2344 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 13, NO. 2, JUNE 2003 Design Optimization of High-Temperature Superconducting Power Transformers Thomas L. Baldwin, John I. Ykema, Cliff L. Allen, and James L. Langston Abstract—As HTS transformers begin to move into utility and commercial applications, engineering must optimize the designs for AC losses and total ownership costs while meeting the performance criteria and application requirements. Transformer manufacturers use cost optimization techniques during the design phase to minimize material costs and satisfy the utility’s loss evaluation requirement. This paper presents the modifications necessary in transformer design and optimization to handle high-temperature superconductors as the winding material. These changes include AC losses, short-circuit stresses, and cooling loads. Design results are presented for a five-legged core, three-phase, 3.5-MVA power transformer optimized for reduced weight and footprint space for electrical distribution onboard ships. Index Terms—High-temperature superconductors, optimization methods, power transformers. I. INTRODUCTION T HE development of high-temperature superconducting power apparatus has made remarkable progress during the last several years. The advantages and potential applications of the new technology to the utility industry have been demonstrated in laboratories and with industrial prototypes . The next step leading to commercialization is developing good design techniques and manufacturing processes. The future success of HTS power apparatus depends on achieving lower AC losses and reducing the total ownership costs. Optimized design techniques play an important role in minimizing cost and losses . On the surface, transformer, design especially for small power transformers, appears to be a simple and rudimentary task. Wire sizes are chosen based on current ratings, core dimensions are selected according to peak flux densities, and so forth . While it is true that transformers can be designed using simple rules, followed by design optimization, care must be taken when migrating from copper or aluminum conductors to HTS materials. Major differences that must be accounted for include the AC loss mechanism, critical currents, and mechanical limits of the conductor. Manuscript received August 5, 2002. This work was supported in part by the U.S. Department of Defense and by the Office of Naval Research. T. L. Baldwin is with the Center for Advanced Power Systems and the FAMU-FSU College of Engineering, Tallahassee, FL 32310 USA (e-mail: [email protected]). J. I. Ykema is with SPD Technologies Inc., L-3 Communications, Philadelphia, PA 19116 USA (e-mail: [email protected]). C. L. Allen is with the Ship Systems, Nothrop Grumman Corporation, Pascagoula, MS 39568 USA (e-mail: [email protected]). J. L. Langston is with the FAMU-FSU College of Engineering, Tallahassee, FL 32310 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TASC.2003.813123 The aim of this paper is to consolidate considerations required in HTS transformer winding design and optimization from various works that have been reported in the literature. II. TRANSFORMER SPECIFICATIONS In this paper, we focus the design scope by considering a class of medium-power, three-phase transformers that are used in shipboard applications. Typical power ratings range from 0.5 MVA to 5.0 MVA, and the nominal primary voltages ratings are 4.16 kV and 6.6 kV at 60 Hz. The nominal secondary voltage ratings range from 120 V to 600 V, representing the utilization voltages. The transformers are located within ship compartments where oil-filled transformers are necessarily prohibited. Ship distribution networks use both three-leg core and five-leg core transformers. In this investigation we use a five-leg warm core with cylindrical windings in individual cryostats. The water-cooled iron core operates at room temperature. The warm-bore cryostat contains the HTS windings. Bi-2223 tape windings are cooled by two cryo-coolers for each of the three phases with conductive metal cooling strips running axially through the coils. The windings are kept under vacuum with no liquid cryogen, which could pose a safety issue in confined spaces. This cooling system permits the winding to operate over a 35 K to 80 K temperature range without substantial design changes. HTS transformer designs must be concerned about the quenching phenomena. Coil quenching may be actively utilized to provide fault current limiting. Otherwise, the design should not allow coil quenching under any expected operating condition. In this design, the operation of the transformer is not interrupted due to coil quenching by short-circuit and inrush currents. III. DESIGN CONSTRAINTS Every transformer must meet performance requirements without damaging itself or the surrounding system when abnormal events occur. Such constraints are generally nonlinear functions of the design variables. The major design constraints for a HTS transformer are discussed next. A. Coil Design We selected concentric cylindrical coils for the windings. The advantage over pancake shape windings is a higher critical current density due to a lower radial flux density of the stray magnetic field. The critical current also increases by a factor of 3.5 when the operating temperature is lowered to 35 K. Achieving 1051-8223/03$17.00 © 2003 IEEE BALDWIN et al.: DESIGN OPTIMIZATION OF HIGH-TEMPERATURE SUPERCONDUCTING POWER TRANSFORMERS Fig. 1. Critical current ratio as a function of the magnetic flux density parallel to the Bi 2223 tape surface. 2345 Fig. 3. Transformer leakage flux of a LH winding transformer. Only the lower half of the windings are shown. The winding and core are cylindrically symmetrical about the core centerline. where is the ampere-turns of the coil and is the coil length. The peak radial magnetic flux density is traditionally not considered, but can be approximated at the end of a coil by (2) Fig. 2. Critical current ratio as a function of the magnetic flux density perpendicular to the Bi 2223 tape surface. Note how rapidly the critical current decreases with increasing flux density when compared with Fig. 1. higher current densities effectively reduces the conductor AC losses. In a concentric winding, the conductor is exposed to a magnetic field oriented perpendicular to the current path. The field , flux compocontains both parallel, , and perpendicular, nents with reference to the tape’s surface. To obtain the needed current density in the Bi-2223 tape, the magnetic design must not exceed a combination of perpendicular and parallel stray flux densities as established from Figs. 1 and 2, which graph the tape’s characteristics . For much of the winding’s length, particularly near the axial center, the flux lines in the leakage channel are parallel to the core axis, as shown in Fig. 3. Near the ends of the windings, some of the flux lines turn and cross into the windings, so that the flux component at the conductor has a radial component. The tape is wound with the wider edge oriented in the axial direction as shown in the insert of Fig. 3. Consequently, the coil’s for the tape. Likewise, the coil’s radial axial field becomes . The peak axial flux density is found in the field becomes middle of the coil’s height and is approximated by (1) where is the width of the tape. The conductor losses are dependent on the transport current and the transversal magnetic fields, and are dominated by hysteresis at 60 Hz. From the self, parallel, and perpendicular magnetic fields three loss components are summed to give the total loss. The self-field loss is given by the Norris equation for a superconductor with elliptical cross-section  (3) is the critical current of the where is the frequency and tape. The loss due to the magnetic field parallel to the width of the tape is described by ; (4) is the where is the coefficient of effective conduction area, is the full penetration field flux tape cross-sectional area, and have been fitted to experimental density. Parameters and test data. The loss due to the perpendicular magnetic field is described by ; (5) is a geometrical parameter, and is a critical magwhere netic field. Table I gives values for the Bi-2223 tape. As a comparison, the current density for copper and aluminum conductors ranges from 1 to 4.5 A/mm (rms) with losses on the order of 15 to 60 mW/A-m. To be competitive, 2346 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 13, NO. 2, JUNE 2003 As faults can happen during any part of the cycle, the worst-case transient fault current is used to determine the mechanical stress. From the transformer standards , TABLE I BI-2223 REINFORCED-WIRE CHARACTERISTICS (8) - - is the open-circuit coil voltage, where - and are the real and reactive components of the complex impedance, is the transformer leakage inductance. The resulting and asymmetrical fault current is used to obtain the magnetic leakage field surrounding the coils and the resulting mechanical forces on the windings. C. Leakage Reactance the AC losses in HTS tapes operating at 77 K must be approximately 100 times smaller. The larger factor addresses the coefficient of cooling for removing the heat. B. Conductor Stresses From Large Currents Conductors may experience large electromagnetic and mechanical forces during short-circuit faults and energization conditions. Bi-2223 tapes experience a reduction in the critical current when placed under tension as indicated in Table I. The critical current is reduced by 5% at the critical stress point . The vector product of the leakage flux density and the current flowing in a conductor results in a radial force on the conductor. The inner concentric winding is compressed with a radial inward force, while the outer winding is expanded with an outward force. The outward radial force is transferred into the con, ductor as a tensile hoop stress. The maximum stress, happens in the innermost turn of the outer coil  As the ac resistance of the superconducting wire is small, the series impedance of the HTS transformer is almost entirely made up of the leakage reactance. In power system operation the impedance serves to limit the short-circuit current. The impedance is specified by the utility or by industry standards. HTS transformers will likely be designed to meet such a constraint if it lacks active fault-current limiting technology. The leakage reactance between any two windings can be calculated by finite element methods that directly solve Maxwell’s equations. In practice, simplified calculations have proven adequate, which assume that the amp-turns are uniformly distributed along the windings, and the magnetic field only contains axial components . The leakage reactance is found to be in terms of the winding dimensions, defined in Fig. 4, and number of turns, (6) (9) is the ampere-turns in the primary winding, is the where axial length of the primary coil, is the rms current in the outer coil under tension, and is the diameter of the innermost conductor turn under tension. The maximum applied stress, which may be caused by the inrush or fault current, should not approach the critical stress of the tape . Whenever a transformer is re-energized, the flux change must match the voltage according to Faraday’s law. Compounded by the residual (or remanent) flux, the core is driven strongly into saturation, requiring a very large excitation or inrush current. The inrush can be many times the normal load current, and is approximated by for a two-coil, LH arranged transformer. For a three-coil, LHL winding arrangement, we assume that the widths of the two lowvoltage windings are equal and that the two leakage channels have equal gap distances, (7) is the residual flux density, is the saturation flux where is the effective cross-sectional area of the core, and density, is the area inside the mean turn of the primary coil. For 3% silicon steel, the remanence is close to 90% of the peak . Most transformers designs normally choose flux density, in the range of 1.6 to 1.8 T, about 10% to 20% below the saturation flux density. (10) D. Cooling System The performance of the winding conductors is temperature dependent. Generally, we can achieve reduced AC losses by lowering the operating temperature, but we are penalized with an increased cooling cost for removing the heat. A comprehensive thermal design optimization is presented in , which balances the compactness of the windings with the efficiency of the cooling system. The procedure includes a heat transfer analysis for cooling load estimate and a thermodynamic evaluation for cryogenic refrigeration. An optimum operating temperature exists that minimizes the overall power consumption. The optimal temperature for conduction-cooled concentric BALDWIN et al.: DESIGN OPTIMIZATION OF HIGH-TEMPERATURE SUPERCONDUCTING POWER TRANSFORMERS 2347 the windings. The designs differ fundamentally in the winding configuration, one being a LH design and the other as a LHL design. The LHL provides a lower electrical length and reduced leakage channel flux, resulting in lower AC conductor losses. In addition, the hoop stress on a conductor is reduced considerably with the LHL design, whereas the LH design exceeds the mechanical limit, and therefore not a feasible design. V. CONCLUSION Fig. 4. Relevant dimensions of the LHL concentric windings for calculating the leakage impedance. TABLE II HTS TRANSFORMER DESIGN RESULTS The success of high-temperature superconducting power transformers depends on finding the niche opportunities where the performance is superior to conventional oil-filled or cast-coil power transformers to justify the higher costs. The changes to the design optimization presented in this paper permit tailoring HTS power transformer for special application at the lowest possible cost. A computer-aided design program implements a transformer optimization while incorporating these concepts. The techniques will be expanded to include new characteristics of HTS transformers such as fault current limiting. At present, the techniques are used to develop theoretical designs of HTS transformers for research purposes. These techniques can be applied directly to transformer manufacturing computer-aided design systems. REFERENCES coil windings is in the range of 65 K to 75 K at the cold head of the cryo-cooler. Cooler temperatures are justified, only if the amount of AC loss is substantially reduced. IV. OPTIMIZATION RESULTS Transformer design optimization is a minimization of the cost of materials and losses. The flux density, current density, mechanical force, and size limits, along with having the required leakage impedance, add constraints to the minimization. In classical transformer optimization, a designer uses four to seven independent variables. 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