ON THE METHODOLOGY FOR SELECTING PARAMETERS OF TEST WAVES REPRODUCING SWITCHING IMPACTS ON THE INSULATION OF HIGH-VOLTAGE SYSTEMS

A. A. VOROB'EV, I. I. KALYATSKII, V. F. PANIN

Two characteristic features in the development of modern power engineering—progress in the field of lightning protection and the transition to extra-high-voltage transmission lines—have led to a significant increase in the role of switching overvoltages in establishing the insulation levels of high-voltage systems. A considerable part of the research connected with the operating conditions of insulation under the action of switching overvoltages is directed toward establishing the forms and levels of impulse voltages that would equivalently reproduce these effects on insulation.

Because of the diversity of the conditions under which insulation operates, and the dependence of the electrical strength of insulating media on the voltage waveform, a comprehensive approach is necessary in establishing the forms of test impulse voltages—one that takes into account the degree of danger to the insulation of a given wave shape and the probability of its occurrence. Such an approach should ensure the most economical insulation levels while guaranteeing reliable operation.

It is evident that the advantages of such a comprehensive approach to solving the problem will be realized only when sufficiently complete information is available on the probability distributions of the forms and magnitudes of switching overvoltages, as well as information on the strength of modern insulating media under switching-type effects.

On the basis of measurements on models, as well as recordings of overvoltages directly in systems, some material has been accumulated on the magnitudes and forms of switching overvoltages; however, it requires further expansion and statistical processing.

Most studies of insulation behavior under switching overvoltages are aimed at determining the effects under which the operating conditions of insulation are most severe. Despite successes in establishing the forms of impulse voltages dangerous for line insulation that may occur in extra-high-voltage systems [1, 2], at present no consistent methodology has been developed for determining the forms of the most dangerous impulse effects. Most often, the duration of application of a voltage of a certain...

amplitude, i.e., the duration of the wave and, more rarely, the duration of its front.

An illustration of such an approach is provided, for example, by Fig. 1a, which shows an overvoltage wave arising when a capacitive load is disconnected. According to [3], this wave, in terms of its effect on insulation, is adequately represented by a \(50\ \mu\text{s}\) half-wave having the same amplitude and the same order of duration. Meanwhile, an analysis of present-day ideas about the nature of pre-discharge processes in dielectrics

Fig. 1. On the criterion for selecting the shape of the test wave. 1,a — \(50\ \mu\text{s}\) half-wave; 2,a — overvoltage wave when a capacitive load is disconnected without restrikes in the circuit breaker, according to [3].

and, chiefly, an analysis of the results of direct measurements of the electric strength of various insulating media have made it possible to draw the following conclusion: along with the amplitude and duration of the applied voltage, one of the principal factors affecting the strength of insulation should be regarded as the rate of rise of the voltage across the interelectrode gap. In other words, if two pulses are applied successively to an insulating gap—

pulse of the same amplitude, duration, and polarity, but with a different rate of voltage rise: \(U_A(t)\) and \(U_B(t)\) (Fig. 1,b), then, first, the effect of the voltage on the gap in both cases will be different and, second, it is not necessary that the pulse \(U_B(t)\) be less dangerous for the insulation than \(U_A(t)\)—despite the fact that the acting pulses are related by the condition:

\[ \int_{0}^{\tau} U_A(t)\,dt > \int_{0}^{\tau} U_B(t)\,dt . \tag{1} \]

Indeed, when considering the dependences of the strength of air and of certain solid dielectrics on the steepness of the applied pulse, according to data [4–9], the following regularity may be noted: at a certain steepness of the applied voltage, which we shall call critical, the dielectric has a minimum of the mean breakdown voltage or of the 50-percent voltage.

For positive polarity of the electrode of greater curvature, the presence of a minimum is manifested more distinctly than for negative polarity. At pulse steepnesses in the region of the critical steepness, an increase in the scatter of the values of electric strength is observed. For air, according to [1, 2], the standard deviation of the breakdown voltage at a steepness close to the critical one is 3–4 times greater than at 50-cycle voltage and standard pulses.

Table 1

VALUES OF CRITICAL STEEPNESS FOR AIR INSULATION \((\mathrm{kV}/\mu\mathrm{s})\)

Note. C—rod, P—plane, GI—garland of insulators, ST.I—post insulators, PR—wire.

Figure 2 presents the dependences of the discharge voltages on the steepness \(a\) of a positive impulse for long air gaps of the “rod–plane” type, according to data from [5]. From Fig. 2 it is evident that, with respect to electric strength, gaps \(1\text{–}6\ \mathrm{m}\) long can be characterized by a certain steepness \(a_{\mathrm{B}}\), at which their discharge voltage is minimal. Table 1 presents the values—

Fig. 2. Dependences of the mean discharge voltages of a “rod–plane” gap in air on the impulse steepness, according to [5].

Table 2

VALUES OF CRITICAL STEEPNESS FOR SOLID DIELECTRICS

Note. O—point, Sh—sphere, P—plane.

values of the steepness \(a_B\), obtained in the analysis of the data of [4–7] for gaps of various lengths and with various configurations of the electric field. Despite the differences in the research methodology used in [4–7], the values of the critical steepness are sufficiently close, and its mean value is \(7\) kV/µs. In the region of this steepness, the average discharge voltage is 10–30% lower than at a voltage of 50 Hz.

Figure 3 presents analogous dependences for solid dielectrics according to the data of [8, 9]. In the investigated range of thicknesses, \(0,05 \div 34\) mm, the impulse breakdown voltages of solid dielectrics are characterized by a critical steepness \(a_T\), at which the electric strength is minimal. From Table 2, which presents the values of the critical steepness obtained in the analysis of the data of [8–10], the value of \(a_T\) can be estimated as \(20 \div 30\) kV/µs. At these steepnesses, the breakdown voltages are 10–20% lower than the short-time strength under dc voltage and under a voltage of 50 Hz [8, 9]. The phenomenon of reduction of impulse electric strength at steepnesses in the critical region is considered in [11, 12]. It should also be noted that solid dielectrics have a lower impulse strength in comparison with their strength under dc voltage, as noted in [13–15].

In this connection it is of interest to note that in [16], where the electric strength of transformer oil was measured, a decrease was obtained

impulse breakdown voltage as compared with breakdown at constant voltage by 23–40%.

In studies of the strength of liquid dielectrics under impulses of duration \(10^{-6} \div 10^{-2}\) sec and with a front of up to several microseconds, many authors established that each interelectrode gap in a liquid, with respect to electrical strength, can be characterized by a certain time of voltage application characteristic of it, often called [17] the discharge-formation time. For application times shorter than the indicated value, the discharge voltages rise sharply; for longer times they remain approximately at one level, i.e., in the volt-second characteristic there is a “step” covering, in time, a duration of up to \(2 \div 4\) orders [19, 20]. The value of this time decreases together with the length of the gap under investigation [20].

The facts thus established made it possible to suppose that each type of insulating medium—gaseous, solid, and liquid—can be characterized by its own critical steepness of a linearly rising voltage. At the critical steepness, the most favorable conditions occur for the development of electrical breakdown and, consequently, the minimum impulse strength.

To confirm the considerations set forth above, experiments were performed on the breakdown of technically pure transformer oil [21, 22]. Breakdown of “point–plane” gaps of length \(5 \div 25\) cm was carried out on an exponential front of a voltage impulse. The variation of the average rate of rise of the voltage in the interelectrode gap was effected within the limits \(0.7 \div 2200\) kV/μsec. In Fig. 4(a, b) the dependences of the breakdown voltage on the steepness of the impulse are presented for gaps of length \(5 \div 25\) cm. A characteristic feature of the dependences is the presence of a minimum of electrical strength in the steepness region \(80 \div 200\) kV/μsec (for positive polarity of the point). It should be noted that for transformer oil, as for air, at steepnesses in the critical region a large instability of the discharge voltages is observed: the standard deviation reached a value of 20% [22]. Its usual values on impulses are \(4 \div 10\)% [19].

Thus, an analysis of the information available in the literature on the strength of dielectrics under impulses with fronts of different duration, as well as the results of experiments on breakdown of gaps in transformer oil on an oblique-front voltage wave, give grounds for asserting that the establishment of the forms of impulse voltages most dangerous for high-voltage insulation can be based on the use of the dependence of dielectric strength on the rate of voltage rise, i.e., proceeding from the condition of the existence of a critical steepness.

To implement the above, it appears necessary to carry out investigations of the strength of various insulating media on oblique-front impulses in order to establish the critical steepnesses characterizing each medium. The critical rates of voltage rise thus found, when determining dangerous effects on insulation of different voltage classes, can be obtained by using different durations of the impulse front.

Let the critical steepnesses for air \((a_{\mathrm{B}})\), solid \((a_{\mathrm{T}})\), and liquid \((a_{\mathrm{Ж}})\) media—\(a_{\mathrm{B}}, a_{\mathrm{T}}, a_{\mathrm{Ж}}\)—be determined. In view of the fact that for air, solid, and liquid insulation the influence of the duration

Fig. 3. Dependences of the breakdown voltages of solid dielectrics on the steepness of the impulse; B — according to [8], A — according to [9].

Graph labels:

• \(U_p(90\%),\ \text{kV}\)
• Lead, \(10\ \text{mm}\), (A)
• Lead, \(15\ \text{mm}\), (A)
• Organic glass, \(2\ \text{mm}\), (A)
• Paraffin, \(34\ \text{mm}\), (A)
• Paraffin, \(34\ \text{mm}\), (A)
• NaCl, \(10\ \text{mm}\), (A)
• NaCl, \(0.15\ \text{mm}\), (B)
• NaCl, \(0.7\ \text{mm}\), (B)
• \(a_{\mathrm{av}},\ \dfrac{\text{kV}}{\mu\text{s}}\)

Fig. 4. Discharge voltages of “point–plane” gaps in transformer oil as a function of the voltage steepness, according to [21, 22];
1,a; 3,a — for positive polarity of the point,
2,a; 4,a — for negative polarity of the point.

Note. The data in Fig. 4,a were obtained for breakdown of gaps in the initial part of an exponential impulse.

the applied voltage (for the given front steepness) is different; each type of insulation should be considered separately.

Air insulation. The following considerations must be taken into account.

In accordance with the results of investigations by many laboratories throughout the world, set forth in [6], the most dangerous waves for purely air insulation of systems under any atmospheric conditions are waves with a front duration that provides a steepness close to the critical one.

In [1] it was established that the levels of line insulation of transmission lines at voltages of 300 kV and above must be determined with allowance for the reduction of its strength on waves with a sloping front, but not by wet-discharge voltages.

A positive solution to the problem of protecting insulation against waves with a gentle front, used at present by arresters, does not appear possible, since with weak nonuniformity of the field of the arrester spark gap the impulse coefficient in the region of critical steepness remains equal to 1, while for insulating gaps it decreases to 0.7 [1].

The protection proposed in [23]—of at least the insulation of the most valuable apparatus—by applying to the protected gap a steep impulse synchronized with a gentle voltage wave seems unrealistic for techno-economic reasons, at least in the near future.

On the basis of the foregoing, it is clear that a systematic reduction in the levels of line insulation, due to progress in the field of overvoltage limitation, is impossible without taking into account the reduced strength of insulation under waves with front steepnesses close to critical. In this connection it also becomes necessary to establish the forms and levels of the most dangerous stresses.

Figure 4a presents the volt-time characteristic for a certain high-voltage device with air insulation. The straight line \(a_{\mathrm{B}}=\mathrm{const}\) is the line of critical steepness, which determines the values of the front duration of the most dangerous impulse stresses for insulating structures of different voltage classes.

If one proceeds from the mean discharge voltage of the given structure at a voltage of \(50\ \mathrm{cps}\)—\(U_{\mathrm{p}50}\) (amplitude value), then the mean discharge voltage at the critical steepness of the front, \(U_{\mathrm{pk}}\), is determined according to the expression:

\[ U_{\mathrm{pk}}=U_{\mathrm{p}50}\cdot K_{\mathrm{B}} . \tag{2} \]

where \(K_{\mathrm{B}}\) is the impulse coefficient in a nonuniform field for interelectrode gaps in air at voltage steepnesses close to critical.

The greatest probability of flashover of the structure under the action of an impulse with amplitude \(U_{\mathrm{pk}}\) occurs at the critical steepness \(a_{\mathrm{B}}\) of the impulse front and, consequently, at the front length

\[ \tau_{\mathrm{fr}}=\frac{U_{\mathrm{pk}}}{a_{\mathrm{B}}}. \tag{3} \]

If it is taken into account that for gentle waves the mean discharge voltage is close to the 50-percent flashover voltage [24], especially for front durations of \(10^{-5}\div 10^{-4}\ \mathrm{sec}\), then it is evident that an impulse with amplitude \(U_{\mathrm{pk}}\) and front \(\tau_{\mathrm{fr}}\) will ensure a probability of insulation flashover close to 50%. The voltage to which it is necessary to limit stresses with critical steepness is

\[ U_{\mathrm{ak}}= \frac{ U_{\mathrm{p}50}\cdot k_{\mathrm{B}}\cdot \left(1-\frac{2\sigma_{\mathrm{k}}}{100}\right) }{ k_{\mathrm{zk}} }, \quad \text{where} \tag{4} \]

\[ \sigma_{\mathrm{k}} \]
is the standard deviation of the discharge voltage at critical steepness.

\(k_{\mathrm{zk}}\) is the safety factor, established, for example, from the condition of instability of the discharge voltages of the spark gaps of protective arresters. Thus, \(U_{\mathrm{ak}}\) is the minimum

discharge voltage corresponding to the lower boundary of the scatter region of the discharge voltage, usually taken as \(4\sigma\).

The greatest hazard of the action at amplitude \(U_{\mathrm{ak}}\) is ensured by the front duration corresponding to the critical steepness \(a_{\mathrm{B}}\).

Fig. 5. On establishing the most dangerous actions of impulse voltage on various insulating media. 5a—for air, 5b—for solid insulation, 5c—for liquid insulation. \(U_{\mathrm{B}}(t)\), \(U_{\mathrm{t}}(t)\), \(U_{\mathrm{ж}}(t)\) are the time courses of the most dangerous impulse voltages for the various insulating media. The remaining designations are explained in the text.

\[ \tau_{\phi}=\frac{U_{\mathrm{ak}}}{a_{\mathrm{v}}}. \tag{5} \]

When establishing the parameters of the wave of the test action, the voltage impulse with amplitude \(U_{\mathrm{ak}}\) and front \(\tau_{\phi}\) should be taken as the reproducing action corresponding to the minimum withstand level.

Solid insulation. In Fig. 4,b the volt-time characteristic is presented for a device with a solid dielectric. The straight line \(a_{\mathrm{t}}=\mathrm{const}\) is the line of critical steepness. Under the same initial assumptions as for air, the corresponding expressions for the amplitude and front duration of the minimum withstand action are obtained in the form:

\[ U_{\mathrm{ak}}= \frac{ U_{\mathrm{p}50}\cdot k_{\mathrm{t}}\cdot \left(1-\frac{2\sigma_{\mathrm{k}}}{100}\right) }{ k_{\mathrm{e k}} }, \quad \text{and} \tag{6} \]

\[ \tau_{\phi}=\frac{U_{\mathrm{ak}}}{a_{\mathrm{t}}}, \tag{7} \]

where \(k_{\mathrm{t}}\) is the impulse coefficient of solid insulation at the critical steepness.

Liquid insulation. It is necessary to note the difference between insulating media with respect to the dependence of strength on impulse steepness. The strength of air insulation, as the impulse steepness decreases below steepnesses close to the critical one, tends toward the strength level at 50 Hz. According to the results of a number of studies [8, 9, 11, etc.], the strength of solid insulation, increasing at steepnesses below the critical one, subsequently remains approximately at one level, at least up to the maximum time limit of switching overvoltages—\(2 \div 5 \cdot 10^{-2}\) sec, according to [25].

The strength of technically pure transformer oil, after some increase in the steepness range \(100 \div 30\) kV/µsec, decreases with a further decrease in steepness, and at steepnesses of several kV/µsec, corresponding to times of \(10^{-4} \div 10^{-3}\) sec, is only 40% higher than the strength at 50 Hz (Fig. 4,c). It follows from this that the most dangerous actions for liquid insulation should be selected from the condition of critical steepness and the required impulse duration; the latter may be neglected with respect to solid and air insulation. The wave duration sufficient to reproduce the action with the greatest danger may be estimated as \(5 \cdot 10^{-3} \div 10^{-2}\) sec. For the purpose of unifying tests, these values may also be used when selecting the duration of dangerous impulses for air and solid insulation. To determine quantitative relationships, additional measurements are necessary of the 50-percent discharge voltages of gaps in transformer oil on a long impulse with different front steepness. However, regardless of the quantitative relationships, the expressions for the amplitude and front duration of the minimum withstand impulse action will remain similar to (4), (6) and (5), (7), i.e.,

\[ U_{\mathrm{ak}}= \frac{ U_{\mathrm{p}50}\cdot k_{\mathrm{zh}}\cdot \left(1-\frac{2\sigma_{\mathrm{k}}}{100}\right) }{ k_{\mathrm{e k}} }, \quad \text{where} \tag{8} \]

\(K_{\mathrm{zh}}\) is the impulse coefficient of the oil gap at the critical steepness,

\(K_{\mathrm{zk}}\) is the safety factor under switching overvoltages, and

\[ \tau_{\mathrm{f}}=\frac{U_{\mathrm{ak}}}{a_{\mathrm{zh}}}. \tag{9} \]

Let us consider the use of the relations obtained above as applied to the available test voltages at \(50\ \mathrm{Hz}\) for insulating structures with different voltage classes. We express \(U_{\mathrm{r}50}\) in terms of the test voltage \(U_{\mathrm{isp}50}\).

\[ U_{\mathrm{r}50}= \frac{U_{\mathrm{isp}50}\cdot K_{\mathrm{z}50}} {1-\dfrac{2\sigma_{50}}{100}}, \quad \text{where} \tag{10} \]

\(K_{\mathrm{z}50}\), \(\sigma_{50}\) are the safety factor and the standard deviation at \(50\ \mathrm{Hz}\).

Let us denote the critical steepness by \(a_{\min}\), and the impulse coefficient at \(a_{\min}\) by \(K_{\min}\).

Then

\[ U_{\mathrm{ak}}= \frac{ U_{\mathrm{isp}50}\cdot K_{\mathrm{z}50}\cdot K_{\min} \left(1-\dfrac{2\sigma_{\mathrm{k}}}{100}\right) }{ K_{\mathrm{zk}}\left(1-\dfrac{2\sigma_{50}}{100}\right) }. \tag{11} \]

The relative amplitude of the minimum withstand impulse is

\[ U_{\mathrm{ak}}^{*}= \frac{ K_{\mathrm{z}50}\cdot K_{\min} \left(1-\dfrac{2\sigma_{\mathrm{k}}}{100}\right) }{ K_{\mathrm{zk}}\left(1-\dfrac{2\sigma_{50}}{100}\right) }, \tag{12} \]

where

\[ U_{\mathrm{ak}}^{*}=\frac{U_{\mathrm{ak}}}{U_{\mathrm{isp}50}}. \]

For one type of insulating medium and different voltage classes, one may expect that \(U_{\mathrm{ak}}^{*}\ne \mathrm{const}\), i.e.,

\[ U_{\mathrm{ak}}^{*}=l(i,s), \quad \text{where} \tag{13} \]

\(i\) is the voltage class,

\(s\) is the type of insulating medium.

Knowing \(U_{\mathrm{ak}}^{*}\) for any voltage class for a given insulating medium, one can determine the parameters of the minimum withstand actions of the structures under consideration:

\[ U_{\mathrm{ak}}(i,s)=U_{\mathrm{isp}50}(i,s)\cdot U_{\mathrm{ak}}^{*}(i,s) \tag{14} \]

and

\[ \tau_{\mathrm{f}}(i,s)= \frac{U_{\mathrm{isp}50}(i,s)\cdot U_{\mathrm{ak}}^{*}(i,s)} {a_{\min}}. \tag{15} \]

The minimum withstand impulse actions established in this way for various insulating media make it possible to determine the parameters of test impulse voltages satisfying the conditions of high economy of the used

insulation while ensuring the guaranteed reliability of its operation. The quantitative relations that are necessary for calculation and selection of insulation can be obtained on the basis of an analysis of the probability distributions of the shapes and multiples of switching overvoltages.

CONCLUSIONS:

1. A methodology has been considered for determining impulse test voltages that characterize the strength of insulating structures under switching overvoltages.

2. The existence has been established of critical steepnesses of linearly rising voltage that characterize the electric strength of insulating media.

3. A procedure has been proposed for determining the most dangerous impulse actions for various insulating media and different voltage classes, taking into account the substantial reduction in electric strength at critical impulse steepnesses.

4. On the basis of establishing the most dangerous impulse actions and analyzing the probability distributions of the shapes and levels of switching overvoltages, an approach has been proposed for selecting the most economical test voltages.

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