On the Method of Sequential Intervals

G. A. Sipaylov

The method of sequential intervals was developed and applied in practice by a number of scholars, beginning with Euler, a full member of the St. Petersburg Academy of Sciences.

Euler begins his book Differential Calculus, published by the St. Petersburg Academy of Sciences in 1755, with an excellent exposition of the doctrine of the calculus of finite differences and of the application of this doctrine to finding sums of integral powers of natural numbers, and so on.

Euler’s method is very simple in conception; in practice, however, it either requires too much time when the interval is very small, or is too inaccurate when the interval is large.

Subsequently, in order to obtain more accurate results, Euler’s method was supplemented by successive approximations, and the entire computational method came to be called the method of successive approximations.

As applied to equations of oscillatory motion, the method of successive approximations was developed by the Russian scholars M. V. Ostrogradsky (Memoirs of the Academy of Sciences, Series VI, Vol. III, 1838) and A. M. Lyapunov in his dissertation The General Problem of the Stability of Motion — Kharkov, 1892.

Academician A. N. Krylov devoted great attention to the development and dissemination of methods of approximate computation [1].

In the preface to the monograph by Sh. E. Mikeladze, Numerical Methods for Integrating Differential Equations with Partial Derivatives (published by the Academy of Sciences, 1935), Academician A. N. Krylov writes:

“Practice is interested only in a solution possessing only that degree of accuracy which is needed in the given question for practical purposes and which corresponds both to the accuracy of the data and to the accuracy of the original equations themselves, as well as to those assumptions that were made in compiling these equations.

The practitioner needs the final result, of course, in numerical form, in a form directly applicable to the matter at hand. The very process of obtaining the result is of no interest to the practitioner—the less this process requires qualified labor, the better.”

The remarkable Russian scientist S. A. Chaplygin, who, together with his teacher, the “father of Russian aviation” N. E. Zhukovsky, created a new science—aerodynamics—was engaged, in addition to work in the field of aerodynamics, in many other questions of mechanics and applied mathematics. He gave his own method, now become classical, for the approximate solution of differential equations.

“Chaplygin’s method” represents one of the most outstanding achievements of Soviet science in the field of applied mathematics.

In the preface to his book A New Method of Approximate Integration of Differential Equations ^1), in which the author’s separate works on approximate integration are collected together, S. A. Chaply—

^1) The book was reissued in 1950.

writes: “Approximate integration of differential equations is one of the fundamental problems of engineering mathematics, and therefore every step in this field, if it provides any new illumination of the process, is of interest. That is why I considered it proper to bring together my works on this question, some of which had appeared as journal articles in the periodical press, and some of which had been published as separate brochures. All these publications have become bibliographic rarities; meanwhile, in my opinion, work in the direction I have outlined ought to be continued.”

At present, the method of sequential intervals has found wide application in the works of Professor V. T. Kasyanov.

V. T. Kasyanov has developed a method for solving differential equations with both constant and variable coefficients for calculating the processes of regulation of magnetic fields, calculating starting, braking, reversal, regulation of speed and load of direct-current motors, etc. [2].

By this method a solution is obtained with sufficient accuracy in the first approximation. This is achieved by directly substituting the mean value of the variable quantity into the original equation in the form of the sum of the initial value and half the increment of this quantity over the time interval under consideration.

Since successive approximations are absent, the calculation method is called the method of sequential intervals, and not the method of successive approximations.

If variable coefficients enter into the calculation equation (for example, in a calculation taking saturation into account, etc.), then the mean value of the variable coefficient in the interval under consideration is again chosen not by successive approximations, but by extrapolation.

With such procedures, the method of sequential intervals, without losing anything in calculation accuracy in comparison with the method of successive approximations, has been freed from the main shortcoming of successive approximations—the large number of computational operations.

In order to present more clearly the advantages of the method of sequential intervals over other calculation methods in solving practical problems, one may consider a simple example.

Suppose it is required to determine the increase of current in a circuit containing \(r\) and \(L\) when it is connected to a constant voltage \(u\).

The voltage balance equation in this case will be:

\[ u = ri + L \frac{di}{dt}. \tag{1} \]

Equation (1) is reduced to the form

\[ \frac{u}{r} = i + \frac{L}{r}\frac{di}{dt}. \tag{2} \]

Denoting

\[ \frac{u}{r} = i_0 \quad \text{and} \quad \frac{L}{r} = T, \]

we obtain

\[ i_0 = i + T \frac{di}{dt}. \tag{3} \]

As is known, the solution of this equation gives the dependence \(i=f(t)\) in the form of the function

\[ i = i_0 \left(1 - e^{-\frac{t}{T}}\right). \tag{4} \]

To solve equation (3) by the method of sequential intervals, it is necessary to write it in the form of finite increments, i.e., to replace a small segment of the curve by a straight line,

\[ i_0=i_m+T\,\frac{\Delta i}{\Delta t}, \tag{5} \]

here \(\Delta t\) is a small interval of time,

\(\Delta i\) is the increment of the current \(i\) over this interval of time, and

\[ i_m=i' + \frac{\Delta i}{2}, \tag{6} \]

where \(i'\) is the value of the current \(i\) at the beginning of the time segment \(\Delta t\). Substituting the value (6) into equation (5), we obtain

\[ i_0=i' + \frac{\Delta i}{2}+T\,\frac{\Delta i}{\Delta t}, \]

whence

\[ \Delta i=\Delta t\,\frac{i_0-i'}{T+\frac{\Delta t}{2}}. \tag{7} \]

After substituting into equation (7) the numerical value of \(T\) and the chosen value of \(\Delta t\), one obtains a simple calculation formula

\[ \Delta i=c\,(i_0-i'), \tag{8} \]

where \(c\) is a constant coefficient.

For practical calculations it is sufficient to take \(\Delta t \simeq \dfrac{T}{2}\), and only in rare cases, when special accuracy of the calculations is required, should \(\Delta t\) be reduced to \(\dfrac{T}{3}\) or \(\dfrac{T}{4}\).

Let us consider a numerical example of solving equation (3) by both methods. Let \(i_0=9.4\); \(T=0.83\). Determine \(i=f(t)\).

The curve \(i=f(t)\) can be constructed from four points, carrying the calculation to \(t \simeq 2T\). Taking \(\Delta t \simeq \dfrac{T}{2}=0.4\) sec and substituting the given parameters into equation (7), we obtain the calculation formula

\[ \Delta i=\frac{0.4(9.4-i')}{0.83+\frac{0.4}{2}}=0.388(9.4-i'). \tag{9} \]

When calculating by the classical method, we use the calculation formula

\[ i=9.4\left(1-e^{-\frac{t}{0.83}}\right). \tag{10} \]

We enter the results of the calculation in Table 1.

Table 1

As Table 1 shows, the approximate curve over the entire range is located above the exact curve by approximately \(0.5\%\) of \(i_0\).

As for the laboriousness of the computational work, it is characterized by the following data. To fill in one horizontal row of the calculation table by the method of sequential intervals, three simple arithmetic operations must be performed: subtraction, multiplication, and addition.

To fill in one horizontal row of the calculation table by the classical method, six more complex operations must be performed: division, multiplication, finding a number from its logarithm, division, subtraction, multiplication.

From the example considered and from a number of other examples [2], it follows that, for practical calculations, the method of sequential intervals gives a sufficient degree of accuracy. As regards the amount of computational work, even in the simplest cases it has a clear advantage over exact methods for the numerical solution of differential equations, and this advantage increases as these equations become more complicated.

If in the example under consideration we take \(\Delta t \simeq \dfrac{T}{4} = 0.2\), then equation (7) takes the form

\[ \Delta i = 0.215(9.4 - i). \tag{11} \]

The results of the calculation are entered in Table 2.

Table 2

Consequently, at \(\Delta t \simeq \dfrac{T}{4}\) there is practically no difference between the approximate and exact solutions, while the number of computational operations, even in this case—i.e., when calculating eight points by the method of sequential intervals—is smaller than when calculating four points by the classical method.

The method of sequential intervals is the most general method for analyzing the dynamic stability of electrical systems. For example, by means of the method of sequential intervals one can determine the limiting clearing time of a short circuit, take into account the action of voltage regulators, the change in armature reaction over time, etc. [3].

A great advantage of the method of sequential intervals is the possibility of expressing the course of a process over time in the case when the coefficients of the differential equation characterizing the process change with time, or are even functions of another variable quantity that depends on time.

For example, suppose that in equation (1) the self-inductance \(L\) is a variable quantity depending on saturation, \(L=f(i)\) (Fig. 1). On another scale, the curve in Fig. 1 will represent the dependence \(T=f(i)\), since \(r=\mathrm{const}\). In this case, to determine the rise of current in a circuit when it is switched on to a constant voltage \(u\), one can use the calculation equation (7), provided that the value of \(T\) for each in-

interval will correspond to its mean value from the curve in Fig. 1, i.e.

\[ \Delta i=\Delta t\,\frac{i_0-i'}{T_m+\frac{\Delta t}{2}}, \tag{12} \]

where

\[ T_m=\frac{T'+T''}{2}, \]

\(T'\) is the time constant at the beginning of the interval under consideration,
\(T''\) is the time constant at the end of the interval \(\Delta t\).

Let us consider a numerical example, for which we shall use the data of the example already analyzed: \(i_0=9.4\); \(T=f(i)\) (Fig. 1); \(T'=0.83\). For greater clarity we take \(\Delta t=0.2 \simeq \frac{\tau}{4}\).

We enter the results of the calculation in Table 3. In contrast to Table 1, in the present case it is expedient to add a column showing the change in \(T\) in passing from one interval to another.

For the sake of detail in the sample calculation, we shall add to Table 3 two more columns, recording \(\Delta t\) and \(T_m\) for each interval. In addition, to make the table easier to read, we introduce intermediate rows (in ordinary calculations these columns are not added and intermediate rows are not made).

Table 3

In the first row we enter the initial conditions.

To calculate the second row, we substitute into formula (12) the initial values from the first row, \(i'=0\) and \(T'=0.83\); however, the value of \(T''\) still remains unknown, and therefore we calculate the second row by the method of successive approximations, while the subsequent rows will be calculated directly by extrapolation [2].

Thus, substituting in the second row, in place of the mean value \(T_m\), its initial value \(T'\), we find, in the first approximation,

\[ \Delta i=0.2\,\frac{9.4-0}{0.83+0.1}=2.02. \]

The resulting current at the end of the first interval will be:

\[ i=i'+\Delta i=0+2.02=2.02. \]

To this current, according to the curve of Fig. 1, there corresponds \(T''=0.70\).

Consequently,

\[ T_m=\frac{0.83+0.70}{2}=0.76. \]

Substituting this value \(T_m\) into formula (12), we find, in the second approximation,

\[ \Delta i=2.18 \quad \text{and} \quad i=2.18, \]

to which, according to the curve of Fig. 1, there corresponds the value \(T''=0.69\).

Fig. 1                Fig. 2

It follows from what has been obtained that the discrepancies with the values found in the first approximation are small, and therefore, taking into account the approximate character of the determination of the parameters, the calculation in the third approximation need not be carried out.

The quantities found in the second approximation are entered in the second row of Table 3 as final.

Before proceeding to the calculation of the next, third, row, V. T. Kasyanov [2] recommends using the data of Table 3 to construct the initial segment of the curve \(i=f(t)\) (Fig. 2). Extrapolating this curve over the time interval \(\frac{\Delta t}{2}\), and taking into account the character of variation of the curve \(T=f(t)\) (Fig. 1), we obtain an approximate value of \(T_m\) for calculating the next row, and so on.

If the applied voltage is sinusoidal, it is more convenient to carry out the calculation by the formula obtained directly from equation (1), which, in the form of finite increments, takes the form:

\[ u_m=r\left(i'+\frac{\Delta i}{2}\right)+L\,\frac{\Delta i}{\Delta t}, \tag{13} \]

whence

\[ \Delta l=\frac{u_m-ri}{\dfrac{r}{2}+\dfrac{L}{\Delta t}}. \tag{14} \]

Here

\[ u_m=\frac{u'+u''}{2}, \]

\(u'\) is the instantaneous value of the voltage at the beginning of the interval \(\Delta t\) under consideration; for the first interval \(u'\) is specified by the initial condition,

\(u''\) is the instantaneous value of the voltage at the end of the interval \(\Delta t\) under consideration.

For practical calculations in this case, depending on the required accuracy, the time interval \(\Delta t\) may be taken equal to \(\dfrac{1}{12}\) or \(\dfrac{1}{18}\) of the period, which at a frequency of 50 cycles corresponds to \(\dfrac{1}{600}\) or, respectively, \(\dfrac{1}{900}\) sec.

It is important to note the circumstance that the calculated values are obtained closer to the actual ones if the mean value of the voltage is taken not as the half-sum of the values at the beginning and at the end of the interval \(\Delta t\) under consideration, but as that voltage which actually occurs in the middle of the interval. This voltage corresponds to the instant of time \(t+\dfrac{\Delta t}{2}\) and is determined by the mean angle.

For example, let at the instant of switching on the voltage pass through zero, i.e., at \(t=0\), \(\alpha_0=0\) and

\[ u'=u\sin(2\pi ft+\alpha_0)=0, \]

then at the end of the first interval, for \(\Delta t=\dfrac{1}{600}\) sec, we shall have

\[ u''=u\sin\frac{\pi}{6}=0.5u \]

and the arithmetic mean

\[ u_m=\frac{u'+u''}{2}=\frac{0+0.5}{2}\,u=0.25u, \]

whereas the actual value of the voltage in the middle of the interval under consideration will be:

\[ u_m=u\sin\left(\frac{0+\dfrac{\pi}{6}}{2}\right)=0.259u. \]

For the second interval, respectively, the arithmetic mean is

\[ u_m=\frac{u\sin\dfrac{\pi}{6}+u\sin\dfrac{\pi}{3}}{2}=0.683u, \]

actual mean

\[ u_m = u \sin \left( \frac{\frac{\pi}{6} + \frac{\pi}{3}}{2} \right) = 0.707\,u \]
and so on.

If the calculation equations contain periodically varying coefficients, as is the case, for example, in the study of single-phase asynchronous motion with a single-phase rotor, then, in order to obtain calculated data closer to the actual values, one should substitute into the calculation equations the mean value of the variable coefficient, proceeding not from the arithmetic mean of the initial and final values, but from the actual value of the coefficient in the middle of the interval under consideration. However, when the mean value of the angle, for example with a sinusoidally varying coefficient, is close to or equal to \(90^\circ\), then the actual values of the coefficient should be reduced to the arithmetic mean between its greatest and smallest values in the interval under consideration. In other respects, the calculation with a sinusoidal applied voltage and variable coefficients proceeds in the same manner as shown in the examples given above.

LITERATURE

1. Krylov A. N. a) Leonard Euler. Report at the ceremonial meeting of the Academy of Sciences of the USSR on October 5, 1933. b) Mathematics. Collected Works of Academician A. N. Krylov, vol. 3, 1949.

2. Kas'yanov V. T. Taking into account the influence of the transverse armature reaction on transient processes in uncompensated direct-current machines, Elektrosila, No. 5, 1948.

3. Zhdanov P. S. Stability of Electric Power Systems. Gosenergoizdat, 1948.