MATHEMATICAL MODELING OF A SYNCHRONOUS GENERATOR OPERATING IN CONJUNCTION WITH A CYCLOCONVERTER

G. A. Sipailov, V. N. Bazarov, A. V. Loos

(Presented by the scientific seminar of the departments of electrical machines and apparatus and general electrical engineering)

The need to obtain a low-frequency voltage (from fractions to ten hertz) leads to the use of a system consisting of a synchronous generator (SG) and a counter-parallel controlled rectifier—cycloconverter (CC). In this case the output-voltage curve, being an envelope composed of segments of the sinusoids of the phase voltages of the supply SG, can change its shape from rectangular to nearly sinusoidal. With a rectangular output voltage, operation of the CC is carried out at a control angle equal to zero. The general diagram of the system is shown in Fig. 1a. The presence of an automatic excitation regulator, which stabilizes the output voltage, allows the system to function normally at variable rotational speed of the SG. In this case the frequency at the output remains constant, and its stability is achieved by the corresponding calculation of the minimum SG frequency [1]. It is quite clear that the switching-on of the second branch of the CC can take place only after the load current has fallen to zero, in order to avoid a short circuit of the SG. During commutation from one branch of the CC to another, transient processes occur in the SG, which must be taken into account when selecting and calculating both the generator itself and the power section of the CC.

Operation of the SG in conjunction with a CC containing semiconductor valves is characterized by complex electromagnetic processes caused by the nonlinearity of the characteristics of the controlled valves and by the magnetic couplings of the generator’s multiloop circuits. An exact analytical calculation of such systems is currently impossible. However, there exist a number of methods that make it possible to approximately analyze electromagnetic processes in complex converter installations.

The analytical method using equivalent circuits has become most widespread in the literature [2, 3]. The supply generator is replaced by an emf source with the equivalent inductance of the machine and the active resistance of the winding included in each phase. The essence of such a replacement consists in replacing the complex equations of the SG transient processes by simpler ones acceptable for analytical investigations.

The use of difference equations is more convenient for analyzing the steady-state modes of the SG and CC, since in this case the duration

of conduction and commutation remains constant. In principle, an analytical investigation of the system by the complete system of equations for the SG and the CC is possible by the matching method, but the practical applicability of the method is determined by the complexity of the differential equations, their high order, the presence of periodic coefficients, and the unwieldiness of the calculations. The problem of matching the solutions at the interval boundaries is also difficult.

Thus, a brief survey of analytical methods shows that analysis of transient processes in the system under study is analytically difficult and requires the use of computing equipment.

Studies using digital computers, however, entail unjustifiably large expenditures of machine time and difficulty in processing the results. It is therefore more convenient to use analog computing equipment.

Mathematical modeling of the SG and the CC makes it possible to reduce the number of assumptions used. However, the investigation of transient processes in this case as well presents certain difficulties. The principal ones are associated with modeling the CC, since the known methods of modeling a valve on the basis of resistances in the model lead to the appearance of closed loops with an odd number of amplifiers and a total transfer coefficient greater than unity. The system model becomes unstable. Modeling with the use of the external characteristic of one branch of the CC does not take into account the effect of rectifier commutations on the SG and does not permit investigation of the CC itself.

The model of the SG and the CC developed by us makes it possible to solve many problems in the investigation of transient conditions. From the standpoint of modeling, the most convenient is the use of differential equations in the \(d, q\) axes. To carry out linear transformations, multiplication blocks are introduced into the model. The differential equations of the SG, under generally accepted assumptions, have the form [4].

\[ \left. \begin{aligned} p\Psi_d &= -U_d - \omega \Psi_q - r i_d,\\ p\Psi_q &= -U_q + \omega \Psi_d - r i_q,\\ p\Psi_f &= U_f - r_f i_f,\\ p\Psi_{Dd} &= -r_{Dd} i_{Dd},\\ p\Psi_{Dq} &= -r_{Dq} i_{Dq},\\[4pt] i_d &= \frac{x_f x_{Dd}-x_{ad}^{2}}{\Delta}\,\Psi_d -\frac{x_{ad}(x_{Dd}-x_{ad})}{\Delta}\,\Psi_f -\frac{x_{ad}(x_f-x_{ad})}{\Delta}\,\Psi_{Dd},\\[4pt] i_f &= \frac{x_d x_{Dd}-x_{ad}^{2}}{\Delta}\,\Psi_f -\frac{x_{ad}(x_{Dd}-x_{ad})}{\Delta}\,\Psi_d -\frac{x_{ad}(x_d-x_{ad})}{\Delta}\,\Psi_{Dd},\\[4pt] i_{Dd} &= \frac{x_d x_f-x_{ad}^{2}}{\Delta}\,\Psi_{Dd} -\frac{x_{ad}(x_f-x_{ad})}{\Delta}\,\Psi_d -\frac{x_{ad}(x_d-x_{ad})}{\Delta}\,\Psi_f,\\[4pt] i_q &= \frac{x_{Dq}}{\Delta'}\,\Psi_q -\frac{x_{aq}}{\Delta'}\,\Psi_{Dq},\\[4pt] i_{Dq} &= \frac{x_q}{\Delta'}\,\Psi_{Dq} -\frac{x_{aq}}{\Delta'}\,\Psi_q, \end{aligned} \right\} \qquad (1) \]

where

\[ \begin{gathered} \Delta = x_d x_f x_{Dd} + 2x_{ad}^{3} - x_{ad}^{2}(x_d+x_f+x_{Dd}),\\ \Delta' = x_q x_{Dq} - x_{aq}^{2}. \end{gathered} \]

Equations for a CC branch:

\[ \begin{gathered} U_a=\frac{1}{3}(2U_1-U_3-U_5);\qquad U_b=\frac{1}{3}(2U_3-U_5-U_1),\\ U_a+U_b+U_c=0,\\ i_n R_n=U_1+U_2=U_3+U_4=U_5+U_6,\\ i_n=i_1+i_3+i_5=i_2+i_4+i_6,\\ i_a=i_1+i_2;\qquad i_b=i_3+i_4;\qquad i_c=i_5+i_6,\\ i_a+i_b+i_c=0 \end{gathered} \tag{2} \]

where 1, 2, 3, 4, 5, 6 are the numbers of the valves.

The voltage drop across a valve is determined by

\[ U_{bK}=R_{bK}i_K, \]

where

\[ R_{bK}= \begin{cases} 0 & \text{for } i_K \geq 0,\\ \infty & \text{for } i_K < 0; \end{cases} \]

\[ K=1,2,3,4,5,6. \]

To obtain the voltages \(U_d, U_q\) and the currents \(i_a, i_b\), we shall use the well-known formulas of linear transformation:

\[ i_a=i_d\cos\omega t+i_q\sin\omega t;\qquad i_b=i_d\cos\left(\omega t-\frac{2\pi}{3}\right)+i_q\sin\left(\omega t-\frac{2\pi}{3}\right), \]

\[ U_d=\frac{2}{3}\left[ U_a\cos\omega t+U_b\cos\left(\omega t-\frac{2\pi}{3}\right)+U_c\cos\left(\omega t+\frac{2\pi}{3}\right) \right], \]

\[ U_q=\frac{2}{3}\left[ U_a\sin\omega t+U_b\sin\left(\omega t-\frac{2\pi}{3}\right)+U_c\sin\left(\omega t+\frac{2\pi}{3}\right) \right]. \tag{3} \]

The system as a whole operates in two basic modes: steady-state (mode one) and transient (mode two). For valve five we have

\[ U_{b5}=i_c R'_c, \tag{4} \]

where \(R'_c\) is the resistance inserted into the phase. The operating logic of the CC is most clearly reflected by the linear diagrams of valve loading (Fig. 1, b, c). As can be seen from the diagrams, the CC operates either in a two–three-valve mode, or in a mode in which the valves operate in groups of three.

On the basis of equations (1, 2, 3, 4), a model of the SG and CC operating into an active load was constructed (Fig. 2).

In mode one, any branch of the CC is connected and the SG operates through the rectifier in steady-state mode. The SG model is compiled from equations (1, 3) in the axes, and consists of two parts: the SG model (node I) and the model of the phase-current converter (node II).

The model of the rectifier with load (node III), the voltage converter (node IV), and the mode-switching unit (node V) are compiled from equations (2, 3, 4).

The outputs of the rectifier model are the phase voltages \(U_a, U_b, U_c\), which, after transformation, are introduced into the SG model. The mode-switching unit (node IV) is intended to switch off one branch of the CC and switch on another branch. Since both branches of the CC cannot operate simultaneously, after the current in the load falls to zero, the model of the same branch is switched on again.

Fig. 1. a — schematic diagram of the SG–CC system;
b, c — linear diagrams of CC valve loading for the two- and three-valve modes, respectively

The model of the mode-switching unit operates as follows. The shutoff signal arrives from the master pulse generator. Triode \(T_1\) opens and OPR-5 transfers the contacts. One pair of contacts blocks OPR-5 through the contacts of OPR-6; the other pair energizes the winding of OPR-6 and OPR-7 through the normally closed contacts of OPR-1 and OPR-2 when the current \(i_c\) passes through zero. When OPR-6 operates, the interlock is removed from OPR-5, but the latter is then blocked through the contacts of OPR-3 and OPR-4. The contacts of OPR-7 open the main circuit of phase “C” and connect resistances \(R'_c\) into it. The current \(i_c \approx 0\).

After the current in phase “B,” and hence in the load, has fallen to zero, OPR-3 removes the interlock from OPR-5, which removes power from OPR-7. The circuit returns to its initial position (mode one) until the next pulse arrives at triode \(T_1\).

The model of the switching-unit block is based on the physical processes occurring during rectifier operation. The proposed model makes it possible to implement a wide range of tasks related to the operation of both the generator and the rectifier. These tasks include, first of all, the determination

Fig. 2. Mathematical model of the SG–CC.

behavior of the SG when rectifiers are switched on and off, emergency modes of the entire system, the influence of the variable rotational speed of the generator, the influence of salient-pole construction, and a number of others.

As an example of the possibility of solving the listed problems on the proposed model, let us consider the behavior of the phase currents \(i_A, i_B, i_C\), the current in the load, and the current in the damper winding along the axis when the rectifier is disconnected and then switched on again. The results of the solution on the MN-14 analog computer are shown in Fig. 3.

Fig. 3. Oscillograms of the solution on the MN-14 analog computer for currents

It is evident from the figures that when control is removed from the rectifier (which corresponds, in the model, to switching a large resistance \(R'_c\) into phase “C”), a transient process takes place in the SG. The valves of the two conducting phases continue to conduct current in the load down to zero. In this case the current in phase “a” decays faster than in the steady-state mode, while the current in phase “b,” on the contrary, decays more slowly. The process by which the current decays to zero depends on the inductances of the SG and on the magnitude of the current (i.e., on the value of \(R_{\mathrm{n}}\)).

At the instant of disconnection, the SG operates in an unbalanced mode, since current flows in two phases, while the third phase is disconnected. A transient process also occurs in the damper system, which is reflected in the waveform of the current in the damper winding.

To verify the correctness of the solution on the analog computer, experimental studies were carried out on a test installation. The prototype SG was built in the dimensions of an AK-51-4 asynchronous machine, and the CC used PTL-50-8 thyristors.

a)

b)

c)

Fig. 4. Experimental oscillograms of the SG–CC system

Comparison of the oscillograms of the solution on the analog computer and those obtained on the experimental installation makes it possible to conclude that the modeling methodology is correct and that the model is serviceable in the study of all possible operating modes of the SG and CC. The effectiveness of using the model increases especially when studying modes that cannot be carried out on the experimental installation.

REFERENCES

1. Yu. P. Grigoriev, Yu. M. Inkov, Z. S. Iospa, V. P. Feoktistov. The problem of centralized power supply for passenger trains under electric and diesel traction. Elektrichestvo, 1971, No. 6.

2. I. A. Glebov. Excitation Systems of Synchronous Generators with Controlled Rectifiers. Publishing House of the Academy of Sciences of the USSR, 1960.

3. V. I. Emelyanov. Three-phase bridge circuit. Proceedings of the Research Institute of Direct Current, No. 8, 1961.

4. I. M. Postnikov. Generalized Theory and Transient Processes of Electrical Machines. Tekhnika, 1966.