APPLICATION OF THE METHOD OF SUCCESSIVE DIFFERENTIATION IN CALCULATING TRANSIENT PROCESSES IN ELECTROMECHANICAL PULSE SOURCES

A. V. LOOS

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

The transient processes of electromechanical pulse sources—for example, single-phase shock generators, valve-type pulse generators, and others—are described by systems of differential equations with periodic coefficients, from which it is impossible to free oneself by any transformations. Studies of transient processes in electrical machines in the general case of asymmetry are based on the use of the principle of constancy of flux linkages, the application of integral equations, approximate solution methods, etc. [1, 2, 3].

In some cases the equations of transient processes of electromechanical pulsed energy sources can be reduced to equations with constant coefficients; however, the need to consider the case of two or more winding systems on the rotor requires the solution of a cubic equation or characteristic equations of higher degrees with complex coefficients, which is impossible in algebraic form [4]. The need to account for saturation of the magnetic circuit and changes in the rotor speed complicates the solution of such problems to an even greater degree. In these cases the most acceptable approach is the use of analytical methods of approximate solution.

Among analytical methods for the approximate integration of systems of differential equations, integration by means of power series using the method of successive differentiation is very widespread [5]. This method is applicable both for solving systems of linear differential equations with constant and variable coefficients and for solving nonlinear problems. The desired particular solution is represented in the form of an expansion in a Taylor series. The effectiveness of the method’s application depends to a large extent on the researcher’s ability to use a priori information about the physical nature of the problem being solved.

Indeed, if one sets up a system of differential equations for an electromechanical pulse source, taking the currents as the unknown functions, then it is known in advance that the solutions will be rapidly oscillating functions. Obviously, representing them in the form of a Taylor series would require a large number of terms, i.e., the solution would be extremely cumbersome. It is more advantageous to write the differential equations of transient processes not for currents, but for flux linkages. This is due to the fact that the flux linkages of the windings change—

are much smaller in time, since they are, as a rule, monotonically varying functions, for a sufficiently accurate representation of which in the form of a Taylor-series expansion only a few terms are required. After determining the flux linkages, the currents are found by solving ordinary algebraic equations.

As an example, let us consider the use of the method of successive differentiation for calculating transient processes in a valve-type pulse generator.

The load current of a valve generator can be calculated from the envelope curve of the phase currents obtained when a synchronous generator is suddenly connected to a symmetrical three-phase active load. The value of the equivalent symmetrical active load is determined by the relation \(R_{\mathrm{e}} = {}^{2}/_{3} R_{\mathrm{n}}\) [6]. Thus, in order to calculate the load-current curve and the phase currents, it is necessary to solve the complete system of differential equations of a synchronous generator when it is connected to a symmetrical active load.

When determining the armature current, the external active resistance may be combined with the active resistance of the stator, \(r = R_{\mathrm{e}} + r_{\mathrm{c}}\). The equations of the transient processes of a synchronous generator in the \(d, q\) axes have the form:

\[ p\Psi_d = -U_d - \omega \Psi_q - r i_d , \tag{1} \]

\[ p\Psi_q = -U_q + \omega \Psi_d - r i_q , \tag{2} \]

\[ p\Psi_f = U_f - r_f i_f , \tag{3} \]

\[ p\Psi_{Dd} = -r_{Dd} i_{Dd} , \tag{4} \]

\[ p\Psi_{Dq} = -r_{Dq} i_{Dq} , \tag{5} \]

where

\[ i_d = \frac{x_f x_{Dd} - x_{ad}^{2}}{\Delta}\Psi_d - \frac{x_{ad}(x_{Dd}-x_{ad})}{\Delta}\Psi_f - \frac{x_{ad}(x_f-x_{ad})}{\Delta}\Psi_{Dd}, \tag{6} \]

\[ i_q = \frac{x_{Dq}}{\Delta'}\Psi_q - \frac{x_{aq}}{\Delta'}\Psi_{Dq}, \tag{7} \]

\[ i_f = \frac{x_d x_{Dd} - x_{ad}^{2}}{\Delta}\Psi_f - \frac{x_{ad}(x_{Dd}-x_{ad})}{\Delta}\Psi_d - \frac{x_{ad}(x_d-x_{ad})}{\Delta}\Psi_{Dd}, \tag{8} \]

\[ i_{Dd} = \frac{x_d x_f - x_{ad}^{2}}{\Delta}\Psi_{Dd} - \frac{x_{ad}(x_f-x_{ad})}{\Delta}\Psi_d - \frac{x_{ad}(x_d-x_{ad})}{\Delta}\Psi_f, \tag{9} \]

\[ i_{Dq} = \frac{x_q}{\Delta'}\Psi_{Dq} - \frac{x_{aq}}{\Delta'}\Psi_q, \tag{10} \]

\[ \Delta = x_d x_f x_{Dd} + 2x_{ad}^{3} - x_{ad}^{2}(x_d + x_f + x_{Dd}), \tag{11} \]

\[ \Delta' = x_q x_{Dq} - x_{aq}^{2}. \tag{12} \]

An analytical solution of the system of equations (1–12) in general form is absent. An attempt to obtain calculation relations for the currents of a synchronous generator in the presence of active resistances in the stator circuit was made in [3]. However, the author made an error physically associated with the inadmissibility of the assumption of constancy of the flux linkages along the longitudinal and transverse axes in a rotating machine in the presence of active resistance in the stator circuit. This error was pointed out in [4], where an exact solution was obtained for the case of one system of windings on the rotor, and the impossibility of applying ordinary methods of solution when considering two or more systems of windings on the rotor was shown. Therefore, the example considered here is of considerable interest.

Substituting (6–10) into (1–5) and taking into account that \(U_d = U_q = 0\), we obtain the transient-process equations, written with respect to the flux linkages in normal Cauchy form:

\[ p\Psi_d=-\omega\Psi_q-\frac{r}{\Delta}\left[(x_f x_{Dd}-x_{ad}^2)\Psi_d-x_{ad}(x_{Dd}-x_{ad})\Psi_f-\right. \]
\[ \left.-x_{ad}(x_f-x_{ad})\Psi_{Dd}\right], \tag{13} \]

\[ p\Psi_q=\omega\Psi_d-\frac{r}{\Delta'}(x_{Dq}\Psi_q-x_{aq}\Psi_{Dq}), \tag{14} \]

\[ p\Psi_f=U_f-\frac{r_f}{\Delta}\left[(x_d x_{Dd}-x_{ad}^2)\Psi_f-x_{ad}(x_{Dd}-x_{ad})\Psi_d-\right. \]
\[ \left.-x_{ad}(x_d-x_{ad})\Psi_{Dd}\right], \tag{15} \]

\[ p\Psi_{Dd}=-\frac{r_{Dd}}{\Delta}\left[(x_f x_d-x_{ad}^2)\Psi_{Dd}-x_{ad}(x_f-x_{ad})\Psi_d-\right. \]
\[ \left.-x_{ad}(x_d-x_{ad})\Psi_f\right], \tag{16} \]

\[ p\Psi_{Dq}=-\frac{r}{\Delta'}(x_q\Psi_{Dd}-x_{aq}\Psi_q). \tag{17} \]

Let us assume that, before being connected to the load, the synchronous generator was operating at no load with excitation current \(i_{f0}\); then the initial conditions at \(t=0\) are

\[ \Psi_{d0}=i_{f0}x_{ad},\qquad \Psi_{f0}=i_{f0}x_f,\qquad \Psi_{Dd0}=i_{f0}x_{ad},\qquad \Psi_{Dq0}=0,\qquad \Psi_{q0}=0. \]

For the adopted initial conditions, the solution for \(\Psi_d,\Psi_q,\Psi_{Dd},\Psi_f,\Psi_{Dq}\) can be represented in the form of a Maclaurin-series expansion

\[ \Psi_d=\Psi_{d0}+\frac{p\Psi_{d0}}{1!}t+\frac{p^2\Psi_{d0}}{2!}t^2+\ldots+\frac{p^n\Psi_{dn0}}{n!}t^n+\ldots . \tag{18} \]

Similarly for the flux linkages \(\Psi_q,\Psi_f,\Psi_{Dd},\Psi_{Dq}\). The initial values of the derivatives of the flux linkages in equations of the form (18) are not difficult to find, for the known initial conditions, by successive differentiation of equations (13–17). After substituting the initial values of the flux linkages and their derivatives into equations of the form (18), we obtain:

\[ \Psi_d=i_{f0}x_{ad}\left[1-\frac{1}{2}\omega^2t^2+\frac{1}{6}r\omega^2\left(\frac{x_{Dd}}{\Delta'}+\frac{x_f x_{Dd}-x_{ad}^2}{\Delta}\right)t^3+\ldots\right], \tag{19} \]

\[ \Psi_q=i_{f0}x_{ad}\left\{\omega t-\frac{1}{2}\frac{r x_{Dq}\omega}{\Delta'}t^2-\frac{1}{6}\omega\times\right. \]
\[ \left.\times\left[\omega^2-\frac{r}{(\Delta')^2}\left(x_{Dq}^2r+x_{aq}^2r_{Dq}\right)\right]t^3+\ldots\right\}, \tag{20} \]

\[ \Psi_f=i_{f0}x_{ad}\left[-\frac{x_f}{x_{ad}}-\frac{1}{6}\frac{r_f x_{ad}(x_{Dd}-x_{ad})\omega^2}{\Delta}t^3+\ldots\right], \tag{21} \]

\[ \Psi_{Dd}=i_{f0}x_{ad}\left[1-\frac{1}{6}\frac{r_{Dd}x_{ad}(x_f-x_{ad})\omega^2}{\Delta}t^3+\ldots\right], \tag{22} \]

\[ \Psi_{Dq}=i_{f0}x_{ad}\left(\frac{1}{2}\frac{r_{Dq}x_{aq}}{\Delta'}\omega t^2-\frac{1}{6}\frac{x_{aq}^2r_{Dq}}{\Delta'}\omega t^3+\ldots\right). \tag{23} \]

The convergence of the solutions for \(\Psi_d,\Psi_q,\Psi_f,\Psi_{Dd},\Psi_{Dq}\) can be determined by investigating the remainders of the Maclaurin-series expansions (19–23):

\[ R_n(\Psi_d)=\frac{t^{n+1}}{(n+1)!}\,p^{(n+1)}\Psi_d(\xi t), \tag{24} \]

where \(0<\xi<1\).

Similarly, for \(\Psi_q, \Psi_f, \Psi_{Dd}, \Psi_{Dq}\). From the found values of the flux linkages, using equations (6—10), it is not difficult to find the currents \(i_d, i_q, i_f, i_{Dd}, i_{Dq}\). By the formulas of linear transformations we determine the phase currents:

\[ i_a = i_d \cos \omega t - i_q \sin \omega t, \tag{25} \]

\[ i_b = i_d \cos \left(\omega t - \frac{2\pi}{3}\right) - i_q \sin \left(\omega t - \frac{2\pi}{3}\right), \tag{26} \]

\[ i_c = -i_a - i_b . \tag{27} \]

The load current of the rectifier pulse generator is found as the sum of the instantaneous values of the phase currents \(i_a, i_b, i_c\) having the same sign.

By the method under consideration, a calculation was performed of the transient processes of a rectifier pulse generator with the parameters:

\[ x_d = x_q = x_{Dd} = x_{Dq} = 1{,}05; \quad x_{ad} = x_{aq} = 1; \quad x_f = 1{,}2; \quad r_c = r_f = \]

\[ = r_{Dd} = r_{Dq} = 0{,}02; \quad R_H = 0{,}05 . \]

Figure 1 presents the calculated curves of the phase currents \(i_a, i_b, i_c\) and the load current \(i_H\). Comparison of the analytical calculations with the results obtained on the MN-14 analog computer in an investigation using the complete system of equations gives

Fig. 1. Calculated current curves of the generator phases and load

good convergence [6]. Estimation of the convergence of the solution by investigating the remainder term of the expansion in the Maclaurin series (24) also shows that the maximum calculation error does not exceed \(5 \div 7\%\).

The method of successive differentiation can be applied to the analysis of transient processes in electromechanical pulse sources whose equations contain variable coefficients. The investigation of transient processes described by nonlinear differential equations also presents no fundamental difficulties when this method is used; however, its application in this case may lead to cumbersome expressions. For the correct choice of the form of the initial system of differential equations, it is necessary in all cases to use a priori information about the physical picture of the processes, which greatly simplifies the solution.

References

1. I. I. Treshchev. Methods for the Study of Alternating-Current Machines. “Energiya,” 1969.

2. A. I. Vazhnov. Fundamentals of the Theory of Transient Processes in Synchronous Machines. Gosenergoizdat, 1960.

3. Ch. Konkordia. Synchronous Machines. Gosenergoizdat, 1959.

4. E. Ya. Kazovskii. Transient Processes in Alternating-Current Electric Machines. Publishing House of the Academy of Sciences of the USSR, 1962.

5. L. E. Elsgolts. Differential Equations and the Calculus of Variations. “Nauka,” 1969.

6. G. A. Sipailov, A. V. Loos, Yu. I. Ryabchikov. Study of transient processes in a rectifier pulse generator. Proceedings of TPI. Present collection.