transformer engineering design and practice 1_phần 8 doc

231 6 Short Circuit Stresses and Strength The continuous increase in demand of electrical power has resulted in the addition of more generating capacity and interconnections in power systems. Both these factors have contributed to an increase in short circuit capacity of networks, making the short circuit duty of transformers more severe. Failure of transformers due to short circuits is a major concern of transformer users. The success rate during actual short circuit tests is far from satisfactory. The test data from high power test laboratories around the world indicates that on an average practically one transformer out of four has failed during the short circuit test, and the failure rate is above 40% for transformers above 100 MVA rating [1]. There are continuous efforts by manufacturers and users to improve the short circuit withstand performance of transformers. A number of suggestions have been made in the literature for improving technical specifications, verification methods and manufacturing processes to enhance reliability of transformers under short circuits. The short circuit strength of a transformer enables it to survive through- fault currents due to external short circuits in a power system network; an inadequate strength may lead to a mechanical collapse of windings, deformation/ damage to clamping structures, and may eventually lead to an electrical fault in the transformer itself. The internal faults initiated by the external short circuits are dangerous as they may involve blow-out of bushings, bursting of tank, fire hazard, etc. The short circuit design is one of the most important and challenging aspects of the transformer design; it has been the preferential subject in many CIGRE Conferences including the recent session (year 2000). Revision has been done in IEC 60076–5 standard, second edition 2000–07, reducing the limit of change in impedance from 2% to 1% for category III (above 100 MVA rating) transformers. This change is in line with the results of many Copyright © 2004 by Marcel Dekker, Inc. Chapter 6232 recent short circuit tests on power transformers greater than 100 MVA, in which an increase of short circuit inductance beyond 1% has caused significant deformation in windings. This revision has far reaching implications for transformer manufacturers. A much stricter control on the variations in materials and manufacturing processes will have to be exercised to avoid looseness and winding movements. This chapter first introduces the basic theory of short circuits as applicable to transformers. The thermal capability of transformer windings under short circuit forces is also discussed. There are basically two types of forces in windings: axial and radial electromagnetic forces produced by radial and axial leakage fields respectively. Analytical and numerical methods for calculation of these forces are discussed. Various failure mechanisms due to these forces are then described. It is very important to understand the dynamic response of a winding to axial electromagnetic forces. Practical difficulties encountered in the dynamic analysis and recent thinking on the whole issue of demonstration of short circuit withstand capability are enumerated. Design parameters and manufacturing processes have pronounced effect on natural frequencies of a winding. Design aspects of winding and clamping structures are elucidated. Precautions to be taken during design and manufacturing of transformers for improving short circuit withstand capability are given. 6.1 Short Circuit Currents There are different types of faults which result into high over currents, viz. single- line-to-ground fault, line-to-line fault with or without simultaneous ground fault and three-phase fault with or without simultaneous ground fault. When the ratio of zero-sequence impedance to positive-sequence impedance is less than one, a single-line-to-ground fault results in higher fault current than a three-phase fault. It is shown in [2] that for a particular case of YNd connected transformer with a delta connected inner winding, the single-line-to-ground fault is more severe. Except for such specific cases, usually the three-phase fault (which is a symmetrical fault) is the most severe one. Hence, it is usual practice to design a transformer to withstand a three-phase short circuit at its terminals, the other windings being assumed to be connected to infinite systems/sources (of constant voltage). The symmetrical short circuit current for a three-phase two-winding transformer is given by (6.1) where V is rated line-to-line voltage in kV, Z T is short circuit impedance of the transformer, and Z S is short circuit impedance of the system given by Copyright © 2004 by Marcel Dekker, Inc. Short Circuit Stresses and Strength 233 (6.2) where S F is short circuit apparent power of the system in MVA and S is three-phase rating of the transformer in MVA. Usually, the system impedance is quite small as compared to the transformer impedance and can be neglected, giving an extra safety margin. In per-unit quantities using sequence notations we get (6.3) where Z 1 is positive-sequence impedance of the transformer (which is leakage impedance to positive-sequence currents calculated as per the procedure given in Section 3.1 of Chapter 3) and V pF is pre-fault voltage. If the pre-fault voltages are assumed to be 1.0 per-unit (p.u.) then for a three-phase solid fault (with a zero value of fault impedance) we get (6.4) The sequence components of currents and voltages are [3] (6.5) For a solid single-line-to-ground fault on phase a, (6.6) I a0 =I a1 =I a2 =I a /3 (6.7) where Z 2 and Z 0 are negative-sequence and zero-sequence impedances of the transformer respectively. For a transformer, which is a static device, the positive and negative-sequence impedances are equal (Z 1 =Z 2 ). The procedures for calculation of the positive and zero-sequence impedances are given in Chapter 3. For a line-to-line fault (between phases b and c), (6.8) (6.9) Copyright © 2004 by Marcel Dekker, Inc. Chapter 6234 Since a three-phase short circuit is usually the most severe fault, it is sufficient if the withstand capability against three-phase short circuit forces is ensured. However, if there is an unloaded tertiary winding in a three-winding transformer, its design must be done by taking into account the short circuit forces during a single-line-to-ground fault on either LV or HV winding. Hence, most of the discussions hereafter are for the three-phase and single-line-to-ground fault conditions. Based on the equations written earlier for the sequence voltages and currents for these two types of faults, we can interconnect the positive-sequence, negative-sequence and zero-sequence networks as shown in figure 6.1. The solution of the resulting network yields the symmetrical components of currents and voltages in windings under fault conditions [4]. Figure 6.1 Sequence networks and for a double-line-to-ground fault, (6.10) Copyright © 2004 by Marcel Dekker, Inc. Short Circuit Stresses and Strength 235 The calculation of three-phase fault current is straight-forward, whereas the calculation of single-line-to-ground fault current requires the estimation of zero- sequence reactances and interconnection of the three sequence networks at the correct points. The calculation of fault current for two transformers under the single-line-to-ground fault condition is described now. Consider a case of delta/star (HV winding in delta and LV winding in star with grounded neutral) distribution transformer with a single-line-to-ground fault on LV side. The equivalent network under the fault condition is shown in figure 6.2 (a), where the three sequence networks are connected at the points of fault (corresponding LV terminals). The impedances denoted with subscript S are the system impedances; for example Z 1HS is the positive-sequence system impedance on HV side. The impedances Z 1HL , Z 2HL and Z 0HL are the positive-sequence, negative-sequence and zero-sequence impedances respectively between HV and LV windings. The zero-sequence network shows open circuit on HV system side because the zero-sequence impedance is infinitely large as viewed/measured from a delta side as explained in Chapter 3 (Section 3.7). When there is no in-feed from LV side (no source on LV side), system impedances are effectively infinite and the network simplifies to that given in figure 6.2 (b). Further, if the system impedances on HV side are very small as compared to the inter-winding impedances, they can be neglected giving the sequence components and fault current as (fault assumed on a phase) Figure 6.2 Single-line-to-ground fault on star side of delta/star transformer Copyright © 2004 by Marcel Dekker, Inc. Chapter 6236 (6.11) (6.12) Now, let us consider a three-winding transformer with an unloaded tertiary winding (HV and LV windings are star connected with their neutrals grounded, and tertiary winding is delta connected). The interconnection of sequence networks is shown in figure 6.3 (a). A single-line-to-ground fault is considered on phase a of LV winding. Since it is a three-winding transformer, the corresponding star equivalent circuits are inserted at appropriate places in the network. In the positive-sequence and negative-sequence networks, the tertiary is shown open- circuited because it is unloaded; only in the zero-sequence network the tertiary is in the circuit since the zero-sequence currents can flow in a closed delta. If the pre- fault currents are neglected, both the sources in positive-sequence network are equal to 1 per-unit voltage. The network gets simplified to that shown in figure 6.3 (b). The positive-sequence impedance is Z 1 =(Z 1 HS+Z 1H +Z 1L )//Z 1LS (6.13) where Z 1HS and Z 1LS are positive-sequence system impedances, and Z 1H and Z 1L are positive-sequence impedances of HV and LV windings respectively in the star equivalent circuit. Figure 6.3 Single-line-to-ground fault in three-winding transformer Copyright © 2004 by Marcel Dekker, Inc. Short Circuit Stresses and Strength 237 Similarly, the negative-sequence and zero-sequence impedances are given by Z 2 =(Z 2HS +Z 2H +Z 2L )//Z 2LS (6.14) Z 0 =([(Z 0HS +Z 0H )//Z 0T ]+Z 0L )//Z 0LS (6.15) The impedances Z 1 and Z 2 are equal because the corresponding positive-sequence and negative-sequence impedances in their expressions are equal. The total fault current is then calculated as I f =3/(Z 1 +Z 2 +Z 0 ) (6.16) The fault current in any of the windings is calculated by adding the corresponding sequence currents flowing in them in the three sequence networks. For example, the current in phase a of HV winding is sum of the currents flowing through the impedances Z 1H , Z 2H and Z 0H of the positive-sequence, negative-sequence and zero-sequence networks respectively. The tertiary winding current is only the zero-sequence current flowing through the impedance Z 0T . An unloaded tertiary winding is used for the stabilizing purpose as discussed in Chapter 3. Since its terminals are not usually brought out, an external short circuit is not possible and it may not be necessary to design it for withstanding a short circuit at its own terminals. However, the above analysis of single-line-to-ground fault in a three-winding transformer has shown that the tertiary winding must be able to withstand the forces produced in it by asymmetrical fault on LV or HV winding. Consider a case of star/star connected transformer with a delta connected tertiary winding, in which a single-line-to-ground fault occurs on the LV side whose neutral is grounded. If there is no in-feed from the LV side (no source on the LV side), with reference to figure 6.3, the impedances Z 1LS , Z 2LS and Z 0LS will be infinite. There will be open circuit on the HV side in the zero-sequence network since HV neutral is not grounded in the case being considered. If these modifications are done in figure 6.3, it can be seen that the faulted LV winding carries all the three sequence currents, whereas the tertiary winding carries only the zero-sequence current. Since all the three sequence currents are equal for a single-line-to-ground fault condition (equation 6.7), the tertiary winding carries one-third of ampere-turns of the faulted LV winding. As explained in Chapter 3, an unloaded tertiary winding is used to stabilize the neutral voltage under asymmetrical loading conditions. The load on each phase of the tertiary winding is equal to one- third of a single-phase/unbalanced load applied on one of the main windings. Hence, the rating of the unloaded tertiary winding is commonly taken as one-third of the rating of the main windings. In single-line-to-ground fault conditions, the conductor of the tertiary winding chosen according to this rule should also help the tertiary winding in withstanding forces under a single-line-to-ground fault Copyright © 2004 by Marcel Dekker, Inc. Chapter 6238 condition in most of the cases. This is particularly true for the case discussed previously in which the neutral terminal of one of the main windings is grounded (in this case the tertiary winding carries one-third of ampere-turns of the faulted winding). For the other connections of windings and neutral grounding conditions, the value of zero-sequence current flowing in the tertiary winding depends on the relative values of impedances of windings and system impedances in the zero-sequence network. For example, in the above case if the HV neutral is also grounded, the zero-sequence current has another path available, and the magnitude of zero-sequence current carried by LV, HV and tertiary windings depends on the relative impedances of the parallel paths (Z 0T in parallel with (Z 0HS +Z 0H ) in figure 6.3). Hence, with the HV neutral also grounded, the forces on the tertiary winding are reduced. As seen in Chapter 3, the stabilizing unloaded tertiary windings are provided to reduce the third harmonic component of flux and voltage by providing a path for third harmonic magnetizing currents and to stabilize the neutral by virtue of reduction in the zero-sequence impedance. For three-phase three-limb transformers of smaller rating with star/star connected windings having grounded neutrals, the tertiary stabilizing winding may not be provided. This is because the reluctance offered to the zero-sequence flux is high, which makes the zero- sequence impedance low and an appreciable unbalanced load can be taken by three-phase three-limb transformers with star/star connected windings. Also, as shown in Appendix A, for such transformers the omission of stabilizing winding does not reduce the fault current drastically, and it should get detected by the protection circuitry. The increase in zero-sequence impedance due to its omission is not significant; the only major difference is the increase in HV neutral current, which should be taken into account while designing the protection system. The removal of tertiary winding in three-phase three-limb transformers with both HV and LV neutrals grounded, eliminates the weakest link from the short circuit design considerations and reduces the ground fault current to some extent. This results in reduction of the short circuit stresses experienced by the transformers and associated equipment. Hence, as explained in Section 3.8, the provision of stabilizing winding in three-phase three-limb transformers should be critically reviewed if permitted by the considerations of harmonic characteristics and protection requirements. The generator step-up transformers are generally subjected to short circuit stresses lower than the interconnecting autotransformers. The higher generator impedance in series with the transformer impedance reduces the fault current magnitude for faults on the HV side of the generator transformer. There is a low probability of faults on its LV side since the bus-bars of each phase are usually enclosed in a metal enclosure (bus-duct). But, since generator transformers are the most critical transformers in the whole network, it is desirable to have a higher safety factor for them. Also, the out-of-phase synchronization in generator transformers can result into currents comparable to three-phase short circuit Copyright © 2004 by Marcel Dekker, Inc. Short Circuit Stresses and Strength 239 currents. It causes saturation of the core due to which an additional magnetizing transient current gets superimposed on the fault current [5]. Considerable axial short circuit forces are generated under these conditions [6]. The nature of short circuit currents can be highly asymmetrical like inrush currents. A short circuit current has the maximum value when the short circuit is performed at zero voltage instant. The asymmetrical short circuit current has two components: a unidirectional component decreasing exponentially with time and an alternating steady-state symmetrical component at fundamental frequency. The rate of decay of the exponential component is decided by X/R ratio of the transformer. The IEC 60076–5 (second edition: 2000–07) for power transformers specifies an asymmetry factor corresponding to switching at the zero voltage instant (the worst condition of switching). For the condition X/R>14, an asymmetrical factor of 1.8 is specified for transformers upto 100 MVA rating, whereas it is 1.9 for transformers above 100 MVA rating. Hence, the peak value of asymmetrical short circuit current can be taken as where I sym is the r.m.s. value of the symmetrical three-phase short circuit current. The IEEE Standard C57.12.00–2000 also specifies the asymmetrical factors for various X/R ratios, the maximum being 2 for the X/R ratio of 1000. 6.2 Thermal Capability at Short Circuit A large current flowing in transformer windings at the time of a short circuit results in temperature rise in them. Because of the fact that the duration of short circuit is usually very short, the temperature rise is not appreciable to cause any damage to the transformer. The IEC publication gives the following formulae for the highest average temperature attained by the winding after a short circuit, (6.17) (6.18) where θ 0 is initial temperature in °C J is current density in A/mm 2 during the short circuit based on the r.m.s. value of symmetrical short circuit current t is duration of the short circuit in seconds Copyright © 2004 by Marcel Dekker, Inc. Chapter 6240 While arriving at these expressions, an assumption is made that the entire heat developed during the short circuit is retained in the winding itself raising its temperature. This assumption is justified because the thermal time constant of a winding in oil-immersed transformers is very high as compared to the duration of the short circuit, which allows us to neglect the heat flow from windings to the surrounding oil. The maximum allowed temperature for oil-immersed transformers with the insulation system temperature of 105°C (thermal class A) is 250°C for a copper conductor whereas the same is 200°C for an aluminum conductor. Let us calculate the temperature attained by a winding with the rated current density of 3.5 A/mm 2 . If the transformer short circuit impedance is 10%, the current density under short circuit will be 35 A/mm 2 (corresponding to the symmetrical short circuit current). Assuming the initial winding temperature as 105°C (worst case condition), the highest temperature attained by the winding made of copper conductor at the end of the short circuit lasting for 2 seconds (worst case duration) is about 121°C, which is much below the limit of 250°C. Hence, the thermal withstand capability of a transformer under the short circuit conditions is usually not a serious design issue. 6.3 Short Circuit Forces The basic equation for the calculation of electromagnetic forces is F=L I×B (6.19) where B is leakage flux density vector, I is current vector and L is winding length. If the analysis of forces is done in two dimensions with the current density in the z direction, the leakage flux density at any point can be resolved into two components, viz. one in the radial direction (Bx) and other in the axial direction (By). Therefore, there is radial force in the x direction due to the axial leakage flux density and axial force in the y direction due to the radial leakage flux density, as shown in figure 6.4. Figure 6.4 Radial and axial forces Copyright © 2004 by Marcel Dekker, Inc. [...]... transformers are given in the IEC standard This method is a sort of comprehensive design review with the involvement of users and may cover review of calculations of short circuit currents and stresses for various types of faults, choice of particular type of material used, safety margins and quality control of manufacturing processes The design review at different stages of design and manufacturing is a very... choices, design margins and adequacy of manufacturing processes They have to demonstrate the validity of their dimensioning rules by reference to similar transformers having passed the test or by reference to the tests on representative models A comparison of stress and strength values of the transformer with these other transformers/models should be done The guidelines for the identification of similar transformers... combined paper and pressboard insulation system, Ep is modulus of elasticity of paper, and Eb is modulus of elasticity of pressboard The terms Lp, Lb and Leq represent thickness of paper, thickness of pressboard and total equivalent thickness of paper and pressboard respectively The eigen values (λ) are calculated [16] from the equation (6.32) where K1 and K2 are the stiffness values of bottom and top end... reliability of transformers under short circuits In order to improve the short circuit performance of transformers, the purchaser or his representative should get involved in reviewing and assessing the quality of design and manufacturing processes at few important stages during the execution of the whole contract There is a lot of scope for cooperation between the transformer manufacturer and purchaser... improvement in technical specifications and design review In many cases, the users may have confidence about the capability of manufacturers based on the design criteria limits and results of short circuit tests obtained on model coils as well as full size transformers over a period of time A standardized calculation method for the demonstration of the ability to withstand the dynamic effects of a short... be taken at the specification, design and production stages of transformers for improving the short circuit strength are described below 6.11.1 System configuration and transformer specification [1] 1 2 3 4 5 6 7 8 9 Limited extension of sub-transmission networks thereby reducing short circuit levels in the system High impedance grounding of the neutral of distribution and subtransmission networks Specification... HV side) and space, since a transformer, with one HV winding and two LV windings, substitutes two double-winding transformers of half the power rating The arrangement also results in a considerable reduction in the values of short circuit currents in the two separately supplied circuits decreasing the required rating of circuit breakers The split-winding transformers are usually step-down transformers... the stresses and displacements produced by the short circuit forces The dynamic analysis, although quite complex, is certainly desirable which improves the understanding of the whole phenomenon and helps designers to enhance the reliability of the transformers under short circuit conditions The dynamic behavior is associated with time-dependence of the instantaneous short circuit current and the corresponding... large displacements and eventual failure of transformers Hence, the dynamic analysis of mechanical system consisting of windings and clamping structures is essential and has been investigated in detail by many researchers The transformer windings, made up of large number of conductors separated by insulating materials, can be represented by an elastic column with distributed mass and spring parameters,... having mass and elasticity The applied electromagnetic forces are oscillatory in nature and they act on the elastic system comprising of winding conductors, insulation system and clamping structures The forces are dynamically transmitted to various parts of the transformer and they can be quite different from the applied forces depending upon the relationship between excitation frequencies and natural . are equal to 1 per-unit voltage. The network gets simplified to that shown in figure 6.3 (b). The positive-sequence impedance is Z 1 =(Z 1 HS+Z 1H +Z 1L )//Z 1LS (6 .13 ) where Z 1HS and Z 1LS are. winding. Design aspects of winding and clamping structures are elucidated. Precautions to be taken during design and manufacturing of transformers for improving short circuit withstand capability are. practically one transformer out of four has failed during the short circuit test, and the failure rate is above 40% for transformers above 10 0 MVA rating [1] . There are continuous efforts by manufacturers and