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Transient phenomena have probably provided transformer design engineers with their most interesting and stimulating challenge. For many years the very elusiveness of the subject coupled with the difficulties often met with in reproducing in the laboratory or test room the identical conditions to those which occur in practice undoubtedly provided the most significant aspect of that challenge. Until the advent of computers quantitative calculations were often very difficult since, under extremely abnormal conditions (for instance, when dealing with voltages at lightning frequencies and with supersaturation of magnetic circuits), the qualities of resistance, inductance and capacitance undergo very material temporary apparent changes compared with their values under normal conditions. A considerable amount of connected investigation has been carried out on transient phenomena of different kinds, by many brilliant investigators, and it is largely to these that we owe our present knowledge of transients.
A number of individual papers have been presented before technical engineering institutions in the UK, the USA and Europe, and these have formed valuable additions to the literature of the subject. We cannot hope, in a volume of this nature, to cover anything approaching the whole field of the subject, but we have here endeavored to present a brief survey of the chief disturbances to which transformers are particularly liable.
The transients to which transformers are mainly subjected are:
• Impact of high-voltage and high-frequency waves arising from various causes, including switching in.
• System switching transients with slower wavefronts than the above.
• Switching in current rushes.
• Short-circuit currents.
It is not intended to discuss specifically the results of faulty operations, such as paralleling out of phase or the opening on load of a system isolator link, as the resulting transients would be of the nature of one or more of those mentioned above.
Impact of high-voltage and high-frequency waves
Transformer windings may be subject to the sudden impact of high-frequency waves arising from switching operations, atmospheric lightning discharges, load rejections, insulator flashovers and short circuits, and, in fact, from almost any change in the electrostatic and electromagnetic conditions of the circuits involved. An appreciable number of transformer failures occurred in the past, particularly in the earlier days of transformer design, due to the failure of inter turn insulation, principally of those end coils connected to the line terminals, though similar insulation failures have also occurred at other places within the windings, notably at points at which there is a change in the winding characteristics. The failures which have occurred on the line-end coils have been due chiefly to the concentration of voltage arising on those coils as a result of the relative values and distribution of the inductance and capacitance between the turns of the coils.
In the early stages when these breakdowns occurred, considerable discussion took place on the relative merits of external protection in the form of choke coils and reinforced insulation of the end coils, but actual experience with external choke coils showed that in many cases their provision did not eliminate the necessity for reinforcement of the end coils, while, on the other hand, added reinforcement of the end coils was itself occasionally still subject to failure, and more frequent breakdown of the interturn insulation occurred beyond the reinforcement. For many years there was in use in the UK a British Standard, BS 422, which provided recommendations for the extent of reinforcement of end turns of higher-voltage windings. Now external protection is provided by means of co-ordinating gaps or surge diverters coupled with the use of insulation co-ordination and winding design has developed to a stage at which more effective measures are available than reinforcement of end turns. This development followed on from a fuller under standing of the response of windings to high-frequency transients and a recognition of the part played by capacitances at these high frequencies.
The following description of the effect of lightning impulses on transformer windings is based on material contained in a book 'Power Transformers for High Voltage Transmission with Special Reference to their Design' by Duncan McDonald (now the late Sir Duncan McDonald), formerly Chief Designer, Transformer Department, Bruce Peebles, published by Bruce Peebles and Company Limited.
L _ Inductance Cs _ Series (turn-to-turn) capacitance Cg _ Shunt (turn-to-ground) capacitance rL _ Loss component of inductance (winding resistance) rs _ Loss component of series capacitance r _ Loss component of shunt capacitance
In assessing its surge behavior the transformer may be represented by an equivalent network possessing capacitance, inductance and resistance elements as shown in FIG. 52. The voltage response of the transformer, the space distribution of potential through its windings at any instant of time, is a function of the magnitude and disposition of these circuit elements and of the nature of the incident voltage. In practice impulse voltages are characterized by a rapid rise to their crest value followed by a relatively slow decline to zero -- by a front of high and a tail of low equivalent frequency. The steeper the front and the flatter the tail of the wave, the more severe its effect on the windings and for this reason, coupled with the analytical convenience and clearer understanding of the principles involved, it is convenient to regard the incident impulse voltage as a unit function wave having a front of infinite and a tail of zero equivalent frequency.
The orthodox explanation of the transient behavior of the windings is based on the time response of the circuit elements to these equivalent frequencies.
At the instant of incidence of the impulse the capacitance elements alone react to the front of the wave establishing an initial distribution of potential which is usually non-uniform (FIG. 53(a)). At the end of the phenomenon, during the tail of the wave, the resistance elements govern the response establishing a final distribution which is usually uniform (FIG. 53(a)). The transitional behavior between the initial and final extremes takes the form of damped transference of electrostatic and electromagnetic energy during which complex oscillations are usually developed (FIG. 53(b)). It can be seen that all parts of the winding may be severely stressed at different instants in time; initially, concentrations of volt age may appear at the line end of the winding; during the transitional period, concentrations may appear at the neutral end whilst voltages to ground considerably in excess of the incident impulse may develop in the main body of the winding.
Generally, under steady-state conditions, equal voltages are induced between turns and consequently, ideally, equal amounts of insulation are required between turns. To utilize this uniformly disposed insulation to best advantage, the volt ages appearing between turns throughout the winding under impulse conditions should also be equal. To approach this ideal, in which oscillation voltages are completely eliminated, the initial distribution, like the final, must also be uniform. Unfortunately, for many years, the basic theories and practice always showed that uniform windings, with a uniform final distribution, inherently exhibited a grossly non-uniform initial distribution. Faced with this paradox, the designer has concentrated in determining by what artifice he might improve the initial distribution whilst striving to maintain winding uniformity.
The initial voltage distribution It will be recalled that the initial distribution is determined wholly by the equivalent capacitance network. Consequently two circuit elements are avail able for controlling and improving the initial response -- the shunt capacitance Cg and the series capacitance Cs. When a unit function wave is applied to the line terminal of a winding whose equivalent network is shown in FIG. 54, the initial distribution of impulse voltage is determined from the differential equation of the capacitance network (FIG. 54(a)). This equation may (for a uniform winding of length L, of uniform interturn capacitance Cs, of uniform turn-to-ground capacitance Cg) be expressed in terms of the instantaneous volt age to ground ex at any point x (measured from the neutral terminal) as:
Solution of Eq. (eqn. 49) may be found in the form:
where the constants of integration A and B are defined by substituting the boundary conditions. In particular, if the winding neutral is solidly grounded ex _ 0 when x _ 0 and from Eq. (eqn. 50)
In addition, ex _ E, the incident surge, when x _ L and, from Eq. (eqn. 50) E _ A(ea _ e_a)
Substituting this value of A in equation (eqn. 51) it is seen that
The former is not very practicable and it is therefore the latter approach which has formed the main strategy for improving the response of transformer windings to lightning surges. Section 4.4 describes the methods which have been developed for increasing the series capacitance for practical windings.
The use of interleaving, which is now one of the most common methods of increasing series capacitance in fact enables near uniform initial distributions to be obtained thus achieving the ideal of utilizing the same uniform interturn insulation structure for both impulse and steady-state withstand. It should be stressed, however, that winding design is a matter of economics and not necessarily one of achieving ideals. Interleaving is an expensive method of winding and where acceptable stress distributions can be obtained without recourse to this method, say, by the use of shields between end sections, designers will always prefer to do this. As unit ratings get larger there will be a tendency for Cg to get smaller relative to Cs anyway due to increase in physical size and increased clearances. In addition, the volts per turn will be greater in a larger unit so that the total number of turns, and hence number of turns per section, will be reduced. The next most critical area after the line-end interturn stress is usually the stress between the first two sections. A reduction in the number of turns per section will help to reduce this. These factors usually mean that impulse stress in a large high-voltage transformer is less than that in one of lesser rating but having the same rated voltage. In the smaller rated unit inter leaving might be essential, whereas for the larger unit it will probably be possible to avoid this.
The final voltage distribution
The form of the final voltage distribution can be calculated in a similar manner to that for the initial distribution. For an incident wave with an infinite tail the capacitance and inductance elements of FIG. 52 appear respectively as open and short circuits and the resulting final distribution is governed wholly by the resistive elements. It will be seen that these resistive elements form a network identical to that of the capacitance network (FIG. 54(a)) if Cs is replaced by
The differential equation for this network may therefore be written:
The solution of this equation, which is of the same form as Eq. (eqn. 49) is clearly given by:
which is a uniform distribution of potential from line to ground.
As in any network consisting of inductances and capacitances, transformer windings are capable of oscillatory response to certain incident disturbances.
When the disturbance has the appropriate properties severe dielectric stresses and, on occasions, failure can result.
In the discussion above relating to lightning impulses the incident disturbance is a once-only occurrence. The oscillatory circuits receive a single burst of energy and return by free oscillations at their natural frequencies to a steady state. Since in most cases the maximum voltage developed in the transformer windings occurs during the first one or two oscillations, the natural frequency and damping of the oscillatory circuits are of only secondary importance.
In contrast, however, certain switching transients may consist of an initial peak voltage followed by an oscillatory component. If the frequency of this oscillation coincides with a natural frequency of the windings a resonance can develop which can take several cycles to reach its maximum amplitude.
The value of this maximum amplitude is dependent on the damping of both the incident transient and of the windings themselves but it can on occasions be greater than the voltage resulting from a lightning impulse. It should be recognized that, unlike the case of designing in resistance to lightning impulses, the solution to resonance problems cannot be achieved by transformer manufacturers acting in isolation. Resonance always requires a passive structure, namely the transformer windings, and an active component represented by the various sources of oscillating voltages in the electrical system.
Resonances became recognized as a cause of dielectric failures in the early 1970s and a number of technical papers dealing with the subject were published over the next decade. The majority of these described specific incidents which had led to the failure of EHV transformers and although the mechanism of failure was ascribed to resonance phenomena the papers generally pro vided little information concerning the source and the nature of the initiating disturbance. In 1979 CIGRE set up a Working Group to deal with resonance problems and to report on the state of the art, including the provision of a description of the response of transformers to oscillating voltages and making a survey of the possible sources of oscillating voltages in electrical systems. The Working Group's findings were presented at the August/September, 1984, session [6.6] and the following notes represent a summary of the salient points from their report. Only power transformers above 110 kV were considered and furnace and other special transformers were excluded. The Working Group also noted that their findings were in line with those of an American IEEE working group dealing with the same subject.
The report described studies carried out on a 405/115/21 kV, 300 MVA sub station transformer having tappings of _13 _ 4.675 kV per step on the HV winding. The arrangement is shown in FIG. 55. The application of a step voltage on an arbitrarily chosen terminal of the transformer will cause the 'net work' to oscillate. In principle, since the step function contains all frequencies, each natural frequency inherent in the network will be excited. The values of the frequencies and their related amplitudes depend on the parameters of the network and the boundary conditions. The total number of natural frequencies is given by the number n of free nodes. The values of the amplitudes are also a function of the location, that is of the ordering number j of a node. Because of the presence of resistances the oscillations are more or less damped, so finally the response at an arbitrary chosen node j to a step voltage US is (disregarding slight phase shifts) given by:
Aj00 describes the final steady-state voltage distribution a is the damping constant.
The response to a standard 1.2/50 full wave (FIG. 55(a)) is similar to the response (eqn. 56) to a step voltage. FIG. 55(b) shows the voltage generated at the free oscillating end of the tapped winding (node m) under the given boundary conditions. From the oscillogram (FIG. 55(b)) it can be seen that there is a dominant natural frequency f i
of about 40 kHz, the related amplitude Ai*,m has a value of about 0.2 per unit. Application of a steady-state sine wave (FIG. 55(c)) with an amplitude UR (1 per unit, that is 420 _ _2/_3) and a frequency fi
* causes resonance and, according to oscillogram (FIG. 55(d)), a volt age with a maximum peak of ARi*,m of 3.85 _ UR is generated at node m. The amplitude is limited to this value due to the inner damping of the transformer d.
The report gives two methods of determining the degree of this internal damping. The first is to vary the frequency of the applied voltage and to make a second measurement, for instance at 0.9 _ f i*. FIG. 56 shows a plot of maximum peak amplitude against frequency. From a measured second peak of 1.04 per unit a ratio of 1.04/3.85 _ 0.27 is derived and plotting this on the curve of FIG. 56 gives a value for d of 0.85, which is stated by the report to accord with the value which can be estimated from the impulse response.
The second method makes use of another mathematical relationship which can be used to determine the ratio:
Values of vmax are shown in FIG. 57 plotted for varying values of inner damping, d, and external damping, that is the damping of the applied oscillatory voltage, ?. For a value of vmax of 3.85/0.2 _ 19.25 and ? _ 1 (i.e. no external damping) it can again be seen that this gives a value of d _ 0.85. FIG. 57 also permits the time, expressed as number of cycles t at which the maximum amplitude occurs to be determined.
Formula (eqn. 57) also enables a comparison to be made of the voltage stresses in the case of resonance with those generated during impulse testing, especially if a natural frequency is dominating the impulse voltage response. Comparing oscillograms (FIG. 57(b)) and (FIG. 57(d)) reveals that the voltage generated during undamped resonance conditions is significantly higher than that under impulse test conditions. In order that this stress should not be exceeded by a resonance excitation of the same amplitude (i.e. 1 per unit) external damping ? must be less than 0.9. Hence, from the curve of FIG. 57 vmax _ 8.7. In this case the resonance voltage would be:
which is in good agreement with oscillogram (FIG. 57(f )).
Another way of describing the resonance response is the comparison with the voltage under rated conditions. This relationship is quantified by the so-called q-factor.
is the actual turns ratio of the node j.
For the example for which rm _ 0.2, a q-factor of qi*,m _ 19.25 can be calculated. In this case either vmax or the q-factor permit the estimation of the maximum stresses developed under resonance condition at node m. However, as the dominating natural frequency of the regulating winding also influences the other windings it might be inferred that a resonance identified on one arbitrary node on the tapping winding might be indicative of other high stresses in other windings. FIG. 58 shows the calculated spatial distribution of the amplitudes Ai*, j throughout the HV winding. Although the q-factors for volt ages to ground are rather moderate or even zero, high q-factors result between certain parts of the winding. From the gradients of the spatial amplitude distribution q-factors of up to 22 can be calculated and such values have indeed been reported.
In summary, the Working Group came to the following conclusions as regards resonances within transformer windings:
• There is a close interdependence between impulse voltage response and resonance response.
• Amplitudes of harmonics can therefore to a certain degree be influenced by controlling the initial voltage distribution of a standard impulse wave for individual windings.
• There must be an awareness of the fact that transferred oscillations from other windings cannot be suppressed and may cause severe stresses.
• Internal damping is a decisive factor on the resonance response.
• q-factors may be misleading and should not be used in assessing transformer behavior.
Determination of resonance response
The Working Group considered that a very detailed analysis was necessary to get precise information about the resonance of a particular transformer. Three different approaches are possible - calculation, measurement or a combination of the two. The calculable number of harmonics depends on the degree of subdivision of the equivalent network. To get sufficient information about the spatial amplitude distribution demands a large number of elements. Elaborate computer programs have to be used but the accuracy of the results still depends on the validity of the parameters inserted. From FIG. 56 it can be seen that a deviation from the resonance frequency of only a few percent, considerably less than the margin of error in many instances, can affect the apparent amplitude by a large amount. In addition, at the present time there are no exact methods available for determining damping factors, the computation has to be based on empirical values and is therefore of limited accuracy. On the other hand to obtain a full assessment of the resonant response from measurements is very laborious, costly and even risky.
There is the problem of making tappings on inner windings, and measurements taken out of the tank are inaccurate due to the difference in the permittivities of air and oil. Hence, the compromise solution of performing a calculation and checking this by means of measurements taken at easily accessible points may prove to be the best option. Even this approach will be costly and should be adopted only if it is considered that a problem may exist.
Sources of oscillating voltages in networks
The Working Group also reported their conclusions concerning the sources of oscillations in networks. They found that their existence stems from one of three possible sources:
Oscillations created by lightning need only be considered if this causes a switching operation or triggers a fault. Faults comprise single phase to ground faults and two- or three-phase short circuits with or without ground fault involvement. Switching may be initiated by the operator or automatically by the sys tem protection.
The Working Group investigated 21 categories of incidents including remote and close-up faults, clearance of faults, reclosing onto faults, energization of a transformer-terminated line, de-energization of an unloaded and loaded transformer, with and without pre-strikes or re-ignitions as appropriate. Their analysis revealed that in only three of these categories was there a likelihood of oscillations which might coincide with a natural frequency of the transformer.
• Polyphase close-up faults on a single line.
• Energization of a short transformer-terminated line from a strong bus.
• Repetitive re-ignitions during the de-energization of a transformer loaded with a reactive load.
These are defined as occurring at a distance of less than 15 km from the transformer, while the line itself is considerably longer. The transformer is likely to be struck by a dangerous oscillatory component only in those cases where one line is connected to the transformer (FIG. 59) and a two- or three-phase fault occurs at the critical distance l, given by:
where c is the velocity of the travelling wave, which is about 300 km/ms for overhead lines and 150 km/ms for cables.
Energization of a transformer-terminated line
Switching in a short line through a circuit-breaker fed from a strong busbar (FIG. 60) creates standing waves which can be within the critical frequency range. Their frequency can be calculated from Eq. (eqn. 59), where l corresponds to the length of the line.
Breaking of small inductive currents ( 1 kA), in particular the interruption of magnetizing currents of transformers, may cause oscillations, but these are in the kHz range and strongly damped, therefore these do not create a risk of resonance. The interruption of reactive loaded transformer currents can cause repetitive re-ignitions at nearly constant time intervals. If the repetition frequency coincides with one of the lower natural frequencies of the transformer, resonance may result. A typical configuration for which this can happen is the case of an unloaded three-phase transformer with a reactor connected to the tertiary winding.
Very fast transients
The majority of switching transients occurring on the system will have slower wavefronts and lower peak voltages than those resulting from lightning strikes and will therefore present a less severe threat to the insulation of the transformer HV windings. The exceptions are certain transients which can arise as a result of switching operations and fault conditions in gas-insulated substations (GIS). These are known as very fast transient overvoltages (VFTOs). The geometry and dielectric of GIS (metallic sheath, coaxial structure and short dielectric distances) lend themselves well to the generation and propagation of VFTOs. Studies of the characteristics of VFTOs have indicated that typically these might have rise times of 20 ns and amplitudes of 1.5 per unit. In the worst condition a rise time of 10 ns and an amplitude of 2.5 per unit is possible. The steep fronted section of the wave is often followed by an oscillatory component in the frequency range 1-10 MHz, the precise value being dependent on the travelling wave characteristics of the GIS system.
The VFTOs arriving at a transformer winding are more difficult to predict since magnitude and front-time depends on the transformer parameters and the precise nature and length of the winding connection to the GIS. In the worst case, the front-time will be only slightly increased and the amplitude increased by possibly 30 percent.
When the VFTOs reach the transformer windings two problems can arise.
Their very much higher frequency compared with standard impulse waves results in high intersection stresses which are usually concentrated in the sections near to the line end. These stresses cannot be controlled by interleaving in the same way as can lightning impulse stresses. The second problem is the production of part-winding resonance resulting from the oscillatory wavetail of the VFTO. This can create oscillatory voltages within the end sections of the transformer windings, producing intersection stresses many times greater than those resulting from lightning impulses.
In attempting to predict the response of transformer windings to VFTOs it is necessary to represent the winding structure in a similar way to that employed in performing calculations of impulse voltage distribution, in that capacitances predominate, however it is no longer sufficient to consider a simple network having constant values of series and shunt capacitance, Cs and Cg, respectively.
In their paper presented to the 1992 CIGRE Summer Meeting, Cornick and others used multiconductor transmission line theory to produce a turn by turn mathematical model of a 40 MVA 220 kV partially interleaved winding.
FIG. 61 shows the type of network considered, in which the capacitance of each turn is taken into account and, in order to predict the resonant frequencies, they also take into account the inductance network. Though laborious, the method lends itself well to computer calculation and, because it is the end sections which are known to be critical, computing time can be reduced by restricting the solution to the end, say the first four, sections of the winding.
The authors compared their predicted intersection and interturn voltages with measurements made on the actual winding following application of the output from a recurrent surge generator producing front chopped impulses. These had a prospective front-time of 1.2 µs chopped at that time. Voltage collapse time was 230 ns, relatively slow for a VFTO. FIG. 62 shows the arrangement of the winding end sections and the applied, predicted and measured intersection voltage between the end two sections. The resonance frequency predicted by calculation was 2.12 MHz and that obtained by measurement 2.22 MHz, considered by the authors to represent good agreement.
Apart from noting the high level of the intersection voltages observed, the above paper makes little general recommendation regarding the need for protection when connecting transformers directly to GIS, or the form which any protection might take. The authors do note, however, that the use of inductances, suitably damped, in series with the transformer windings might be justified in specific cases.