justified by the fact that the higher current harmonics do not have any appreciable influence on the stability of synchronous machines. In the subsequent exposition it is assumed that, in emergency operating conditions of the system, the inverter maintains the usual order of valve firing, two and three at a time, and that the inverted current is ideally smoothed.

Fig. 1.

In stability calculations it is widely practiced to replace a group of generators, and even entire power stations, by an equivalent generator, if there is confidence that the rotors of these generators will oscillate identically in the transient regime. However, application of the method of successive intervals makes it possible to combine even nonsynchronously oscillating machines into an equivalent generator, but such combining must be carried out separately for each calculation interval. Taking this circumstance into account, we shall combine, by known formulas, the electric system into an equivalent generator with e.m.f. $\dot E_u$ and equivalent reactance $x_c$. Such a reduction of the power system will be carried out in each calculation interval.

As a result we obtain a simple circuit consisting of an inverter, operating through a transformer and an equivalent resistance, against the counter-e.m.f. $E_u$ (Fig. 1).

In converter theory, very often the voltage at the terminals of the primary winding of the power transformer is taken as the counter-e.m.f. of the inverter, while the leakage reactances of this transformer are taken as the anode reactances. But this will be correct only in the case of an insignificant power of the converter installation. In our case, however, the inverter power is comparable with the power of the system and, proceeding in this way, one can introduce a large error. With sufficiently large inverter power, the equivalent e.m.f. of the system $\dot E_u$ must be taken as the counter-e.m.f., and the anode reactances will depend on the impedance of the system $z_c$ [9].

The fundamental harmonic of the voltage on the buses of the inverter substation $\dot U_0$ will differ from the equivalent e.m.f. of the system both in magnitude and in phase.

It is known that the formulas for determining the inverted current, commutation angles, power, higher harmonics, etc., include the counter-e.m.f. of the inverter and the firing advance angle $\beta$ [1, 4]. This angle is measured from the point of intersection of the line-to-line counter-e.m.f. But the control-pulse circuit operates from the voltage at the inverter input, which differs from the counter-e.m.f. by the amount of the voltage drop in the equivalent impedance of the system:

$$ \dot U_0 = \dot E_u + \dot I_u z_c . \tag{1} $$

This dependence can also be traced on the vector diagram shown in Fig. 2.

In the diagram the following notation is used:

\(\dot E_u\) — equivalent counter-e.m.f. of the inverter;

\(\dot U_0\) — fundamental-frequency voltage at the buses of the inverter substation;

\(\psi\) — angle between the vectors \(\dot E_u\) and \(\dot U_0\);

\(\alpha_z\) — the angle supplementing the impedance angle of the equivalent resistance of the system to \(\dfrac{\pi}{2}\).

Directly from the vector diagram one may write:

\[ \sin \psi = \frac{I_u z_c}{E_u}\cos\left(\alpha_z-\beta_0+\frac{\gamma}{2}\right). \tag{2} \]

If, however, the control-pulse circuit operates from the fundamental-harmonic voltage at the inverter output, then, with the angle \(\beta_0\) set to a definite value, the ignition advance angle will in fact change continuously as the inverter current changes and will be equal to:

\[ \beta=\beta_0+\psi. \tag{3} \]

It is known that the displacement angle of the fundamental harmonic of the inverter current relative to the counter-e.m.f. is given by the formula

\[ \varphi_u=\beta-\frac{\gamma}{2}. \tag{4} \]

Fig. 2.

Taking equality (3) into account, the last formula assumes the form:

\[ \varphi_u=\beta_0+\psi-\frac{\gamma}{2}. \tag{5} \]

Thus, in order to determine the phase of the inverter current, it is necessary to determine the commutation angle \(\gamma\).

The problem of determining the commutation angle is studied in sufficient detail in many works [1, 4, 5], but most authors, as already noted, do not take into account the equivalent e.m.f. of the system and its reactance. This deficiency can be eliminated if the overlap of the anodes is regarded as an interphase short circuit in the system beyond the inverter transformer. In this case the problem of determining the commutation angle will reduce to calculating the current of a two-phase short circuit in the system. It is known that this current is given by the following formula [2]:

\[ I_k=\frac{\sqrt{3}E_u}{x_{1э}+x_{2э}}, \tag{6} \]

where \(E_u\) is the phase equivalent e.m.f. of the system, found earlier;

\(x_{1э}\) — resulting positive-sequence reactance with respect to the point of the short circuit.

\(x_{2э}\) — resulting negative-sequence reactance with respect to the same point.

The active resistance of the elements of the system is much smaller than the reactive resistance and may be neglected.

If the linear equivalent counter-e.m.f. of the inverter varies according to the law:

\[ e_u=\sqrt{6}\,E_u\sin\omega t, \tag{7} \]

then the commutation current at the instant of anode overlap will have the following form:

\[ i_k=\frac{\sqrt{6}E_u}{x_{1э}+x_{2э}}\sin\left(\omega t-\frac{\pi}{2}\right). \tag{8} \]

From the theory of inversion [1, 4] it is known that anode overlap begins at \(\omega t_1=180-\beta\) and ends at \(\omega t_2=180-\beta+\gamma\), and that during the overlap time the current \(i_k\) manages to change by the amount \(I_d\).

Consequently,

\[ I_d=i_k(t_2)-i_k(t_1), \]

or

\[ I_d=\frac{\sqrt{6}E_u}{x_{1э}+x_{2э}}[\cos(\beta-\gamma)-\cos\beta]. \tag{9} \]

In our reasoning the aperiodic component of the two-phase short-circuit current is not taken into account. However, the role of this component in determining the current \(I_d\) is small. According to the data of N. N. Shchedrin [2], aperiodic currents during short circuits in systems have a time constant of the order of \(0.05 \div 0.1\) sec and disappear \(0.2 \div 0.3\) sec after the beginning of the fault. The anode overlap during commutation lasts only \(0.0005 \div 0.0015\) sec, and over this short interval of time the aperiodic current changes insignificantly.

Taking equality (3) into account, the last formula is written in the following form:

\[ I_d=\frac{\sqrt{6}E_u}{x_{1э}+x_{2э}}[\cos(\beta_0+\psi-\gamma)-\cos(\beta_0+\psi)]. \tag{10} \]

In order to find the commutation angle \(\gamma\) of interest to us, it is necessary to solve the system of equations (2) and (10). Both equations are transcendental, and in order to solve the system the trigonometric functions may be expanded in a series, restricting it to the first two terms. As a result we obtain the following system:

\[ \frac{2I_d(x_{1э}+x_{2э})}{\sqrt{6}E_u}=2\gamma\beta_0-2\gamma\psi-\gamma^2; \]

\[ \psi-\frac{\psi^3}{6}=\frac{I_u z_c}{2E_u}\left[2-(\alpha_z-\beta_0)^2-\gamma(\alpha_z-\beta_0)-\frac{\gamma^2}{4}\right]. \]

Solving the system with respect to \(\gamma\), after simplifications we obtain:

\[ \gamma=\frac{M-\sqrt{M^2-\frac{1.045\,(x_{1э}+x_{2э})}{z_c}N}}{N}, \tag{11} \]

where

\[ M=1+\frac{\beta_0E_u}{I_u z_c}-0.5(\alpha_z-\beta_0)^2; \]

\[ N=\frac{E_u}{I_u z_c}+\alpha_z-\beta_0. \]

If the active resistances of the generators, transmission lines, etc., are not taken into account, then \(\alpha_z=0\), and instead of \(z_c\) there will appear \(x_c\). Now the angle \(\psi\) is easily found from the formula:

\[ \psi=\arcsin\frac{I_u z_c}{E_u}\cos\left(\alpha_z-\beta_0+\frac{\gamma}{2}\right). \tag{12} \]

The voltage of interest to us on the buses of the inverter substation can be written as:

\[ \dot U_0=\dot E_u+z_c I_u e^{j\left(\varphi E+\beta_0+\psi-\frac{\pi}{2}\right)}=U_0 e^{j\varphi_0}. \tag{13} \]

In this formula it is necessary to determine the magnitude of the inverter current \(I_u\).

The current delivered by the inverter to the electric system it supplies has, as is known, a nonsinusoidal waveform. When the valves are connected in a double three-phase bridge circuit, the inverter current has the form shown in Fig. 3.

Fig. 3.

The higher harmonics of the inverter current are of no interest in the present case. The fundamental harmonic can be found by expanding the function in a Fourier series. To simplify the problem, we replace the curvilinear portions by straight lines, as shown by the dashed line in Fig. 3. Then the amplitude of the fundamental harmonic of the inverter current will be calculated by the formula

\[ \sqrt{2}\, I_u=\sqrt{A_1^2+A_2^2}, \tag{14} \]

where

\[ A_1=\frac{I_d}{\pi\gamma}\left(3\sin\gamma-\sqrt{3}\cos\gamma+\sqrt{3}\right); \]

\[ A_2=\frac{I_d}{\pi\gamma}\left(3\cos\gamma+\sqrt{3}\sin\gamma-3\right). \]

Calculations show that, when the commutation angle varies over the widest limits from zero to \(45^\circ\), with the value of the inverted current \(I_d\) unchanged, the amplitude of the first harmonic \(I_u\) changes negligibly, only within 3%. On this basis it may be assumed that the inverter current does not depend on the commutation angle and is determined by the magnitude of the current \(I_d\) by the following relation (for a bridge circuit):

\[ I_u=0.78 I_d. \tag{15} \]

Above, some formulas were given by means of which it is possible to find the voltage at the point of connection of the inverter under emergency conditions, which in turn makes it possible to calculate the swing process of synchronous machines.

At present, stability calculations of electric systems are performed by the method of successive intervals. The advantages of this method are its clarity, the possibility of studying the process in time, and the simplicity of accounting for quantities that vary in an unsteady state. This widely known technique can also be extended to systems in which there are powerful converter installations. We shall briefly set forth the scheme for calculating the stability of such systems.

1. The entire process of machine swinging is divided into equal, sufficiently small time intervals.

2. The normal operating condition of the system is calculated, and the operating inverted current \(I_d\), the e.m.f. behind the transient reactance of all generators of the system, and their powers are found.

3. At the first instant after the fault, the e.m.f.s of the generators behind the transient reactance remain unchanged. Taking this property into account, we find the equivalent counter-e.m.f. of the inverter at the beginning of the first calculation interval. For this circuit, by known formulas, we reduce the system to an equivalent generator.

The rectifier–inverter link, as a rule, operates in the mode of constant current \(I_d\), which at the beginning of the first interval retains its pre-fault value. However, with another control system it is also possible to determine the value of the inverted current in the emergency regime \([6, 8]\).

1. The resulting resistances of the positive- and negative-sequence networks with respect to the point of overlap of the inverter anodes are determined.

2. The commutation angle \(\gamma\) and the displacement angle \(\psi\) are calculated for the beginning of the first interval \((11), (12)\).

3. The phasor of the fundamental harmonic of the voltage \(\dot U_0\) of the inverter-substation buses is found \((13)\).

4. The power delivered by the individual generators at the beginning of the first calculation interval is determined. For this purpose one may use any of the known formulas, in particular the following:

\[ P_r = \frac{(E'_d)^2}{z_{11}} \sin \alpha_{11} + \frac{E'_d U_0}{z_{12}} \sin(\varphi_d - \varphi_0 - \alpha_{12}). \]

1. The excess power on the shafts of the machines at the beginning of the first interval is found:

\[ \Delta P_{(0)} = \frac{P_0 - P_{(0)}}{P_0} \]

and so on, as is set forth in the course on stability \([3]\). After the value of the generators’ e.m.f. at the end of the first interval has been found, the calculation of the second interval can be carried out, again finding the equivalent counter-e.m.f. of the inverter, and so forth.

Conclusion

This article has briefly set forth the basic premises for calculating the dynamic stability of electrical systems with powerful inverter installations. However, this method can also be successfully extended to systems in which there are powerful rectifier installations.

The use of the method of successive intervals makes it possible to take into account the time variation of the generators’ e.m.f. and of the inverted current. The most powerful converter installations will be used for the purposes of transmitting energy by direct current. The presence of a direct-current transmission line and especially of smoothing reactors may, under certain conditions, considerably complicate the calculation, but its principles are fully preserved. The complication will arise from the need to calculate, for each interval, the magnitude of the inverted current, while taking into account all elements of the direct-current transmission line and the characteristics of the regulators of both the rectifier and the inverter.

A very difficult problem arises in the case of an unsymmetrical short circuit in a power system with converters. The chief obstacle here is that the question of the unsymmetrical operation of controlled converters has in general been studied very little. This problem, to a certain extent, can also be solved on the basis of the same principles as the case of symmetrical short circuits \([8]\).

The proposed method has been confirmed by calculation examples and by experimental data, which are not presented in this article.

Questions of nonstationary regimes in power systems with converter installations are very complex, since they combine two difficult problems: on the one hand, the problem of the stability of electrical systems and transient regimes in direct-current lines; on the other—

the problem of asymmetrical and non-steady-state regimes of controlled converters. But the need to solve them has already become urgent [7].

The present work may be regarded as one of the attempts to find, for engineering purposes, an acceptable solution to problems of this kind.

References

1. Kaganov I. A., Inverting Direct Current into Three-Phase Current, Gosenergoizdat, 1941.

2. Shchedrin N. N., Short-Circuit Currents in High-Voltage Systems; Methods of Calculation, ONTI, 1935.

3. Zhdanov P. S., Stability of Electrical Systems, Gosenergoizdat, 1948.

4. Chernyshev M. A., Current Inversion at Traction Substations, Transzheldorizdat, 1950.

5. Gikis A. F., Anode-current interruption. Elektrichestvo, 1948, No. 8.

6. Krug K. A., Chervonenkis Ya. M., Nakhodkin Yu. D., Pomerantsev M. S., Regulation of current, voltage, and transmitted power in the transmission of electric power by high-voltage direct current, Academy of Sciences of the USSR, G. M. Krzhizhanovsky Power Engineering Institute, report, 1948.

7. Meiman L. R., Theoretical problems of modern electrical engineering. Elektrichestvo, 1955, No. 7.

8. Lukutin V. A., Certain questions of the dynamic stability of electrical systems with powerful converter installations, dissertation, Tomsk, 1955.

9. Lukutin V. A., Accounting for generator parameters in determining the commutation angle. Published in the present collection.