Lightning impulse strength models

The wide variety of lightning stroke characteristics, together with the modifi cation effects that the line components introduce, stresses line insulation with a diversity of impulse-voltage waveshapes. While insulation coordination is based on impulse strength deter-mined for standard impulse voltages (see Section 2.6.2), it is important to be able to evaluate insulation performance when stressed by nonstandard lightning impulses. The following paragraphs summarize the main characteristics of these models and provide recommen-dations for the model to be used. The proposed approaches for predicting the dielectric strength of air gaps under lightning overvoltages usually allow the calculation of both the minimum breakdown voltage and the time-to-breakdown.

A. Voltage–time curves

They give the dependence of the peak voltage of the specifi c impulse shape on the time-to-breakdown, see Figure 2.52. Volt-time (or time-lag) curves are determined experimen-tally for a specifi c gap or for an insulator string, and may be represented with empirical equations, applicable only within the range of parameters covered experimentally [119]. In practice, measurements can be affected by several factors: impulse front shape, front times of the applied standard lightning impulse, gap distance and gap geometry, polarity, and internal impedance of the impulse generator (due to the predischarge currents in the gap).

There are special cases when the use of these curves can be advantageous. The IEEE simplifi ed backfl ash method takes advantage of the insensitivity of the time-to-breakdown to waveshapes of some insulator strings by performing overvoltage analysis at a fi xed time of 2 μs [84]. At times-to-breakdown shorter than 3 μs, the nonstandard impulse strength can be more that 10% higher than the standard impulse strength. Therefore, more

Flashover during the wave front

Flashover at the peak voltage

Flashover during the tail Volt–time characteristic

No flashover

Time

Voltage

FIGURE 2.52

Volt–time characteristic.

representative descriptions must be used for this type of application. The standard volt–

time curves do not apply to multiple fl ashover studies and their accuracy might be poor when long times-to-breakdown or low probability fl ashovers (i.e., taking place on the tail of the surge) are being studied.

B. Integration methods

Their aim is to predict insulation performance as a function of one or more signifi cant parameters of the nonstandard voltage waveshape [120–128]. Their common basic assump-tions are that

There is a minimum voltage

• V0 that must be exceeded before any breakdown can

start or continue.

The time-to-breakdown is a function of both the magnitude and the time duration

•

of the applied voltage above the minimum voltage V₀.

There exists a unique set of constants associated with breakdown for each

insula-•

tor confi guration.

In the most general formulation, different weights can be given to the effects of voltage magnitude and time. The dielectric breakdown of the insulation is obtained then from the following equation:

( )

=

∫

0

( ) 0 d

t

n

t

D v t V t (2.135)

where

t₀ is the time after which the voltage v(t) is higher than the required minimum voltage V₀ (also known as reference voltage)

t_c is the time-to-breakdown

The constant D is known as disruptive effect constant.

Different values for V₀, n, and D have been proposed, but each proposal refers to a par-ticular set of results. If n = 1, the method is know as the equal-area law, see Figure 2.53.

V

t D

V₀

t₀ t_c

FIGURE 2.53 Integration method.

Although easy to use, these methods can be applied to specifi c geometries and voltage shapes only. This law has been applied in several typical gap confi gurations of overhead lines and substations, and fi ts the test results better than other simplifi cations; however, for nonuniform gaps, the value of V0 that gives the best conformity between test results and the equal-area law does not coincide with results derived from measurements.

C. Physical models

They consider the different phases and their dependence on the applied voltage and com-pute the time-to-breakdown as the time that is necessary for completion of all the phases of the discharge process [74,127,129–131].

As detailed above, when the applied voltage exceeds the corona inception voltage, stream-ers propagate and cross the gap after a certain time if the voltage remains high enough.

The streamer propagation is accompanied by current impulses of signifi cant magnitude.

Only when the streamers have crossed the gap, the leaders can start their development.

Usually the leader velocity increases exponentially. Both the steamer and the leader phase can develop from only one or from both electrodes. When the leader has crossed the gap, or when the two leaders meet, the breakdown occurs.

Most studies have been performed with double-exponential type impulses. Although the presence of oscillations or other abrupt changes in the applied voltage does not change the overall behavior of the breakdown process, they can disturb the leader propagation, causing discontinuous breakdown development, as the voltage can fall below that neces-sary for leader propagation; the result is a discharge where the leader propagates and stops at each cycle, as modulated by the presence of the oscillations.

The time-to-breakdown can be expressed as the sum of three components:

= + +i s 

t t t t (2.136)

where

ti is the corona inception voltage

ts is the time the streamers need to cross the gap or to meet the streamers from the oppo-site electrode

tℓ is the leader propagation time

Corona phase: Corona inception occurs at time

• ti when the applied voltage has

reached a convenient value, which depends on electrode geometry, gap distance, and rate of rise of the applied voltage. This voltage can be computed and ti derived if the impulse shape of the voltage is known. In the case of gaps with nonuni-form fi eld distribution (most of the practical air insulations), the inception voltage is below the breakdown voltage, and this time can be usually neglected without introducing large errors.

Streamer propagation phase: It starts at corona inception and is completed when

•

the gap is fully crossed by streamers. The corresponding time depends on the applied voltage; the minimum value necessary for streamers to cover the whole gap is not far from the CFO when standard lightning impulses are applied. At this voltage, the time the streamers need to cross the gap is maximum, and it decreases as voltage is raised. This time is almost independent of voltage polarity, electrode confi guration, and gap clearance, but it is strongly dependent on the ratio E/E₅₀, where E is the average fi eld in the gap at the applied voltage V and E₅₀ is the

average fi eld at CFO. In almost all the proposed models, it is, however, assumed that the streamer phase is completed when the applied voltage has reached a value which gives an average fi eld in the gap equal to E50. The duration of the streamer phase, ts, can be estimated as follows [74]:

= −

s 50

1 1.25 E 0.95

t E (2.137)

For geometries different from the rod–plane confi guration, where two stream-ers are present, the resulting velocity of the streamer has to be interpreted as the average velocity of an equivalent streamer that would cross over the same gap in the same time but starting from one electrode only.

Leader phase: The length of the leader at the end of

• t_s is a small percentage of the

total gap length. The propagation of the leader is described in terms of the instanta-neous values of its velocity. The leader propagation time is generally calculated from its velocity, which depends on the applied voltage and the leader length. Various expressions have been proposed for the leader velocity. The experimental evidence shows that the leader velocity increases proportionally to the length of the gap which is not yet covered by the leader, and can be described by the following equation:

⎛ ⎞

v(t) is the actual (absolute value) voltage in the gap, which does not generally correspond to the theoretical value, due to the current fl owing into the circuit.

For all confi gurations, the following equation has been proposed for calculation of the leader velocity [130]

⎡ ⎤

A simpler approach had been proposed in [131]

⎡ ⎤ and insulator type, see Table 2.11.

Leader current: During the discharge development a current fl ows in the circuit so

•

that the voltage applied at the gap is not the voltage from the unloaded generating

circuit. The value v(t) has to be the actual voltage applied to the gap, due to the volt-age drop caused by current fl owing during leader propagation. The actual applied voltage can be determined when the circuit characteristics and the current fl owing during the leader propagation are known.

The physical model is valid for a large variety of impulse shapes and can be used in the evaluation of dielectric strength of a variety of geometries. The leader progression model (LPM) shown in Equation 2.140 has proved to have adequate accuracy for most calculations. The integration methods have comparable accuracies but more restricted application in relation to waveshapes. The empirical methods can give a good accuracy when they are used within their validity limits (i.e., when specifi c data are used for a spe-cifi c insulator or gap, together with a careful application of the model). The use of volt–time curves works well in the short time-to-breakdown domain (2–6 μs). Based on the above, it can be concluded that no single approach alone can be recommended for all applications.