Mathematical Modeling of a Synchronous Generator under Rectifier Load
Abstract Generated abstract
This paper addresses the modeling of transient processes in a three-phase synchronous generator feeding an active load through a bridge full-wave rectifier, a case for which common approximate methods neglect harmonics and transient current components. The authors formulate the generator equations in d, q coordinates and develop a rectifier representation based on phase-current polarity and valve operating logic, avoiding unstable algebraic loops associated with diode resistance models. The resulting analog-computer model determines generator currents, rectifier valve quantities, load voltage, and transformed terminal voltages while retaining rectifier-induced harmonics. An example of sudden load connection computed on an MN-14 analog machine is reported to agree well with experimental results, and possible extensions include magnetic saturation and rotor-speed variation.
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Mathematical Modeling of a Synchronous Generator Under Rectifier Load
A. V. Loos, Yu. I. Ryabchikov
(Submitted by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
To generate powerful short-duration pulses of electrical energy, a number of systems use the pulsed operating mode of a three-phase synchronous generator through a bridge two-half-period rectifier feeding an active load (Fig. 1).
At present, for studying the operating modes of a synchronous generator under rectifier load, approximate methods have found the greatest application. Their use presupposes neglect of the higher harmonics of currents and voltages on the alternating-current side, and, on the rectified-current side, consideration only of average values over the repetition period. Nor are the transient and subtransient components of the currents of the synchronous generator taken into account [1, 2, 3].
However, an exact determination of the dynamic properties of the synchronous-generator–rectifier–load system requires solving the complete system of nonlinear differential equations with variable coefficients.
Mathematical modeling of a synchronous generator under rectifier load makes it possible to reduce the number of assumptions adopted. However, the study of transient processes in this case as well presents certain difficulties.
The principal ones are connected with the development of a model of a three-phase two-half-period rectifier, since known methods of modeling a diode on the basis of resistances in the model of the device under study lead to the appearance of closed loops composed of an odd number of amplifiers with a large gain factor. The presence in the solution circuit of such algebraic-type loops causes drift and interference in the solution [4]. Known methods of combining mathematical modeling with physical modeling of the rectifier have likewise not found wide application because of the great difficulties associated with the stability and tuning of current sources [5].
A consequence of the reasons indicated is the widely used simplified modeling of transient processes in a synchronous generator under rectifier load, which consists in replacing the equations of the real rectifier by the equation of its external characteristic [6]. Such a representation leads to neglect of the higher harmonics of currents and voltages caused by the presence of the rectifier, and is unacceptable in solving many problems.
Fig. 1. Operating diagram of a synchronous generator under rectifier load
Below, in modeling a synchronous generator operating under a rectifier load, transient processes in the rectifier are taken into account. The complete system of equations of the machine–valve cascade consists of the equations of the generator, the rectifier, and the load circuit. The equations of the synchronous generator are most simply represented in the coordinate system \(d, q\):
\[ \begin{aligned} - U_d &= p\Psi_d + \omega \Psi_q + r i_d,\\ - U_q &= p\Psi_q - \omega \Psi_d + r i_q,\\ U_f &= p\Psi_f + r_f i_f,\\ 0 &= p\Psi_{Dd} + r_{Dd} i_{Dd},\\ 0 &= p\Psi_{Dq} + r_{Dq} i_{Dq}, \end{aligned} \tag{1} \]
where
\[ \begin{aligned} \Psi_d &= x_d i_d + x_{ad} i_f + x_{ad} i_{Dd},\\ \Psi_q &= x_q i_q + x_{aq} i_{Dq},\\ \Psi_f &= x_{ad} i_d + x_f i_f + x_{ad} i_{Dd},\\ \Psi_{Dd} &= x_{ad} i_d + x_{ad} i_f + x_{Dd} i_{Dd},\\ \Psi_{Dq} &= x_{aq} i_q + x_{Dq} i_{Dq}. \end{aligned} \tag{2} \]
In the system of equations presented, saturation of the magnetic circuit along the paths of the main magnetic flux and leakage fluxes is not taken into account. However, taking these into account does not present particular difficulties and, if necessary, can be carried out by known methods [5]. When considering electromechanical transient processes, the equation of rotor motion must be added to system (1), (2).
As a result of solving equations (1), (2), it is possible to obtain the currents \(i_d\) and \(i_q\) of the synchronous generator (node 1, Fig. 2) by means of equations of linear transformations.
\[ \begin{gathered} i_a=i_d\cos\Theta+i_q\sin\Theta,\\ i_b=i_d\cos\left(\Theta-\frac{2\pi}{3}\right)+i_q\sin\left(\Theta-\frac{2\pi}{3}\right),\\ i_c=-i_a-i_b. \end{gathered} \tag{3} \]
we obtain the phase currents of the generator \(i_a, i_b, i_c\) (node II, Fig. 2).
The problem of modeling the rectifier is reduced to determining, from the obtained phase currents of the generator \(i_a, i_b, i_c\), the currents and voltages in the various elements of the rectifier and in the load. Taking into account the impossibility of operation of a three-phase two-half-period circuit in four-valve modes, i.e., the impossibility of simultaneous operation of two valves of one phase, it is possible to specify the logic of operation of the rectifier by means of the OPR blocks of the MN-14 analog machine. Indeed, the phase current, for example \(i_a\), depending on its polarity, is always equal to the current in one of valves 1 or 2. When valve 2 is operating, the voltage drop across valve 1 is determined from the expression
\[ U_{v1}=-U_n-U_{v2} \tag{4} \]
If the characteristics of ideal diodes are represented in the form
\[ r_k= \begin{cases} 0, & \text{for } i_k>0,\\ \infty, & \text{for } i_k\leqslant 0, \end{cases} \]
where
\[ k=1,2,3,4,5,6, \]
then expression (4) takes the form
\[ U_{v1}=-U_n \tag{5} \]
Analogous relations can also be obtained for the remaining currents and voltages of the rectifier. The voltage drop across the load is found from the relation:
\[ U_n=(i_1+i_3+i_5)R_n=i_nR_n. \tag{6} \]
From the known voltage drops across the valves \(U_{v1}, U_{v3}, U_{v5}\), it is not difficult to find the voltage at the rectifier terminals:
\[ \begin{gathered} U_a=\frac{1}{3}(2U_{v1}-U_{v3}-U_{v5}),\\[4pt] U_b=\frac{1}{3}(2U_{v3}-U_{v1}-U_{v5}),\\[4pt] U_c=-U_a-U_b. \end{gathered} \tag{7} \]
The rectifier model is represented in Fig. 2 by node V. To determine the unknown voltages \(U_d\) and \(U_q\) from the obtained \(U_a, U_b, U_c\), we use the equations of linear transformations:
\[ U_d=\frac{2}{3}\left[ U_a\cos\Theta+U_b\cos\left(\Theta-\frac{2\pi}{3}\right)+U_c\cos\left(\Theta+\frac{2\pi}{3}\right) \right]. \]
\[ U_q=\frac{2}{3}\left[ U_a\sin\Theta+U_b\sin\left(\Theta-\frac{2\pi}{3}\right)+U_c\sin\left(\Theta+\frac{2\pi}{3}\right) \right]. \tag{8} \]
The scheme for solving equations (8) is represented by node III in Fig. 2.
The harmonic functions of the angle \(\Theta\) required for the solution are obtained from a sinusoidal-oscillation generator (node IV, Fig. 2).
In the developed model of a synchronous generator operating with a rectifier load, there are no unstable elements. Its implementation on the basis of an analog computing machine made it possible to re-
Fig. 2. Mathematical model of a synchronous generator under rectifier load
to solve a wide range of problems in the development and study of machine–valve cascades.
Figure 3 presents an oscillogram of the solution for the sudden connection of a three-phase synchronous generator, through an uncontrolled rectifier, to an active load. A comparison of the results of calculations on the MN-14 with experimental investigations shows good agreement.
Fig. 3. Oscillogram of the solution at \(MN = 14\); \(i_d\) — load current, \(i_a\) — phase current, \(U_a\) — phase voltage.
The developed method for modeling a synchronous generator under rectifier load can be improved by introducing allowance for saturation of the generator magnetic circuit and for variation of the rotor speed, which will significantly broaden the capabilities of the model presented.
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