Experimental and Mathematical Modeling Studies on Current Distribution in High Temperature Superconducting DC Cables Venkata Pothavajhala

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2014 Experimental and Mathematical Modeling Studies on Current Distribution in High Temperature Superconducting DC Cables Venkata Pothavajhala

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COLLEGE OF ENGINEERING

EXPERIMENTAL AND MATHEMATICAL MODELING STUDIES ON CURRENT

DISTRIBUTION IN HIGH TEMPERATURE SUPERCONDUCTING DC CABLES

VENKATA POTHAVAJHALA

A Thesis submitted to the Department of Electrical and Computer Engineering in partial fulfillment of the requirements for the degree of Master of Science

Degree Awarded: Summer Semester, 2014

Venkata Pothavajhala defended this thesis on June 24, 2014. The members of the supervisory committee were:

Chris Edrington Professor Directing Thesis

Lukas Graber Committee Member

Petru Andrei Committee Member

The Graduate School has verified and approved the above-named committee members, and certifies that the thesis has been approved in accordance with university requirements.

I dedicate this work to my parents and to my research supervisor

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ACKNOWLEDGMENTS

I would like thank Professor Chris S. Edrington for being an excellent academic advisor and guiding me through the MS program and helping me choose my course work and planning my research. I would like to express my deepest appreciation and sincere gratitude to my research supervisor Dr. Sastry Pamidi for providing me the opportunity to work under his guidance and for his continuous support throughout my research. It is a pleasure working under such a great scientist. I’m thankful to him for spending the time for our weekly discussions though he is very busy with other projects, proposals and meetings. He is always ready to help students and colleagues professionally and also personally in all possible ways. I am greatly thankful to Dr. Lukas Graber who provided much support for my research. He also gave me many tips and suggestions on how to work and present in a professional way. I thank him for spending time to meet almost every week to discuss my research. Dr. Chul H. Kim is another person who continuously supported me right from the beginning of my work at CAPS. I express my sincere appreciation to him for training me in the laboratory on the experimental techniques. I understand that it is not easy to teach every time a new student joins the group to make him/her understand how things work. Dr. Kim was very patient in doing so. He also spent much time every week in helping me to present my work in a useful and understandable way. It would not be possible for me to successfully publish in the IEEE Transactions on Applied Superconductivity without the strong help and guidance from all the above mentioned researchers. I also strongly believe that the applied superconductivity and cryogenics group at CAPS is a great place to do research and grow professionally and personally. I am thankful for the Office of Naval Research and Center for Advanced Power Systems for providing me the funding and state of the art research facilities and wonderful opportunities to learn about latest research in electrical power technologies. I wholeheartedly thank my parents for providing me the opportunity to pursue Masters. Especially, I’m grateful to my mother who provided moral support and motivation required for my study and research work. It’s her constant encouragement that helped me in moving towards my goal.

TABLE OF CONTENTS

LIST OF TABLES ...... vii LIST OF FIGURES ...... viii ABSTRACT ...... x 1. INTRODUCTION ...... 1 1.1 Discovery of Superconductivity ...... 1 1.1.1 The Meissner effect...... 2 1.1.2 Electrodynamics and thermodynamics of superconductivity ...... 3 1.1.3 Quantum mechanics of superconductivity ...... 4 1.2 Types of Superconductors ...... 6 1.2.1 Low temperature superconductors ...... 6 1.2.2 High temperature superconductors ...... 6 1.2.3 First generation (1G) superconductors...... 8 1.2.4 Second generation (2G) superconductors ...... 8 2. APPLICATIONS ...... 10 2.1 Digital Electronics ...... 10 2.2 Medical Applications ...... 10 2.3 Electric Power ...... 11 2.3.1 Machines ...... 11 2.3.2 Power transformers ...... 11 2.3.3 Fault current limiters ...... 12 2.3.4 Magnetic energy storage ...... 12 2.3.5 Magnetic levitation trains ...... 12 2.4 Particle Physics ...... 13 2.5 Fusion Technology...... 13 2.6 Space Applications...... 14 3. SUPERCONDUCTING POWER CABLES ...... 15 3.1 Overview ...... 15 3.2 Research Objective ...... 18 4. RESULTS OF CABLE MEASUREMENTS ...... 21 4.1 Measurement of Critical Current and Index Value of Tapes in 2G HTS Cable ...... 21 5. EXPERIMENTS ON DISTRIBUTION OF CURRENT IN 2G HTS CABLES ...... 23

5.1 Experimental Procedure ...... 23 5.1.1 Measurement of critical current ...... 24 5.1.2 Measurement of index (n) value ...... 24 5.1.3 Measurement of contact resistances...... 24 5.1.4 Measurement of current through individual tapes ...... 25 5.2 Mathematical Model ...... 25 5.3 Simulations ...... 27 5.3.1 Effects of variation of contact resistance ...... 27 5.3.2 Effects of variation of critical current ...... 28 5.3.3 Effects of variation of index value ...... 29 5.4 Analysis Using Monte Carlo Simulations...... 30 5.4.1 Introduction to Monte Carlo ...... 30 5.4.2 Monte Carlo simulations ...... 31 5.5 Discussion of Results ...... 33 6. EFFECT OF LONGITUDINAL VARIATIONS IN CHARACTERISTICS OF INDIVUDUAL TAPES ON THE PERFORMANCE OF SUPERCONDUCTING CABLE ...... 34 6.1 Longitudinal Variation in Critical Current ...... 35 6.2 Longitudinal Variation in Index Value ...... 38 6.3 Results and Discussion ...... 39 7. CONCLUSION ...... 41 8. FUTURE WORK / OUTLOOK ...... 42 APPENDIX A. MEASUREMENT METHOD FOR DC CRITICAL CURRENT OF SUPERCONDUCTING POWER CABLES ...... 43 Terms and Definitions...... 43 Critical temperature ...... 43 Critical current ...... 43 Index (n)-value ...... 43 Procedure to Determine Critical Current and Index Value ...... 43 APPENDIX B. COPYRIGHT PERMISSION ...... 45 REFERENCES ...... 46 BIOGRAPHICAL SKETCH ...... 59

LIST OF TABLES

Table 1: Discovery of superconducting materials ...... 6

Table 2: High temperature superconducting materials and their critical temperatures ...... 8

Table 3: Ic and n values of several segments of tapes taken from HTS cable ...... 21

Table 4: Measurement of critical current and index value of sample HTS tapes manufactured by SuperOX and Sunam...... 22

Table 5: Characteristics of tape used in the experiment ...... 25

Table 6: Mean and standard deviation of contact resistances, Ics and n values ...... 32

Table 7: The value of coefficient (z) for different confidence levels ...... 37

Table 8: Confidence interval of Icm for different standard deviations ...... 37

Table 9: The ratio of apparent critical current to mean critical current of the cable for different standard deviations of tape Icis ...... 40

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LIST OF FIGURES

Figure 1: Sudden change in resistance (on y axis in Ω) of mercury with decrease in temperature (on x axis in K) ...... 2

Figure 2: Superconductor in magnetic field (a) above critical temperature and (b) below critical temperature ...... 2

Figure 3: (a) Magnetic field (H) versus Temperature (T), 3(b) Representation of Magnetic moment (I) versus applied field (H) for type 1 and type 2 superconductors...... 4

Figure 4: Current versus voltage characteristic of thin film Josephson junction ...... 5

Figure 5: Schematic representation of SQUID, formed with two identical Josephson junctions ... 5

Figure 6: Evolution of superconducting transition temperatures ...... 7

Figure 8: Design configurations of HTS cable ...... 17

Figure 10: (a) Vertical and (b) Horizontal views of superconducting cable (unwrapped) ...... 21

Figure 11: Experimental setup ...... 23

Figure 12: Equivalent circuit of the experimental setup. Tape1, 2, and 3 are 2G HTS tapes and a copper tape are in a parallel network. Sh1, Sh2, Sh3, Sh4 are calibrated shunt resistors for current sensing...... 23

Figure 13: Sample superconducting cable with the solder joints enlarged ...... 24

Figure 14: Equivalent circuit model of the setup in Fig. 11, with the values of the parameters obtained from the experiment...... 26

Figure 15: Comparison of the total current vs. individual tape currents obtained through the experiments and the mathematical model...... 27

viii

Figure 17: Individual tape currents for different Ic: 83.16 A, 87.03 A, 92.87 A, 95.29 A, 99.12 A, 100.81 A, 103.26 A, 112.82 A; n = 30 for all tapes; all tapes have equal contact resistance of 50 nΩ ...... 29

Figure 18: Current distribution for different n values of tapes: 33.8, 31.4, 22.1, 32.5, 26.7, 32.1, 29.9 and 33.9. The contact resistance of each tape was equal to 50 nΩ and the Ics were equal to 100 A ...... 29

Figure 19: Schematic of some decisions made to generate the history of an individual neutron in a Monte Carlo calculation...... 31

Figure 20: Number of cables with at least one of their tapes exceeding their given fractional Ic at (a) 85% (b) at 90% design Ic of the cable, number of tapes which exceeded (c) 95% and (d) 98% of their Ic at 90% cable design Ic...... 32

Figure 21: A model tape in a HTS cable divided into several 1 cm long sections with varying Ic values...... 35

Figure 22: Distribution of critical current of tape sections ...... 36

Figure 23: Percentage decrease in apparent Ic for certain longitudinal variation in Ic at different average Ics ...... 38

Figure 24: Percentage decrease in apparent n for certain longitudinal variation in n ...... 39

Figure 25: The apparent Ic of the cable as function of cable mean Ic for a certain standard deviation in Ici along the length of individual tapes of the cable...... 39

ABSTRACT

High temperature superconducting power cables have the advantage of high current density and low losses over conventional cables. One of the factors that affect the stability and reliability of a superconducting cable is the distribution of current among the tapes of cable. Current distribution was investigated as a function of variations in contact resistance, individual tape critical current (Ic), and index (n)-value of individual tapes. It has been shown that besides contact resistances, variations in other superconducting parameters affect current distribution. Variations in critical current and n-value become important at low contact resistances. The effects of collective variations in contact resistances, individual tape critical current, and n-value were studied through simulations using Monte Carlo method. Using an experimentally validated mathematical model, 1000 cables were simulated with normally distributed random values of contact resistances, individual tape critical current, and n-value. Current distribution in the 1000 simulated cables demonstrated the need for selecting tapes with a narrow distribution in the superconducting parameters to minimize the risk of catastrophic damage to superconducting cables during their operation. It has been demonstrated that there is a potential danger of pushing some tapes closer to their critical current before the current in the cable reaches its design critical current. Mathematical models were also used to study the effect of longitudinal variations in the tape parameters on superconducting cable using Monte Carlo simulations. Each tape of a 30 meter long, 3 kA model cable with 30 tapes was considered to have longitudinal variations in Ic, and n values for every 1 cm section, thus generating particular standard deviation in Ic and n for all 3000 sections of each tape. The results indicate that the apparent critical current and index value (n) of the cable is reduced by a certain percentage depending upon the extent of variation in the characteristics along the length of the tapes.

CHAPTER ONE

INTRODUCTION 1.1 Discovery of Superconductivity

In the year 1882, Heike Kamerlingh Onnes-a Dutch physicist was appointed as a professor in experimental physics at the University of Leiden [1]. This marked the beginning of low temperature physics. He established an ambitious program of research on the properties of gases. Johannes van der Waals was his mentor who developed a theory called the Law of Corresponding States, which explains the behavior of gases. To confirm the predictions of van der Waal’s theory, Onnes had to study gases over a wide range of temperatures. Onnes started with diatomic gases like oxygen, nitrogen and hydrogen but these gases had very low critical temperatures and even low boiling points: 90.15 K for oxygen and 20.15 K for hydrogen. So he started to build a cryogenic laboratory in which the Cailletet compressor was the major piece of apparatus acquired in 1884 and it had to undergo many changes to make it suitable for scientific use [2]. Oxygen was first liquefied at Leiden in 1892 and hydrogen was liquefied in 1906. Under the direction of his brother- Mr. O. Kamerlingh Onnes, the office of commercial intelligence at Amsterdam was able to find and import monazite sand from North Carolina, USA, which was then used to prepare helium gas in house [3]. He designed and built a multistage apparatus to liquefy helium which used liquid hydrogen to precool the helium. The first helium liquefier was constructed in 1908 that produced 0.28 liter liquid/hour [1]. The experiment of liquefying helium began at 5:30 on the morning of July 10, 1908. After thirteen hours, Onnes observed that helium liquefied at about 4 degrees above absolute zero [4]. He later decided to study the electrical resistance of metals at different temperatures. In late 1910, Onnes along with Cornelis Dorsman and Gilles Holst conducted series of resistance measurements which required metals with high purity to measure resistance accurately. They chose mercury for this and observed that it has no resistance at 4.2 K [5]. This transition to superconducting state was abrupt and is shown in Figure 1 Thus on April 28, 1911 superconductivity was discovered.

0.15 Ω

0.125

0.1

Hg 0.075

0.05

0.025 10-5 Ω 0 4.0 4.1 4.2 4.3 4.4 K

Figure 1: Sudden change in resistance (on y axis in Ω) of mercury with decrease in temperature (on x axis in K)

1.1.1 The Meissner effect

In 1933, Walther Meissner and his colleague Robert Ochsenfeld conducted an experiment that was a breakthrough. They cooled a cylinder of pure tin in reserved magnetic field and observed that flux was abruptly expelled from the cylinder when it is below its critical temperature [6] as shown in Figure 2, making it a perfect diamagnet. The expulsion of magnetic field occurs due to the induced superconducting current on the surface of the superconductor [7]. The superconductivity was quenched as the flux re-entered into the cylinder at high fields. This process was reversible as the flux was expelled on reducing the magnetic field. The discovery of the Meissner effect opened new gateways for superconductivity research.

(a) (b)

Figure 2: Superconductor in magnetic field (a) above critical temperature and (b) below critical temperature

1.1.2 Electrodynamics and thermodynamics of superconductivity

German scientists-Fritz London, a theoretical physicist and Heinz London, an experimentalist, studied the effects of electric and magnetic fields on superconductors. They proposed the London theory [8], which predicted the depth to which the magnetic fields can penetrate on the surface of superconductor. The magnitude of this penetration depends on the mass, charge, and density of the superconducting electrons. The two fluid thermodynamic model proposed by Cornelius Gorter and H. B. G. Casimir in Leiden (1934) is related to the above property of electrons in a way that the model talks about the strength of superconductivity in a sample. According to this model, superconductors have two kinds of electrons: superconducting and normal [9]. Another important parameter called the coherence length was proposed by Brian Pippard at Cambridge University in 1953 [10]. His idea was that density of superconducting electrons can only change over a characteristic distance called the coherence length. Superconductors can be divided into two types based upon the above discussion: London penetration depth (λ) and the coherence length (ξ).

Type 1: In Type 1 superconductors, if the strength of applied magnetic field is above a critical field then the superconductivity will be destroyed and the ratio of λ and ξ lies between 0 and 1/√2 [11]. So these materials are either completely superconducting or completely normal [12]. This is normally exhibited by pure metals like aluminum, mercury, and lead [13]. Type 2: Type 2 superconductors have two critical magnetic fields in which the superconducting state is retained even when the magnetic flux vortices penetrate the material and this state is called the mixed state [14]. These materials have two critical fields: lower and upper. The superconductivity is lost above the upper critical field [13]. The ratio of λ and ξ for these superconductors is greater than 1/√2 [11]. Some low temperature superconductors (Nb based alloys) and most of the high temperature superconductors (YBCO and BSCCO) are of type 2 [15]. Flux pinning [16] [17] or trapping is a property of these materials and hence are not perfect diamagnets. This phenomenon along with the Meissner effect, form the basis for magnetically levitated trains [18]. Fig. 3 shows the magnetization and phase diagrams of type 1 and type 2 superconductors [19].

Type 1 Type 2

H H HC2

HC H B≠0 C1

R=0,B=0 R=0,B=0 T TC T c T (a) (a)

-I -I

HC HH HC1 HC2 (b) (b)

Figure 3: (a) Magnetic field (H) versus Temperature (T), 3(b) Representation of Magnetic moment (I) versus applied field (H) for type 1 and type 2 superconductors.

1.1.3 Quantum mechanics of superconductivity

In 1956, Leon N. Cooper proposed that two electrons, though electron-phonon interactions form a bound pairs so that the overall energy of the system is lowered [20]. These electron pairs are called Cooper pairs. A superconductor has many such pairs at low temperatures. According to the Barden, Cooper, and Schrieffer (BCS) theory [10] [21], the superconductor then attains the condensed ground state in which all the electron pairs are in a collective state. The energy required to rupture this state is equal to the total energy of all the electron pairs and hence is much higher than the energy of the oscillating atoms of the material, thus offering no resistance to the overall electron flow. An important manifestation of the quantum mechanics is the tunneling of the electrons through a thin (about 1 nm) junction between two superconductors called as the Josephson effect (1962) [22]. This phenomenon allows a limited current (few microamperes to few milliamperes) to pass through the junction without any voltage appearing [23]. Figure 4 shows the current (µA)-voltage (mV) characteristics of a Josephson junction where a zero-voltage supercurrent is clearly visible.

Current

0 Voltage Figure 4: Current versus voltage characteristic of thin film Josephson junction

The physical behavior of an object at any time can be predicted by its wavefunction. The superconducting electrons share a common wavefunction. So, if we know the amplitude and phase (wave nature) of one superconducting electron, we can estimate the wavefunction of any superconducting electron. This is called as the long range order phenomenon. Consider a superconducting ring placed in a magnetic field. The field tries to change the phase of the electrons in the ring. To preserve the long range order of the wavefunction, the ring produces current that creates a field opposite to the applied field. This current allows only a discrete set of magnetic flux values that will not let the phase be out of order, thus preserving the wavefunction. These quantized fluxes in the superconducting ring are called as fluxons. The flux quantization and the Josephson tunneling, together form the basic operating principle of the Superconducting Quantum Interference Devices (SQUID) [24] used in medicine and high performance electronic devices. A SQUID is formed when two Josephson junctions are connected in parallel to form a loop [25].

Josephson junction

Ф Current

Superconductor

Figure 5: Schematic representation of SQUID, formed with two identical Josephson junctions

1.2 Types of Superconductors

Materials that act as superconductors when cooled to temperatures below 20 K are called low temperature superconductors (LTS), and those that can retain their superconductivity above 25 K are called high temperature superconductors (HTS).

1.2.1 Low temperature superconductors

Mercury was the first superconducting element that was discovered. Its critical

temperature (Tc) is 4.2 K. Tin and lead were the next superconducting elements discovered

raising the Tc to 7.2 K. The discovery of other LTS materials is given in Table 1. Europium was the latest element, discovered in 2009 that is superconducting only under extremely high pressure (80 GPa) [13]. The cryogenic systems for LTS materials require more energy and are more complex than those used for HTS materials and hence LTS, such as niobium-titanium (Nb-

Ti, Tc= 10 K) [26] and Niobium-Tin (Nb3Sn, Tc= 18 K) [27] have limited applications.

Table 1: Discovery of superconducting materials

Element Tc (K) Discovery

Mercury 4.2 1911

Tin 3.7 1912

Lead 7.2 1912

Tantalum 4.4 1928

Thorium 1.4 1929

Niobium 9.2 1930

1.2.2 High temperature superconductors

Cuprate materials have the advantage of exhibiting superconductivity when cooled by

liquid nitrogen unlike LTS materials that require liquid helium. The high Tc originates from cuprate superconductors having layered crystal like structures, which consists of conducting

CuO2 planes. These planes are separated by layers that are formed by other elements and oxygen.

The holes are the usual mobile charge carriers and are believed to reside in the CuO2 planes [28].

Discovery of high temperature superconductors Bernd Matthias discovered many rules to locate new superconducting materials. In 1950s he along with his co-workers studied many materials and patterns in their behavior. They found that superconductivity of a material depends on its number of valence electrons, and the materials with average valence electrons of 5 and 7 electrons per atom have highest transition temperatures [29]. Over the next 20 years scientists discovered more than 20 superconductors using Matthias’s rule, most of them having transition temperature around 20 K. In 1986, Karl Alex Mueller- head of physics lab and professor at the University of Zurich along with Johannes Georg Bednorz discovered superconductors that had transition temperatures above 30 K. These superconductors are called as high temperature superconductors. Figure 6 shows the evolution of transition temperatures subsequent to the discovery of superconductivity [30].

140 K Hg-Ba-Ca-Cu-O (135 K) Tl-Ba-Ca-Cu-O (125 K) 120 Bi-Sr-Ca-Cu-O (110 K) 100 Y-Ba-Cu-O (92 K) 80

Liquid N2

40 La-Ba-Cu-O

20 Nb3Ge Hg NbTi

Liquid He Nb3Sn 0 1900 1920 1940 1960 1980 2000

Figure 6: Evolution of superconducting transition temperatures

The important high Tc cuprate materials are listed in Table 2 [28].

Table 2: High temperature superconducting materials and their critical temperatures

Material Nickname Tc (K)

YBa2Cu3O7 YBCO; YBCO-123; Y-123 92

Bi2Sr2Ca2Cu3O10 BSCCO; BSCC0-2223; Bi-2223 110

Tl2Ba2Ca2Cu2O10 TBCCO; TBCCO-2223; Tl-2223 122

HgBa2Ca2Cu3O8 HBCCO; HBCCO-1223; Hg-1223 133

1.2.3 First generation (1G) superconductors

Materials made of Bismuth such as Bismuth-Strontium-Calcium-Copper-Oxide

(Bi2Sr2Ca1Cu2O10) compounds are the first generation high temperature superconductors. The BSCCO-2212 [31] is very versatile and is the first HTS material used for making superconducting wires. Its critical temperature is 90 K. This wire can be manufactured by different methods such as powder-in-tube, dip-coating and tape casting. BSCC0-2223 HTS has a critical temperature of 110 K and is mostly manufactured by oxide-powder-in-tube method [32].

1.2.4 Second generation (2G) superconductors

The 1G wires were not efficient at magnetic fields greater than 2 T as they required temperatures below 40 K at such higher fields. The wire’s performance decreases after 0.2 T at 77 K and is not reversible [33]. The 2G HTS materials were developed, using Rare earth barium

copper oxide - REBa2Cu3O7 (REBCO). The 2G wires had the advantages of high critical current [33], low cost [34], ability to be produced in long length [35], and high magnetic field performance. American Superconducting Corporation’s (AMSC) 2G wire architecture is shown in Figure 7(a) [36]; it is manufactured using metal organic deposition process. SuperPower Inc. used the ion beam assisted deposition process to manufacture its 2G conductors (Fig. 7(b)) [37]. AMSC uses the Metal Organic Deposition (MOD)/ Rolling Assisted Biaxially Textured Substrates (RABiTS) approach [38]. It is based on a wide strip technology and is ideally suited for the industrial reel-to-reel processing required for a high volume, low cost manufacturing.

(a) (b)

The 2G HTS RE-123 based wires are efficient and reliable in applications such as electric power cables [39], fault current limiters [40], motors, generators [41] [42] and superconducting magnetic energy storage systems [43]. The 2G wires have many system advantages like enabling smaller footprint (weight, volume) and high power density allowing more options to design the rest of the device [44].

The most common and commercially available HTS conductors are BSCCO-2212, BSCCO-

2223, YBCO-123 and MgB2, which are manufactured by American Superconductor Corp. (AMSC), SuperPower- USA and Sumitomo Electric Industries (SEI) - Japan.

CHAPTER TWO

APPLICATIONS

Superconducting wire capable of handling 100 amperes, enough current to supply a family home, is not much thicker than a human hair. By contrast, copper wiring for the same power level must be several mm thick across to avoid overheating. Superconducting wire offers the combination of compactness and lossless current flow that permits us to perform technological marvels that would be impossible with ordinary conductors [32].

2.1 Digital Electronics

Superconducting devices have high intrinsic switching speed and low power dissipation compared to those of semiconductor devices [45]. A new kind of Josephson junction technology called Rapid Single Flux Quantum (RSFQ) logic circuits can generate, pass, memorize and reproduce picoseconds voltage pulses [46]. An RSFQ T-flip flop operating at 770 GHz was demonstrated at 4.2 K [47]. One information bit of an SFQ circuit is stored in a superconducting loop as a flux quantum. The loop includes more than one overdamped Josephson junction and forms a quantum interferometer [48]. A 20 GHz, 8 bit Flux 1 microprocessor was the first RFSQ microprocessor designed and fabricated to study architectural and design challenges [49]. The superconducting electronics have the potential to be used in high end computing devices such as petaflops scale computers for achieving ultra-low power and ultra-high speed. These processors were aimed at achieving 100 GHz on chip clock speed and a data rate of 8 Gbps with 0.8 µm Josephson junctions [50].

2.2 Medical Applications

One of the major applications of superconductors in medicine is as magnets that are used in Magnetic Resonance Imaging (MRI) and particle accelerators for cancer therapy. MRI is based on nuclear magnetic resonance [51]. The change in the magnetic moment of the nuclei in strong magnetic field when subjected to radio frequency resonance is detected and these signals are used to distinguish the tissues. Permanent magnet MRI systems are heavy and usually have field strengths of less than 0.35 T. MRI systems with central field greater than 0.4 T use superconducting magnets of high uniformity and temporal stability as a core component. Whole

body MRIs with fields between 7 T to 9.4 T that would yield higher resolution were installed by General Electric (GE) at University of Illinois at Chicago for advanced medical research [52]. The other important application is SQUID based magnetic gradiometers that are used in Mangnetoencephalography (MEG) and Magnetocardiography (MCG) [53]. Human heart, brain, lungs and other organs emit fields of about 1/100,000,000 to 1/10,000 of earth’s magnetic field [54]. SQUID magnetometers offer superior performance in measuring such low magnetic fields. The better resolution of MCG compared to Electrocardiography (ECG) provides more accurate detection of heart functionality [55]. High Tc SQUIDs were also being developed for biomedical applications [56].

2.3 Electric Power

2.3.1 Machines

Superconducting machines are much smaller, lighter, more efficient and quieter compared to conventional machines [57]. Using HTS wires in electric machines, a current density of 100 A/mm2 is possible instead of 5 A/mm2 in the copper wires of conventional machines thus making the windings much smaller [58]. HTS rotating machines save fuel and space on-board a ship when used for propulsion [59]. Superconductors have losses when carrying alternating currents [60], so most of these machines use superconducting coils for DC field winding on the rotor. Motors with up to 3.7 MW and 1800 rpm were built with HTS field windings [61]. High power density HTS machines were designed for naval applications that include 36 MW- 3000 rpm generator, 36 MW- 120 rpm propulsion motor and 4 MW-7000 rpm power generation modules [62]. Recent attempts were made to use low loss superconducting AC coils for the stator [63].

2.3.2 Power transformers

It is estimated that up to £195,000 can be saved on an 800 MVA transformer by eliminating the I2R losses [64]. HTS power transformers have the advantages of higher power density, lower operating losses, inherent fault current limiting capabilities, lighter weight, smaller footprint and are considered more environmentally friendly [65]. A 5/10 MVA HTS Transformer with increased safety and higher power rating for same footprint as a conventional transformer, was successfully developed and tested under Superconducting Partnership Initiative

(SPI) projects [66]. 2G HTS windings were also used for a transformer design that reduces the AC magnetization losses compared to BSCC0-2223 conductors [67].

2.3.3 Fault current limiters

A superconducting fault current limiter (SFCL) can be used to limit high fault currents from damaging other grid components during faults [68]. SFCLs are basically of two types: resistive and inductive [32] [69]. The resistive SFCL is connected in series in a line and its resistance is zero during normal operation and during fault conditions or when the current exceeds the critical current of the conductor, its resistance increases rapidly to resist the high currents, thus limiting the peak fault current [70]. Inductive SFCL works like a transformer with shorted superconducting secondary winding. During normal operation the resistance of secondary is zero, therefore reducing the reactance of the primary winding. During a fault, the high secondary resistance increases the reactance of the primary winding, thus limiting the fault current [71]. The use of 2G HTS wires for SFCL applications offers many advantages like high index (n)-value, superior electromechanical performance, large surface area for cooling, availability in long lengths, high throughput and low manufacturing cost [72]. Recently a 22.9 kV SFCL has been installed in the distribution grid in Korea [73].

2.3.4 Magnetic energy storage

Superconducting Magnetic Energy Storage (SMES) systems have the potential of becoming the most efficient of all the available storage systems such as batteries, flywheel energy storage, pumped-hydro and capacitors [53]. SMES convert AC from the utility to DC and uses it to store energy in superconducting coil in the form of a magnetic field [74]. The system has four parts: a superconducting coil magnet (SCM), power conditioning system (PCS), cryogenic system (CS), and the control unit (CU) [75]. The stability and quality of the power systems can be increased using SMES.

2.3.5 Magnetic levitation trains

The process in which one object is suspended over other with no forces other than magnetic field is called magnetic levitation (Maglev). The Maglev trains work on the principle of Meissner effect and flux pinning with the help of superconducting magnets. They have many advantages over conventional trains such as: elimination of wheels and tracks thereby reducing

the noise, friction and maintenance costs [76], much lower or zero possibility for derailing [77], and possibility to eliminate gears and bearings. The Maglev trains have achieved top speeds of over 552 km/h in just 100 sec within a distance of 8 km [78].

2.4 Particle Physics

Particle physics explores the inner structure of matter. Accelerators are used to boost particles with high energy and make them collide so that their structure can be studied after they decompose [79]. The charged particles are accelerated by electric fields produced by radio frequency resonant cavities. Strong magnetic fields guide and focus the particle beam [80]. The high energy multi TeV proton accelerators use superconducting magnets to achieve high fields of about 10 T. In 1967, superconducting magnets were proposed for proton synchrotrons and their feasibility was studied [81]. A synchrotron is a circular accelerator in which particles are synchronized with the radio frequency field. The European Organization of Nuclear Research (CERN) developed the Large Hadron Collider (LHC) with a circumference of 27 km in which the head-on collision of protons will reach 14 TeV in the near future [82] with the collision energy of 8 TeV in 2012 [83]. LHC consists of 8000 superconducting magnets of different types of which 1232 superconducting dipoles operate at magnetic field strengths of 8.3 Tesla [84]. Superconducting radio frequency cavity resonators can operate at higher electric field intensity than traditional resonators [85].

2.5 Fusion Technology

Strong magnetic fields can be used to confine plasma for developing fusion energy. Plasma is a high temperature gaseous state with charge neutrality and collective interactions between charged particles and waves [86]. Temperatures around 100,000,000°C and pressures in the megapascal range are required in magnetic fusion reactors to confine plasma for a long time; this can be achieved by superconducting magnets [87]. The International Thermonuclear Experimental Reactor (ITER) is the world’s first experimental fusion reactor [88]. Its magnet system uses Nb3Sn coils that carry 46 kA of current at maximum field of 13 T for its central solenoid [89].

2.6 Space Applications

The High Temperature Superconductivity Space Experiment (HTSSE) was initiated by the Naval Research Laboratory (NRL) in 1988 to develop HTS devices and components to be used in space [90]. The space missions can be benefited by reduction in weight, size and power consumption and at the same time have high performance using HTS [91]. The experiment has two phases, the aim of HTSSE-1 was to test the survivability of simple HTS devices in space and the HTSSE-2 [92] was designed to test more advanced subsystems and cryocoolers that was launched into space in 1999. The first successful experiment had a thin YBCO film integrated with a cryocooler and it orbits around the earth on the TECHSAT II satellite [93].

CHAPTER THREE

SUPERCONDUCTING POWER CABLES 3.1 Overview

It is estimated that about 5% of the power is being wasted in its transmission and distribution [32]. The growing demand for reliable high quality power encourages utilities to react to the changing conditions and to offer new solutions. HTS cable is a promising new technology to address these issues. The two major advantages of using HTS cables are 1. For a given cross-section, HTS cables can carry much high power than conventional copper cables. This is because HTS wire carry currents of about 100 A/mm2 whereas copper (XLPE cable) is usually rated at 1 A/mm2 [94]. So without having to spend for new infrastructure, they provide increased capacity when they are replaced by conventional cables in the existing ducts and tunnels.

The typical weight of a 5 kA superconducting cable system (100 m, MgB2 coaxial cable) is about 165 kg where as a copper cable of similar rating would be 10 tons (20 kg/kA/m) [95]. 2. The I2R losses in HTS cables are negligible compared to conventional cables. Hence using them for high current and low voltage application of power transmission gives us two advantages, it eliminates the requirement of high insulation and we do not need substations to change high voltage to low voltage for the end-users, thus providing the flexibility to relocate substations to inexpensive sites. One of the important factors that determine the cost of superconducting system depends on the power required to run the compressor for refrigeration system. This mainly depends on the efficiency of the refrigeration. Practical refrigeration systems have some heat inputs and their compression and expansion steps are not lossless. So, their efficiency (η) is the ratio of actual work required to ideal Carnot work [96].

�� = = Where �� = �−� = ��

TL is the lower temperature and TH is the higher/ambient temperature of the refrigeration system and COP is the coefficient of performance and SP is the specific power. Other critical point to be considered is that large systems are more efficient than small systems.

The specific power, number of watts of electrical power required to remove 1 W of heat from the superconducting device depends on the operating temperature and the type of cryocooler used. For 50 K operating temperature, the specific power is around 30 for common refrigeration systems.

The total loss in a superconducting DC cable system is the sum of losses in terminations (WTerm), losses in cryogenic system (WCryo) and losses due to AC ripple (WAC Ripple).

= ∗ �� + � ∗ (�� + �� ) Allweins, E. Marzahn [95] conducted a study on technical feasibility and economic benefits of superconducting DC cables made of MgB2 wires for shipboard applications. Taking into account the power requirements for providing cryogenic environment using typical commercial cryocoolers, they concluded that a superconducting cables operating at 20-30 K are more efficient than a corresponding copper cable for lengths > 30 m. For cables shorter than 30 m, termination losses dominate making the superconducting cables less attractive.

As the Navy moves to the all-electric warship, superconducting materials play a key role. The high temperature superconducting materials enable extended range, high efficiency, and high power density naval propulsion [97]. Superconducting systems for fleet implementations were being developed by the Navy from the beginning of the 21st century. USS Higgins is the first ship to use superconducting cables in a degaussing system [98]. This system is used to decrease or eliminate the magnetic field of the ship so that the magnetic mines under water would not detect the ship. High temperature superconducting (HTS) cables are highly efficient in power transmission compared to conventional cables [99]. HTS DC transmission systems have the advantages of high transport current capability and zero resistive loss [100]. Studies show that

HTS DC power cables can be used in future power grids [101]. Recent developments also include design, fabrication and installation of 2GHTS power cables in power grid [102]. Superconducting power cables can also be used in DC electric railway systems to increase the transportation capacity of railway lines [103]. The feasibility of HTS cables as underwater power cables was also studied and the conceptual design was presented in [104]. Superconducting cables can be divided into two types based on the type of dielectric they use: Warm dielectric cables, in which the insulation is applied over the cryostat; and cold dielectric cables, in which the coolant flowing in between two layers of HTS wires also acts as dielectric [32]. The cold dielectric cables are of two types based on their structure: single phase and triax three phase [105]. The cold dielectric cables have the advantages such as high capacity, low AC losses, low inductance and suppression of electromagnetic fields outside cable assembly. The different design configurations of HTS cables are shown in Figure 8.

Concentric phases (Triax)Three separate phases Three phases in one cryo envelop

Figure 8: Design configurations of HTS cable

Oak Ridge National Lab (ORNL) built and tested the first full scale triax design under Ultera in 2002 [106]. It is a 3-m long prototype rated at 3 kA-rms and 15 kV-rms, with three concentric phases made of BSCCO-2223 HTS tapes, separated by layers of cold dielectric. The three main projects of HTS cable demonstrations were 1. 13.2 kV, 3 kA Triax HTS cable of 200 meters installed at American Electric Power in Columbus, OH (2006), whose cable terminations were designed by ORNL [107]. 2. 34.5 kV, 0.8 kA HTS underground cable of 350 m installed at Albany, NY in 2006 [108]. The cable was fabricated by Sumitomo Electric using BSCCO wire and a 30 m section of this was replaced by 2G (YBCO) cable in the phase II of the project [109]. 3. 134 kV 2.4 kA BSCCO HTS cable installation at Long Island Power Authority (LIPA) grid with a length of 660 meters [110]. The goal of LIPA II was to install a replacement phase conductor made by AMSC’s 2G wire [111].

The design of compact three phase concentric 10 kV cable with a capacity of 40 MW produced by Nexans for the Ampacity project is shown in Figure 9. It was used to replace a 1 km high voltage cable between two transformers in Essen, Germany.

Helium gas cooled superconducting power cables were also tested recently [113] that offer flexibility in operating temperatures, high power densities when required, and a reduction in weight compared to liquid nitrogen cooled cables.

“While a superconducting cable acts like a super highway for electrons, a quench is like a jackknifed truck that ties up the entire roadway. Just as one crippled truck can render a major traffic artery useless in a matter of moments, the destruction of superconductivity in one spot in a superconducting cable can bring about its demise” [4]. One of the reasons for quenches is the unequal distribution of current among the tapes of the cable. To investigate this risk, an attempt was made to model superconducting cables that consist of a certain number of superconducting tapes. The model was used to analyze the various parameters on which the current distribution depends on.

3.2 Research Objective

High temperature superconducting DC cables have the advantages of high current density and low losses compared to non-superconducting cables. The power capacity of an HTS DC transmission system can be more than ten times that of a conventional system or use a much lower voltage level to deliver the same energy [114]. DC electric power demand is increasing 18

rapidly, which requires reduction in cost and losses of the power supply systems. Studies also show that superconducting DC distribution systems are feasible and offer a reduction in cost compared to conventional cables over long period [115]. The attractiveness and feasibility of DC superconducting transmission was studied and the findings were published recently by the Electric Power Research Institute [116]. Superconducting power cables are fabricated using a large number of superconducting tapes, usually in multiple layers that form parallel networks for current transmission. Uniform distribution of current among the tapes of the cable is one of the important issues to ensure reliable operation of the cables and to devise risk mitigation techniques [117]. Non-uniform current distribution reduces the rated capacity of the cable and also leads to higher losses [118]. Current sharing is also an important issue in resistive superconducting fault current limiters [119], [120]. Efforts are underway to understand the reasons for non-uniform current distribution by mapping the circumferential magnetic field using hall sensors [121]. Different parameters of superconducting cables are to be considered in understanding non-uniform current distribution, such as differences in the resistance of soldered contacts of the tapes to the terminations of the cable, critical current variations among individual tapes and along the length of each tape, index (n) value of superconducting tapes, radial and longitudinal temperature gradients. Most of these above parameters have natural statistical variations of characteristics resulting from the complex structure of second generation high temperature superconducting (2G HTS) coated conductors that are manufactured in a batch process of multiple steps. The dominant reason for non-uniform current distribution in short cables (< 30 m) is the differences in contact resistances at the ends. All the variations in tape characteristics, the contact resistances, and the variation of contacts with aging have statistical fluctuations. It is necessary to understand the role of the variations in each of the individual parameters listed above and more importantly their combined effect on the current distribution in a manufactured cable. This will help in assessing the risk in operating high power density superconducting DC power cables and to devise methods to mitigate the risk. Superconducting AC power cables will be affected by similar issues, but the focus of this thesis is superconducting DC cables fabricated from 2G HTS tapes. The role of these variations is magnified in cables cooled with gaseous helium because the lower heat capacity of helium gas and the much higher power densities supported by the cables operated in 50-60 K temperature

range can result in higher temperature gradients [122]. The goal of the effort described in the thesis is to assess the role of the combined statistical variations in cable characteristics in non-uniform current distribution manifested by pushing some individual tapes closer to their respective critical current before the current in the cable reached its design critical current. Monte Carlo methods combined with a mathematical model are used to analyze the combined effect of some cable and tape characteristics on current distribution among the tapes in a cable. Experimental results on a simple network of parallel superconducting tapes were used to validate the developed mathematical model that was used to simulate and analyze current distribution in a set of 1,000 cables with statistical variations in some critical parameters.

Also, the critical current (Ic) of commercially available superconducting tapes varies along the length [123] [36]. Commercial tapes from multiple vendors were procured and characterized to assess the extent of variations to be used in the models. It was observed that these longitudinal variations in the superconducting properties would deteriorate the performance of the cable. The effect of the variations in Ic and index (n) value on the cable was studied using Monte Carlo method and presented.

CHAPTER FOUR

RESULTS OF CABLE MEASUREMENTS 4.1 Measurement of Critical Current and Index Value of Tapes in 2G HTS Cable

To investigate the extent of variation in Ics of HTS tapes used for manufacturing commercial superconducting cables, a 30 m long 2G HTS cable manufactured by Southwire® was considered. Two one-meter sections of the HTS cable, one cut from each end of cable were tested for variations in critical current of its tapes. The cable has three layers of HTS tapes and these tapes were manufactured by American Superconducting Corporation (AMSC®). Three

tapes from each layer and from each end of the cable were tested for their Ics. These values were given in Table 3.

(a) (b)

Figure 10: (a) Vertical and (b) Horizontal views of superconducting cable (unwrapped)

The above figure shows the segment of cable used for Ic and n value measurements.

Table 3: Ic and n values of several segments of tapes taken from HTS cable

Layer 1 Layer 2 Layer 3

Ic (A) 97 106 104 102 100 101 98 105 113 Termination 1 n 33 30 32 33 31 30 32 36 31

Ic (A) 98 100 103 98 105 102 99 107 105 Termination 2 n 31 35 29 30 28 29 24 26 25

These results indicate that the critical current of some tapes were less than their design Ic

(100 A). It can be observed that the difference in Ics is as high as 15% and the difference in n values is as high as 50%. Table 3 and 4 shows the measurements of tapes from different manufacturers to investigate the

extent of variation in Ics and n values.

Table 4: Measurement of critical current and index value of sample HTS tapes manufactured by SuperOX and Sunam

Measured Length Measured Ic Measured Manufact Min Ic by Ic of Measured n of Width of section 1 n of urer manufacturer section 2 of section 2 voltage (27 cm) section 1 (27 cm) tap

SuperOx 4 mm 120 A 140 A 30 136 A 29 15 cm

SuperOx 12 mm 350 A 444 A 41 439 A 40 15 cm

Sunam 4 mm 150 A 172 A 40 166 A 38 15 cm

From the above measurements, it is clear that different tapes in a cable have different Ics

and n values, also different sections within a single piece of tape have slightly different Ics and n values. To investigate the effect of these variations, simulations on mathematical model were performed along with their experimental validation.

CHAPTER FIVE

EXPERIMENTS ON DISTRIBUTION OF CURRENT IN 2G HTS CABLES 5.1 Experimental Procedure

A typical superconducting cable has high resistance between the tapes that the current cannot bypass from one tape to another. The distribution of the current among the tapes can only take place at the terminations of the cable where the tapes are soldered to the copper lead. The stability of superconducting cables with multiple strands is highly influenced by the current distribution among the strands [124]. So we designed a simplified setup of a superconducting cable and conducted an experiment to investigate this effect on the cables with multiple tapes. The experimental setup (Figure 11) consisted of three second generation HTS tapes produced by SuperPower Inc. and one copper tape in a parallel network. The copper tape was used to replicate the copper conductor used in HTS cables to serve as a stabilizer [125] providing an alternative path for the current during a quench.

Figure 11: Experimental setup

Copper connector 40 cm Copper connector

V1V2 V3 V4 Tape1 Sh1 V5V6 V7 V8 Tape2 Sh2 V17 V9 V10 V18 V11 V12 Sh3 Tape3 V13 V14 V15 V16 Copper Sh4

100 cm

5.1.1 Measurement of critical current

The critical current (Ic) of each tape was measured by plotting current (I) vs. voltage (V) and the using the 1 µV/cm electric field criterion [126].

5.1.2 Measurement of index (n) value

The n-values were determined from the experimental slopes of the respective log (V) - log (I) -graph by approximating the equation V α In. More details about the measurement are available in [127].

The standards of superconductor measurements and ways of measuring Ic and n value can also be found in [128].

5.1.3 Measurement of contact resistances

The uniform distribution of the current among the tapes of cables also depends upon the distribution of the contact resistances between the tapes and the copper of the termination [129]. The contact resistances of the solder joints (Figure 13) in a cable are usually not equal for all of the tapes because of differences in solder joint quality and aging. After the critical currents and the index value of each tape were measured, all the tapes were connected in a parallel network as shown in Figure 11, and current was ramped up to 600 A. The contact resistances of three 2G HTS tapes and the copper tape were calculated using the voltage drop across the contacts and the current obtained using the respective shunt resistance.

Figure 13: Sample superconducting cable with the solder joints enlarged

5.1.4 Measurement of current through individual tapes

The resistances of the shunts used were measured in an open bath of liquid nitrogen

(LN2) to minimize temperature variations for accurate values of the currents through tapes. There

was about 7% decrease in resistance of the standard shunts from room temperature to LN2 temperature (77 K). The newly obtained values of shunt resistances were used to calculate the current across each tape in the network. The critical currents, n-values and contact resistances of the three HTS tapes and copper tape measured were given below.

Table 5: Characteristics of tape used in the experiment

Tape Critical current (A) Index (n)-value [±5%] Contact resistances (sum of left and right contacts) HTS 1 98 37 60 µΩ HTS 2 110 35 80 µΩ HTS 3 107 39 90 µΩ Copper - - 140 µΩ

5.2 Mathematical Model

A mathematical model [130] was used to calculate the current through individual tapes in

a parallel network given the total current, individual tape Ics, n-values, and contact resistances. For the model, the number of superconducting tapes that can be considered were limited due to the limitations of the number of non-linear equations that can be solved simultaneously in the software package of choice (Maple 16 by Maplesoft). The validity of the model was verified using the experimental results. The values of contact resistances, critical currents and the index numbers were taken from the experimental data and then the mathematical model was used to solve for currents in individual branches. The voltage across each HTS tape was modeled using the electric field criterion and the critical current as:

� � = ( ) �� 25

I1 R1a R hts1 R1b Sh1 Connector R1c 50 µΩ 10 µΩ 233 µΩ 2.1 mΩ I2 R2a R hts2 R2b Sh2 Connector R2c 50 µΩ 30 µΩ 232 µΩ 1.7 mΩ R3a R hts3 R3b Sh3 Connector R3c I0 80 µΩ 10 µΩ 239 µΩ 3.1 mΩ I3 R4a R Cu R4b Sh4 Connector R4c 90 µΩ 0.2 mΩ 50 µΩ 234 µΩ 2.2 mΩ I4 + − V0

Figure 14: Equivalent circuit model of the setup in Fig. 11, with the values of the parameters obtained from the experiment.

Where Ec is the electric field criterion (1 µV/cm), l is the length of cable (40 cm), I is the

current through the tape, Ic is the critical current, and n is index value. The following equations were solved simultaneously with the help of the scientific computing software package Maple™ 16 for obtaining the individual branch currents in Figure 14, using the function ‘fsolve’. This function computes the solution numerically for the given set of equations.

� = � + � + � + � � = ��Ω + �Ω + �Ω + .Ω + � ( ) �� = � �Ω + �Ω + �Ω + .Ω + � = ��Ω + �Ω + �Ω + .Ω + � ��

= ��Ω + .Ω + �Ω + �Ω + .Ω The currents through individual tapes as obtained from the measurements and from the mathematical model against the total current through the network shown in Figure 14 are plotted in Figure 15. As can be seen from Figure 15, the calculated and experimental current values agree validating the cable model. The plots show differences in the extent of current sharing among the tapes of the network that represents a simple cable.

140

120

100

80 Tape 2 Copper 60 Tape 1 40 Tape 3

Individual Individual tape currents (A) 0 0 200 400 600 Total current (A) tape1 tape2 tape3 Copper solid line: model; dotted line : experiment

Figure 15: Comparison of the total current vs. individual tape currents obtained through the experiments and the mathematical model.

It can be observed that the current through the copper tape is higher than the Tape 3. This is because the overall resistance of the branch 3 (3.429 mΩ) is more than branch 4 (2.774 mΩ) as shown in Figure 14, highlighting the importance of contact resistances. Current sharing was

determined by differences of Ic, n-value, and resistance along the electrical path. Since current sharing is dominated by the contact resistance in this experiment, the copper tape performs not any worse than the HTS tapes. Resistances originate from the contacts and usually the resistance of copper lead is arbitrary for cable.

5.3 Simulations

The mathematical model developed was used to study the effect of each of these parameters individually and collectively on the distribution of current among the tapes in superconducting cables. When one individual parameter was being studied, all other parameters were kept constant at the predetermined values. The model cable contains eight tapes with a design current of 800 A based on a critical current of 100 A per tape; slight reduction of cable critical current due to self-field effects was neglected for the study.

5.3.1 Effects of variation of contact resistance

The current distribution in different tapes of an 8-tape cable model with normally distributed random values for the contact resistances with average of 50 nΩ and standard deviation of 20 nΩ was plotted in Figure 16. The contact resistances represent the sum of joint resistances at both ends of the tape connections. The resistance due to a connector (as used in the

experiment) was not included because such a connector does not exist in an actual cable. To analyze the effects of higher contact resistances, the current distribution for higher

magnitudes of contact resistances were calculated with tape Ic equal to 100 A and n values equal to 30 for all tapes in all the cases.

100 90 80 Increasing resistance 70 60 50 40 30 20 Individual Individual tape currents (A) 10 0 0 200 400 600 800 Total current (A)

Figure 16: Individual tape currents for different values of contact resistances: 33.2 nΩ, 37.0 nΩ, 42.9 nΩ, 45.3 nΩ, 49.1 nΩ, 50.8 nΩ, 53.3 nΩ, and 62.8 nΩ; Ic = 100 A for all tapes; n = 30 for all tapes. As expected, at higher values of contact resistances the unbalance in the current distribution was significant at higher currents. It was observed that for lower contact resistances

(contact resistances in the order of nΩ) the current distribution was mostly equal at higher currents because the voltage generated by tapes at high currents dominated the voltages due to contact resistances. When the contact resistances are in the order of µΩ, they create more voltage difference in the tapes at high currents and hence the distribution is significantly uneven at high currents.

5.3.2 Effects of variation of critical current

The mathematical model was solved for different normally distributed random values for

Ic with a mean of 100 A and a standard deviation of 10 A. The results of the model are plotted in

Figure 17. Though the tapes have different Ics, when the contact resistances of them are equal, all of them carry the same current until the total current approaches 8 (number of tapes) times the

least Ic of tapes. The unequal current distribution starts earlier if the contact resistances are of the

order of nΩ. The effect of Ic variations on current distribution were also investigated for varying levels of contact resistances.

120 Increasing Ic 100

20 Individual Individual tape currents 0 0 200 400 600 800 Total current

The current distributions were calculated for different Ic values (as in description of Figure 17) and fixed contact resistances of the tapes for 5 cases: at 5 nΩ, 50 nΩ, 500 nΩ, 5 µΩ, and 50 µΩ. The n value was kept at 30 for all tapes in all the cases. It was observed that as the contact resistances increase, the current distribution was less sensitive to the difference in the critical currents of the tapes. The deviation of the tape current from the mean was more prominent for lower values of contact resistances than for higher.

5.3.3 Effects of variation of index value

The current distribution was obtained for different normally distributed random n values with an average of 30 and standard deviation of 3.

Individual Individual tape currents (A) 50 300 500 700 900 Total current (A)

As seen in Figure 18, the effect of change in n value (within the values chosen) on the current

distribution is much less than the effect of variation of contact resistance and Ic. Also, the current balance equalizes in the high current regime.

5.4 Analysis Using Monte Carlo Simulations

5.4.1 Introduction to Monte Carlo

The Monte Carlo method is a technique of statistical sampling. It has been applied to a vast number of scientific problems over the years. This method was invented by a Polish mathematician- Stanislaw Marcin Ulam in 1946. The idea originated from his attempt to find the chances that a Canfield solitaire laid out with 52 cards will come out successfully [131]. Laying it out a hundred times and counting the number of successful plays seemed to be more practical method than abstract thinking. He tried to interpret processes described by differential equations as equivalent succession of random operations. John von Neumann, Professor of Mathematics at the Institute of Advanced Study, Los Alamos National Laboratory was interested in the world’s first electronic computer – ENIAC (Electronic Numerical Integrator and Computer) built at the Moore School of Electrical Engineering at the University of Pennsylvania and suggested his colleagues at the Lab- Stan Frankel and Nicholas Metropolis to prepare computational model for the thermonuclear reaction to test on ENIAC [132]. John also developed the stored program concept for the ENIAC. Stan Ulam discussed the idea of solving the problem of neutron diffusion using the statistical sampling approach with John von Neumann. This method was named “Monte Carlo” by Metropolis and was its first formulation for an electronic computing machine. The data for solving the neutron diffusion problems are chosen at random to represent a number of neutrons in a chain reacting system. These randomly chosen variables introduced at certain points represent the occurrence of various processes with the correct probabilities and thus the history and progeny of neutrons is determined by detailed calculations of the motions and collisions of these neutrons [133]. The Monte Carlo method can be described as the confluence of deterministic, stochastic and computational methods using computer generated random numbers [134]. Its applications range from neutron-photon transport codes [135] through evaluation of multi-dimensional integrals [136], exploration of properties of high-temperature plasmas and into quantum mechanics of complex systems.

Initial velocity and g and g assumed position of neutron v x from initial conditions v? x?

Length of free gl determined from flight L? material properties

Crossing of material boundaryCollision

g ' determined from Type of collision g determined by l Length of free flight k properties of new k? known branching ratios in new material L’ ? material

Scattering Absorption Fission

Crossing of material boundaryCollision Number and velocities New velocity v’? of new neutrons n?, Chain terminated ’ v1’?,v2 ?... g ' determined from v g , g '… determined from scattering cross sections n3 n v n1 fission cross sections and incoming velocities n2

Figure 19: Schematic of some decisions made to generate the history of an individual neutron in a Monte Carlo calculation.

The Monte Carlo method can also be used to study quantum phenomenon in statistical physics, for example by modeling of many particle systems such as liquids and crystals of helium and the electron gas that exhibit quantum effects at macroscopic level [137]. The different applications of the Monte Carlo method in science and engineering are summarized in [138].

5.4.2 Monte Carlo simulations

When cables are manufactured using tapes with varying properties, current sharing among the tapes is not equal. To analyze the extent of imbalance in the current distribution, 1000 cables were simulated using Monte Carlo method. Randomly generated values were used for the

contact resistance, Ic and n value which were used to solve the equations of the circuit. These values were chosen such that they are close to their average values with which the cables are actually designed. The mean and standard deviation of these generated values are given below. Maple™ 16 was used to generate the random values for the tape parameters and to solve the system of non-linear equations to get the current distribution in individual tapes.

Table 6: Mean and standard deviation of contact resistances, Ics and n values

Mean Standard deviation Contact resistances 50 nΩ 20 nΩ

Ic 100 A 10 A n 30 3

Individual tape currents were calculated at a total cable current of 680 A and 720 A,

which are 85% and 90% of design Ic of the cable (800 A). At 85% of the design cable critical current, out of the 1000 cables simulated, two cables have all their 8 tapes and 1 cable with one

exceeding 95% of their Ic. At 90% design cable Ic, six cables have all eight tapes, two cables

have one tape each, and one cable has two tapes exceeding 98% of their Ic values (Figure 20 (d)).

Such situations could lead to quench in tapes that are closer to their Ic for a longer time. When the extent of variations in the tape parameters increases, these tendencies of multiple tapes close to their critical current also tend to increase. Figure 20 (b) shows the number of cables and

their corresponding range of tapes that exceed (c) 95% and (d) 98% of their Ic at 90% cable

design Ic.

140 140 116 120 120 92 100 90% tape Ic 100 95% of tape Ic

80 95% of tape Ic 80 98% of tape Ic 100% of tape Ic 60 98% of tape Ic 60

40 40

number number of cables number number of cables 20 20 9 3 0 1 0 0 (a) (b)

60 7 52 6 50 6 5 40 4 30 22 3 2 20 2 9 9 1 10 number of cables number number of cables 1 0 0 0 1 4 6 8 1 4 6 8 number of tapes number of tapes (c) (d)

5.5 Discussion of Results

Based on the results of the Monte Carlo analysis, it can be seen that out of the 1000 cables simulated, approximately 10% of the cables fall in the risky zone that some of their tapes carry high currents because of the combined effect of differences in contact resistances, Ic and n, which may lead to their damage. The experimental and model based studies on the current distribution in superconducting DC cable show that it is important to select tapes with a narrow distribution of properties while manufacturing superconducting cables. For short cables, critical current is the second major issue to be considered after the contact resistance of the tapes. When the contact resistances are low (in the order of nΩ), even small variations in the critical current and the n-value of individual tapes can lead to significant variation in the distribution of current at higher fractions of cable design critical current. Significant spread in individual tape parameters can lead to situations in which some tapes carry a current close to their critical current well before the cable as a whole reaches its design critical current. Such an uneven current sharing can lead to catastrophic damage to the cables

CHAPTER SIX

EFFECT OF LONGITUDINAL VARIATIONS IN CHARACTERISTICS OF INDIVUDUAL TAPES ON THE PERFORMANCE OF SUPERCONDUCTING CABLE

High temperature superconducting cables would be prone to mechanical stress [139] [140], such as twisting and bending [141], while transportation or installation. In such situations,

Ic and n values of some sections of the cable are liable to have less than expected values. The effect of longitudinal variations of Ic and n of HTS tapes on the overall performance of the tape is studied using a 30 meter long cable model. The effect of Ic, n value and contact resistances were investigated and the results indicate that uneven current distribution due to differences in these parameters among the tapes would lead to catastrophic damage to the cables [142]. During the processing and operation of 2G HTS tapes, they might be subjected to mechanical, thermal and electromagnetic stress which can result in degradation of Ic. The effect of bending on Ic and minimum bending radius of the tape to avoid degradation with and without stabilizers were studied in [143]. Compression, torsional strains and transverse loading on the tapes of the cable also cause variations in Ic along the length [144] [145] [146]. The importance of the uniformity in the Ic for practical applications of HTS coated conductors was emphasized by measuring the local Ics for various tensile strains in [147]. The Ic of commercially available 2G HTS tapes varies along the length [148] [149]. Sometimes, the variation in Ic along the length could be as high as 10% compared to the average or minimum Ic. The variation of Ic and n value along the length of YBCO coated conductor that is commercially manufactured by MOD/RABiTS process was studied by the Los Alamos National Laboratory in self field and applied external field and were found to be significant [150]. Commercial tapes from multiple vendors were procured and characterized to assess the extent of variations to be used in the models.

This thesis also presents the effects of the longitudinal variations in Ic and n value of individual tapes on the overall performance of the superconducting cables. Commercial tapes from multiple vendors were procured and characterized to assess the extent of variation in Ic and n value and the data were used in the mathematical models presented. Monte Carlo method was used for analyzing the effect of variations in tape parameters on the performance of superconducting cables calculated using mathematical cable model. These simulations were used 34

to understand the nature and extent of the variations on the performance of HTS cable systems.

6.1 Longitudinal Variation in Critical Current

As HTS tapes used for manufacturing superconducting cables have some variations in Ic along their length, a mathematical model was developed for a 30 m cable made of 30

superconducting tapes in parallel configuration. HTS tapes were considered to have different Ics for each centimeter section along the length (Figure 21).

1 cm section Vi

Ik Ik

th i section with critical current Ici

30 m

Figure 21: A model tape in a HTS cable divided into several 1 cm long sections with varying I c values.

Ik is the total current passing through the tape (varying from 1 A to 120 A), Vi is the th voltage produced by i section of the tape having a critical current of Ici and an index-value of ni.

� � � � = ( ) �� Ec is the electric field criterion, 1µV/cm. The total voltage (Vt) produced by the 30 m long tape is sum of voltages produced by N (3000) sections of the tape

�

= ∑ � The apparent critical current�= (Ica) of the whole 30 m tape can be calculated using the electric field of the tape (E)

= N ∑i= i th Here, li is the length of i section of the tape, which is equal to 1 cm in this case.

In designing a HTS superconducting cable with a required Ic, typically the average Ic of

superconducting tape is used. The tape has a range of Ic values and manufacturers have the ability to measure and supply the data for every 5 cm or even every 1 cm.

To test the effect of the sections that have Ic less than the design Ic of the tape, the

following Ici distribution (Figure 22) that has Ici less than the design Ic for 10% of tape sections was considered with a constant n of 30 for all sections.

160 140 120 100 80 60

40 Number of sections of Number 20 0 85 90 95 100 105 110 115 120 Critical current of individual sections (I ) in Amps ci

Figure 22: Distribution of critical current of tape sections

The equivalent critical current Ica of the tape obtained in this case was 96 A. From this

result it was clear that though 90% of the tape sections have Icis more than 100 A, the resulting

Ica of the whole tape was reduced by 4 A. This was because the sections of tape (10%) with Icis lower than 100 A produce more voltage at higher currents causing the electric field to reach its criterion (1µV/cm) before the current reaches its design critical value (100 A). In this case,

operating the tape at 96 A would lead to operational risk as there are some sections with Icis less than 96 A. To study a more realistic case, all the 3000 sections of tape were assigned normally

distributed random values for their Icis with a mean of 100 A (Icm) and varying standard deviation

(Ici) of 1 A, 3 A, 5 A, 7 A and 10 A. The Icis are generated such that their mean has a confidence level of 99% within the confidence interval calculated and shown below for various standard deviations

Confidence interval = mean ± � z= confidence coefficient √� 36

σ=standard deviation N= number for iterations (sections in this case) = 3000

Table 7: The value of coefficient (z) for different confidence levels

Confidence 99.75 99 98 96 95.5 95 90 80 68 50 level % z 3 2.58 2.33 2.05 2 1.96 1.645 1.28 1 0.6745

Table 8: Confidence interval of Icm for different standard deviations

Standard deviation (Ici) Confidence interval for mean of Icis 1 99.95 to 100.04 3 99.85 to 100.14 5 99.76 to 100.23 7 99.67 to 100.32 10 99.5 to 100.4

 �� = √ ∑ �� − �� = The Ica of the tape was calculated for each case of Ici. Monte Carlo method was used to

simulate a large number of such tapes and the average value of Ica was calculated for each

standard deviation in Icis. The percentage decrease in apparent critical current and percentage of

standard deviation (% Ici) in Icis were calculated as in Eq. 5-6 and plotted in Figure 23.

� � � � − � % �� = × ��   �� % = � × � 37

5 current of whole tape of whole current

percentage decrease in apparent critical inapparent decrease percentage 0 2 4 6 8 10 12

percentage of standard deviation of indivudual tape Ic 100 A

Figure 23: Percentage decrease in apparent Ic for certain longitudinal variation in Ic at different average Ics

6.2 Longitudinal Variation in Index Value

The effect of longitudinal variation of n value was also studied taking normally generated

random values for different sections of the tape with mean n- value (nm) of 30. The Ic of the tape was assumed to be constant along the length: 100 A. The individual section voltages were calculated as in Eq. 1. After finding the total voltage generated (as in Eq. 2), the electric field of the tape was calculated (as in Eq. 3) and the resulting E-I curve was fitted using the power law,

extracting the apparent index value (na). Monte Carlo method was used to simulate such tapes

and the average na was calculated.

Figure 24 shows the percentage decrease in apparent n value obtained for certain percentage of

standard deviation in the n value of sections of tape (% ni).

a a n − n % Decrease in n = × n  

i % = × n

14 12 10 8 6 4 2 0 percentage decrease in apparent n inapparent decrease percentage 0 2 4 6 8 10 12 14 percentage of standard deviation in n (%  ) i ni

Figure 24: Percentage decrease in apparent n for certain longitudinal variation in n

Figure 23 and Figure 24 indicate that, the more pronounced the deviation of Ici and n from their average values, the more pronounced is the reduction in their apparent values.

6.3 Results and Discussion

To investigate the effect of tapes having longitudinal variations in their characteristics on the performance of the whole cables, 300 tapes were generated each 30 meter long, with certain

standard deviation in the Ici along their length and with a mean of 100 A. 30 of these tapes were

used to form a model cable with a design Ic of 3 kA. The apparent Ic of the whole cable is nothing

but the sum of Icas of individual tapes in this case because the contact resistances and n values of all tapes are taken to be equal. Ten such model cables were formed with each cable having 30

tapes of certain standard deviation in Ic along their length. The n value of all tapes was assumed to be equal to 30 in this case.

120

100

as function of its total mean Ic mean total itsof functionas 0 Percentage of apparent Ic cable of Ic apparent of Percentage 0 2 4 6 8 10 12 Standard deviation in Ic along the length of tapes Series1

Figure 25: The apparent Ic of the cable as function of cable mean Ic for a certain standard deviation in Ici along the length of individual tapes of the cable. 39

It can be observed that as the deviation in Ic along the length of the tapes increase, the Ica of the cable decreases. This is shown in the below table.

Table 9: The ratio of apparent critical current to mean critical current of the cable for different standard deviations of tape Icis

Cable # (Ica/Icm)% of the cable for different standard deviations of tape Icis

Ici = 1 Ici = 3 Ici = 5 Ici = 7 Ici = 10

1 99.8 98.4 93.53 87.26 74.5 2 99.86 98.53 94.33 88.63 76.7 3 99.82 98.53 92.86 88.83 77.23 4 99.8 98.2 95.36 85.9 72.8 5 99.82 98.8 94.36 90.3 79.5 6 99.81 98.53 94.53 88.7 76.9 7 99.83 98.6 93.5 88.83 77.1 8 99.8 98.4 96.1 87.16 74.43 9 99.84 99 94.43 91.93 81.96 10 99.85 98.2 92.7 85.6 72.33

CHAPTER SEVEN

CONCLUSION

The experimental and model based studies on the current distribution in superconducting DC cable show that it is important to select tapes with a narrow distribution of properties while manufacturing superconducting cables. For short cable lengths, the critical current is the second major issue to be considered after the contact resistances of the tape ends. When the contact resistances are low (in the order of nΩ), even small variations in the critical current and the n value of individual tapes can lead to significant variation in the distribution of current at higher fractions of cable design critical current. Significant spread in individual tape parameters can lead to situations in which some tapes carry a current close to their critical current well before the cable as a whole reaches its design critical current. Such an uneven current sharing can lead to catastrophic damage to the cables. Monte Carlo simulations on superconducting tapes in long cable applications indicate that the longitudinal variations in Ic and n values of the tapes reduce the design/rated capacity of the cable and have a significant effect on its performance. Generally, manufacturers of superconducting tape provide the minimum Ic for a batch of superconductor, but short sectional variations of tape also affect the cable performance in a long length application. This is because the voltage produced by the lower Ic sections would be high enough to push the tape to reach its critical current well before the average Ic of the tape. Hence the sectional property distribution should also be considered for best performance of superconducting power cable design and operation.

CHAPTER EIGHT

FUTURE WORK / OUTLOOK

Mathematical models of cables are useful to analyze the behavior of superconducting cables. Hence these models should be used in design process to accommodate non uniform current distribution in superconducting cables. Complex models that include multi-phase and multi-layer design for superconducting cables are needed for better design and operational logistics. Temperature gradients and magnetic field also effect the current distribution in superconducting cables. So models to investigate these effects are required to be developed. Mathematical models combined with electrical models of superconducting cables will be useful for electric power system design and analysis.

APPENDIX A

MEASUREMENT METHOD FOR DC CRITICAL CURRENT OF SUPERCONDUCTING POWER CABLES Terms and Definitions

Critical temperature

The temperature beyond which the superconducting property is lost by a material in superconducting state is called as the critical temperature. The change from normal state to superconducting and superconducting to normal state is called the superconducting phase transition. Different materials have different critical temperatures.

Critical current

The current that causes a superconductor to reach its electric field criterion when conducted under its critical temperature is called as the critical current. The electric field criterion (Ec) is usually taken as 1 µV/cm.

Index (n)-value

It is the exponent obtained when current (I) - voltage (V) curve is approximated by the equation V α In, in a specific range of electric field strength.

Procedure to Determine Critical Current and Index Value

DC critical current of a superconducting power cable is determined by measuring the voltage of a certain length of cable while running DC current through it. Current is increased from 0 to a value where the voltage increases suddenly. During the current ramp, the I-V characteristics are recorded. The cable should be attached to the current contacting blocks by soldering. This block is a metal to let the test current in and is usually made of copper. Before soldering, the contact areas should be cleaned with an organic solvent such as alcohol. The soldering process must be done at a temperature safe for the cable (HTS tapes, insulation). The current leads are then bolted to the current contacting block. The voltage taps are usually installed on the superconducting cable. After cooling down the sample below its critical temperature, the sample is connected to the DC current source and to the voltage measurement device such as a data acquisition system

controlled by LabVIEW program. A regular speed sweep method or step-hold method is used for the current ramp for the DC critical current measurement. After measuring V, the Electric field (E) of the tape can be calculated by dividing the voltage by the length of the tape. Plot the E vs I for the given tape and the critical current is the current at which the tape reaches its Ec.

APPENDIX B

COPYRIGHT PERMISSION For: Figure 7 and Figure 9 The IEEE does not require individuals working on a thesis to obtain a formal reuse license, however, you may print out this statement to be used as a permission grant: Requirements to be followed when using any portion (e.g., figure, graph, table or textual material) of an IEEE copyrighted paper in a thesis: 1) In the case of textual material (e.g., using short quotes or referring to the work within these papers) users must give full credit to the original source (author, paper, publication) followed by the IEEE copyright line © 2011 IEEE. 2) In the case of illustrations or tabular material, we require that the copyright line © [Year of original publication] IEEE appear prominently with each reprinted figure and/or table. 3) If a substantial portion of the original paper is to be used, and if you are not the senior author, also obtain the senior author’s approval. Requirements to be followed when using an entire IEEE copyrighted paper in a thesis: 1) The following IEEE copyright/ credit notice should be placed prominently in the references: © [year of original publication] IEEE. Reprinted, with permission, from [author names, paper title, IEEE publication title, and month/year of publication] 2) Only the accepted version of an IEEE copyrighted paper can be used when posting the paper or your thesis on-line. 3) In placing the thesis on the author's university website, please display the following message in a prominent place on the website: In reference to IEEE copyrighted material which is used with permission in this thesis, the IEEE does not endorse any of [university/educational entity's name goes here]'s products or services. Internal or personal use of this material is permitted. If interested in reprinting/republishing IEEE copyrighted material for advertising or promotional purposes or for creating new collective works for resale or redistribution, please go to http://www.ieee.org/publications_standards/publications/rights/rights_link.html to learn how to obtain a License from RightsLink.

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BIOGRAPHICAL SKETCH

Venkata AB Pothavajhala was born in Narasaraopet, Andhra Pradesh, India to Chiranjeevi Sarma P. and Srilakshmi Purnima P. on March 14, 1990. He was raised in Hyderabad where he had most of his primary education from Mount Carmel High School and secondary education from Sri Chaitanya Jr. College. Venkata completed his Bachelor’s degree in Electronics and Communication Engineering from Swami Vivekananda Institute of Technology affiliated to Jawaharlal Nehru Technological University. He joined Florida State University in 2012 to pursue his Masters in Electrical Engineering. During his study, he worked as research assistant at Center for Advanced Power Systems (CAPS) in the area of High Temperature Superconductivity under the guidance of Dr. Sastry Pamidi.

He has published two papers during his Master’s Degree program. They are listed below: [1] J.-H. Kim, C. H. Kim, V. Pothavajhala and S. Pamidi, "Current Sharing and Redistribution in Superconducting DC Cable," IEEE Transactions on Applied Superconductivity, vol. 23, no. 3, p. #4801304, 2013.

[2] V. Pothavajhala, L. Graber, C. H. Kim and S. Pamidi, "Experimental and Model Based Studies on Current Distribution in Superconducting DC Cables," IEEE Transactions on Applied Superconductivity, vol. 24, no. 3, p. #48000505, 2014.