ACCOUNTING FOR SWITCHING OVERVOLTAGES IN CALCULATING THE RELIABILITY OF ELECTRICAL INSULATION

V. S. DMITREVSKII

(Presented by the scientific seminar of the Department of Electrical-Insulation and Cable Engineering)

The calculation of the reliability of electrical insulation must take into account the effect of operating and switching overvoltages. The reliability of electrical insulation under the action of voltage \(U\) and time \(\tau\) is found as the probability of failure-free operation during the time \(\tau\).

The multiplicity of voltages \(k = \dfrac{U}{U_{\phi}}\) is a random function and obeys the normal distribution law [1].

The lifetime of insulation can be expressed [2, 3].

\[ \tau = \frac{A}{E^m} \tag{1} \]

or, provided that the dimensions of the structure are specified,

\[ \tau = \frac{B}{U^m}, \tag{2} \]

where \(\tau\) is the time to breakdown of the insulation;
\(A, B\) are constants depending on the type of material and construction;
\(E, U\) are, respectively, the field strength and the voltage acting on the insulation;
\(m\) is an exponent.

It is not difficult to obtain the distribution of the time to breakdown of insulation at some voltage. Using expression 2, one can, by introducing a scale factor, pass from one voltage to another, for example, from the test voltage to the operating voltage. However, for practical calculations it is necessary to know the distribution of the time to insulation failure under the action of operating voltage and switching overvoltages. Assuming that insulation failures under the action of operating overvoltage and switching overvoltages are independent, the probability of failures can be found as the sum of the probabilities of two independent events, i.e. [4].

\[ Q(\tau) = Q_1(\tau) + Q_2(\tau) - 2Q_1(\tau)Q_2(\tau). \tag{3} \]

In this equation \(Q_1(\tau)\) is the probability of insulation failure under the action of operating voltage, and \(Q_2(\tau)\) is the same, but under the action of switching overvoltages.

In order to use the distribution of the time to failure under operating conditions, we introduce the concept of the effective value of the voltage. The effective voltage is such a voltage at which the time to insulation failure is equal to the time to insulation failure

tion under the action of overvoltage amplitudes, provided that the latter follow one after another without pauses. The effective voltage can be found as

\[ U_{\mathrm{e}} = U_{\phi}\sqrt[m]{\int_{1}^{\infty} \kappa^m f(\kappa)\,d\kappa}. \tag{4} \]

Taking into account that the distribution of the overvoltage ratio is normal, we obtain

\[ U_{\mathrm{e}} = U_{\phi}\sqrt[m]{\int_{1}^{\infty} \kappa^m \frac{1}{\sqrt{2\pi s}} e^{-\frac{(\kappa-\bar{\kappa})^2}{2s^2}}\,d\kappa}. \tag{4a} \]

The integral in equation 4a is not expressible in terms of elementary functions. It can be evaluated on computers, or the integral can be replaced by a sum

\[ U_{\mathrm{e}} = U_{\phi}\sqrt[m]{\sum_{1}^{\infty} \kappa^m \frac{1}{\sqrt{2\pi s}} e^{-\frac{(\kappa-\bar{\kappa})^2}{2s^2}}\,\Delta\kappa}. \tag{4b} \]

If the overvoltages are limited by protective apparatus, then the integral in (4a) and the sum (4b) are taken up to the maximum overvoltage ratio admitted by the existing protection.

The duration of each overvoltage is equal to \(t_{\mathrm{p}}\), and the pause between them is \(t_0\).

In the general case \(t_{\mathrm{p}}\) and \(t_0\) are random variables. However, in solving the problem in the first approximation, one may use the mathematical expectations of \(t_{\mathrm{p}}\) and \(t_0\). Then the effective time of voltage action under an overvoltage will be found as:

\[ \tau_0 = \beta \frac{t_{\mathrm{p}}\tau}{t_0}, \tag{5} \]

where \(\tau\) is the insulation service time;
\(\beta\) is the coefficient of acceleration of insulation aging under alternating action of voltage and rest.

Thus, in [5] it is shown that the application of alternating voltage for 30 min with pauses of 15 min gives \(\beta = 2\), while application for 1 min with pauses of 30 min gives \(\beta = 30\).

We have found that, under the action of aperiodic voltage impulses, the number of impulses up to insulation breakdown is 2–3 times greater than the number of half-periods of alternating current.

Using equations (2), (4), and (5), one can obtain \(Q_2(\tau)\), if the distribution of insulation failure times at some voltage is known.

The proposed method can be used for any electrical-insulation structure for which the equation of the life curve (2) is known.

REFERENCES

1. A. E. Artem’ev, N. N. Tikhodeev, S. S. Shur, Statistical Principles for Selecting the Insulation of Power-Transmission Lines, Energiya, 1965.

2. B. T. Renne, “Electrical strength of paper capacitors,” in: Breakdown of Dielectrics and Semiconductors, Energiya, 1964.

3. L. A. Bergal’, E. N. Naidis, S. S. Solomonik, Cables and Wires for Electrical Equipment, Energiya, 1964.

4. A. D. Enifanov, Reliability of Automatic Systems, Mashinostroenie, 1964.

5. T. Yu. Bazhenova, “Use of ionization characteristics for comparative evaluation of certain new types of insulation,” in: Ionization Aging, Corona Resistance, and Methods of Testing High-Voltage Insulation, TsINTI Elektropromyshlennosti, 1963.