UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

High-voltage measurement using the quadrature method and permittivity-shielding Chavez, Patrick Pablo 2002

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2002-751139.pdf [ 14.74MB ]
Metadata
JSON: 831-1.0065370.json
JSON-LD: 831-1.0065370-ld.json
RDF/XML (Pretty): 831-1.0065370-rdf.xml
RDF/JSON: 831-1.0065370-rdf.json
Turtle: 831-1.0065370-turtle.txt
N-Triples: 831-1.0065370-rdf-ntriples.txt
Original Record: 831-1.0065370-source.json
Full Text
831-1.0065370-fulltext.txt
Citation
831-1.0065370.ris

Full Text

H I G H - V O L T A G E M E A S U R E M E N T USING T H E Q U A D R A T U R E M E T H O D A N D  PERMTTTIVrTY-SHIELDLNG by Patrick Pablo Chavez B . A . S c , The University of British Columbia, 1995 M . A . S c , The University of British Columbia, 1997 A THESIS SUBMITTED IN P A R T I A L F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Electrical and Computer Engineering) We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH C O L U M B I A October 2001 © Patrick Pablo Chavez, 2001  In  presenting  degree freely  at  this  the  available  copying  of  department publication  of  in  partial  fulfilment  of  the  University  of  British  Columbia,  I  agree  for  this or  thesis  reference  thesis by  this  for  his thesis  and  scholarly  or for  her  Department  DE-6 (2/88)  Columbia  I  further  purposes  gain  that  agree  may  be  It  is  representatives.  financial  permission.  T h e U n i v e r s i t y o f British Vancouver, Canada  study.  requirements  shall  not  that  the  Library  permission  granted  by  understood be  for  allowed  an  advanced  shall for  the that  without  head  make  it  extensive of  my  copying  or  my  written  Abstract  Using extensive finite element modeling, the quadrature method and permittivityshielding for measuring high voltage using electric field sensors were developed. The quadrature method is a technique that determines the necessary number of sensors that make point-like measurements of the field and the best positioning and weighting of those sensors for any chosen electrode geometry, for a given worst-case field perturbation due to external changes in the field (stray field effects), and for a particular minimum accuracy requirement. As a result, electrodes can be positioned much farther apart for a given number of sensors than was previously known to be possible, which results in lower field stresses in and around the electrodes requiring less insulation than was otherwise possible. Permittivity-shielding is a technique that offers an alternative to metallic shielding for the purpose of isolating electric field measurements from stray field effects. Materials with high dielectric constants, high-enough conductivities, or both supply the shielding by filling a space spanning from one electrode to the other and shield the field inside and nearby that space. With permittivity-shielding, electrodes are not required to be in close proximity with one another, avoiding the need for any special insulation. This kind of shielding also has the added benefit of moderating the field between the electrodes. 138 kV, 230 kV, and 345 kV optical voltage transducers (OVTs) each using three optical field sensors according to the quadrature method were constructed and tested. They meet the highest IEC and ANSI/IEEE accuracy requirements for instrument transformers even in the presence of substation-like changes in local geometry, such as the movement or installation of neighboring equipment. The 138 kV and 345 kV OVTs also use off-the-shelf resistors to supply  ii  resistive permittivity-shielding, and they maintain accuracy in the presence of severe stray field effects, typically caused by the mixture of moisture and pollution on an OVT's surface. These OVTs offer all the inherent advantages of optical sensor technologies but are simpler and safer in design than other OVTs. Their basic structure consists of an off-the-shelf insulator, three small optical field sensors, and possibly resistors (for resistive shielding).  iii  Table of Contents  Abstract  ii  Table of Contents  iv  List of Tables  vii  List of Figures  ix  Acknowledgements  xii  Preface  xiv  Chapter 1  Introduction  1  1.1  Motivation  1  1.2 1.3  Conventional Voltage Transformers Optical Voltage Transducers 1.3.1 Electric Field Shielding - Existing Technique 1.3.2 Electric Field Line Integration - Existing Techniques 1.3.3 Advantages Over Conventional Voltage Transformers  2 3 4 6 9  Tools for Modeling Electrostatic and Quasistatic Fields.  10  Introduction Boundary Integral/Finite Element Analysis Mesh Generation Postprocessing  10 11 15 17  The Quadrature Method  19  Overview Theory Simulations Error in an Applied Quadrature Formula 3.4.1 Positioning Error 3.4.2 Weighting Error 3.4.3 Blurring Error 3.4.4 Quadrature Error 3.4.5 Quasi-Quadrature for Known Position Offsets Special Case - Sensor-Dependent Field Application Conclusion  19 20 27 32 34 36 37 39 47 48 50 52  Chapter 2 2.1 2.2 2.3 2.4 Chapter 3 3.1 3.2 3.3 3.4  3.5 3.6 3.7  iv  Chapter 4 4.1 4.2 4.3 4.4 4.5 Chapter 5 5.1 5.2  5.3 Chapter 6  6.1 6.2  6.3  6.4  6.5  6.6  Permittivity-Shielding  53  Overview Description Application Effect of Frequency Dependence of Resistive Shielding on the Quadrature Method Summary  53 53 65  Optical Electric Field Sensors  75  Introduction Integrated Optic Pockels Cell 5.2.1 Description 5.2.2 Importance of IOPC Shape Miniature Bulk-Optic Pockels Cell  75 75 75 80 82  71 73  Development of Novel High-Voltage Optical Transducers for Metering and Relaying Applications 84 Introduction Integrated-Optic Voltage Transducer - Proof of Concept 6.2.1 Description 6.2.2 Simulation and Test Results 6.2.3 Summary and Conclusion Optical Voltage Transducer 6.3.1 Introduction 6.3.2 Principles of Design and Operation 6.3.3 Laboratory Test Results 6.3.4 Conclusion Study of Pollution Effects 6.4.1 Introduction 6.4.2 Laboratory Pollution Tests 6.4.2.1 Approach 6.4.2.2 Test Setup and Results 6.4.2.2.1 Aluminum Foil Tests 6.4.2.2.2 Pollution Tests. 6.4.3 Simulated Accuracies for Unshielded and Resistively Shielded OVTs in Dryband Conditions 6.4.4 Summary and Conclusion Resistively Shielded Optical Voltage Transducers 6.5.1 O V T Structures 6.5.2 Tests 6.5.2.1 Temperature Rise due to Resistors 6.5.2.2 Aluminum-Foil Test Results 6.5.2.3 Pollution Test Results 6.5.2.4 Ice Test Results 6.5.3 Conclusion Summary. V  84 85 85 89 93 93 93 94 98 105 106 106 107 107 108 110 114 120 125 126 126 127 128 128 130 132 135 136  Chapter 7  Summary and Conclusion  137  Bibliography  141  vi  List of Tables  Table 3.1. Weights and abscissas for different ./V and for a - 0  30  Table 3.2. Percent error for perturbations caused by a vertical ground plane  31  Table 3.3. Percent error for perturbations caused by an electrically floating sphere 50 mm below the most heavily weighted sample point 31 Table 3.4. Percent error of the Riemann sum for perturbations caused by a vertical ground plane for ./V = 100  32  Table 4.1. Voltage error in perturbed systems with varying shielding-tube permittivities. (a) Magnitude error (percent) and (b) phase error (minutes) 63 Table 4.2. (a) Voltage error in perturbed systems with varying ID and fixed 100 MQ shield resistance, (a) Magnitude error (percent) and (b) phase error (minutes).. 66 Table 4.3. (a) Magnitude error in percent and (b) phase error in minutes of iV-sensor standoff voltage transducer polluted with water  70  Table 6.1. Formula sample positions with respect to a and weights normalized to the middle sensor's (Sample #2) weight  89  Table 6.2. Simulation percentage errors in voltage measurement (AV) and percentage changes in local electric field measurements (AE (xi)) for a vertical ground plane 1.5 meters away  90  x  Table 6.3. Laboratory test percentage errors in voltage measurement (AV) and percentage changes in local electric field measurements (AE (xi)) for several external conditions 91 x  Table 6.4. Simulated magnitude errors due to extreme perturbations  97  Table 6.5. Simulated changes in E (xi) due to perturbations  97  x  Table 6.6. Formula sample positions, JC„ with respect to a and weights, cc„ normalized to the middle sensor's (#2's) calculated weight  98  Table 6.7. Magnitude errors due to safe perturbations  104  Table 6.8. Magnitude errors due to unsafe perturbations  104  Table 6.9. Changes in E (xi) due to perturbations  105  x  vii  Table 6.10. Magnitude and phase errors for no-shielding case and for pollution layer resistivites (a) 100 Qcm and (b) 1 k£2cm  121  Table 6.11. Magnitude and phase errors for center-tube-shielding case and for pollution layer resistivites (a) 100 Q c m and (b) 1 kQ-cm 123 Table 6.12. Magnitude and phase errors due to quarter-height dryband using aluminum foil of (a) the three-stage 138 kV O V T , (b) the single-stage 138 kV O V T , and (c) the 345 kV O V T  129  Table 6.13. Relative magnitude and phase changes in the electric field at the sensor locations  129  Table 6.14. Magnitude and phase errors due to pollution on the shed surface of the three-stage 138 kV O V T  131  Table 6.15. Magnitude and phase errors due to pollution on the shed surface of the 345 k V O V T  131  viii  List of Figures  Fig. 1.1. Example of voltage measurement using metallic shielding  6  Fig. 1.2. Example of voltage measurement using (a) continuous integration or (b) piece-wise integration  8  Fig. 2.1. A 2D unbounded BJFE domain  13  Fig. 2.2. Isoparametric mapping  16  Fig. 3.1. (a) The unperturbed two-electrode system, (b) a perturbed system with an infinite vertical ground plane, (c) and a perturbed system with an electrically floating sphere 50 mm below the most heavily weighted sample point (M.H.W.S.P). . . . 28 Fig. 3.2. (a) E distributions and (b) the perturbations for unperturbed and perturbed systems  29  Fig. 3.3. Convergence of an iterative solution process  50  Fig. 3.4. A high-voltage transducer  52  Fig. 4.1. Two-electrode structure (unperturbed system)  55  x  Fig. 4.2. (a) Magnitudes and (b) phases of E for dielectric tubes and (c) magnitudes and (d) phases of E for resistive tubes and for 1 V applied 58 x  x  Fig. 4.3. Perturbed systems involving (a) a ground plane, (b) a grounded sphere, and (c) a uniform resistive layer with an air-gap at the top 59 Fig. 4.4. Unperturbed and perturbed electric field magnitude distributions with and without shielding for an applied I V  60  Fig. 4.5. Voltage sensing standoff with shielding  68  Fig. 4.6. (a) Magnitudes and (b) phases of E for resistive tubes and for 1 V applied  73  Fig. 5.1. An IOPC sensing system  76  Fig. 5.2. An immersion-type IOPC  79  Fig. 5.3. Penetration factors versus height-to-width ratio, h/w  82  Fig. 5.4. BGOPC electric field sensor  83  x  ix  Fig. 6.1. (a) An insulating structure and (b) the same structure with a vertical ground plane 1.5 m away  86  Fig. 6.2. E along the insulator axis for 100 kV applied across the standoff insulator with and without a ground plane present 86 x  Fig. 6.3. The scaling function p representing the introduction of a vertical ground plane 1.5 m away from the standoff insulator  87  Fig. 6.4. The difference between F, the best (in the least-squares sense) 5 degree th  polynomial fit to p, and p  88  Fig. 6.5. Integrated-optic voltage transducer with a vertical grounded screen 1.5 m away. 92 Fig. 6.6. The unperturbed O V T system  94  Fig. 6.7. Axial electric field along the path of integration for an unperturbed O V T and a perturbed O V T with 140 kV applied  95  Fig. 6.8. p for a semi-infinite vertical ground plane 1.6 m away from the OVT's a x i s . . . . 96 Fig. 6.9. High-voltage test set-up  99  Fig. 6.10. O V T (a) magnitude errors and (b) phase errors  101  Fig. 6.11. Grounded screen perturbation 1.6 m away  102  Fig. 6.12. Grounded sphere perturbation 1.2 m away  103  Fig. 6.13. Truck perturbation  103  Fig. 6.14. OVTs at the Ingledow substation, Surrey, B C , Canada  105  Fig. 6.15. Al-foil test setup (prior to foil application)  108  Fig. 6.16. (a) Simulated E  np x  for the rated applied voltage of 80.5 kV and  (b) unperturbed p Fig. 6.17. Insulator with A l foil and a middle dryband  109 110  Fig. 6.18. (a) The magnitude, (b) the phase, and (c) a snapshot (at the peak of the voltage waveform) of p for top dryband 112 Fig. 6.19. (a) The magnitude, (b) the phase, and (c) a snapshot (at the peak of the voltage waveform) of p for middle dryband 113 Fig. 6.20. Polluted insulator 115  6.21. (a) The magnitude, (b) the phase, and (c) a snapshot (at the peak of the voltage waveform) of p for wet pollution 116 6.22. (a) The magnitude, (b) the phase, and (c) a snapshot (at the peak of the voltage waveform) of p for dry pollution 117 6.23. (a) The magnitude, (b) the phase, and (c) a snapshot (at the peak of the voltage waveform) of p for fog-pollution 118 6.24. Experimental E 's, for some of the tested cases using the experimental p and simulated E for the rated applied voltage of 80.5 kV x  119  6.25. O V T covered with ice at -18° C  132  x  unp  6.26. Accuracy of OVT under melting ice: a) magnitude error and b) phase error... 134 6.27. Field changes at sensor locations under melting ice: a) magnitude and b) phase displacement 135  xi  Acknowledgements  First and foremost, I would like to thank my parents, Pablo Chavez and Irene Chavez. At the root of my ability to achieve this work, and to achieve future goals successfully, is the foresight and discipline instilled in me at an early age by my father. The ability to passionately enjoy and explore all of life to the fullest I owe to my mother, and it helps me to shape my goals and gives me the courage to face adversity. This work is a testament to their hard work as parents. I would also like to thank Dr. Nick Jaeger. I believe this thesis to be one of my greatest accomplishments to date, and the quality, completeness, and usefulness of my thesis work are testaments to the thoroughness and diligence in Dr. Jaeger's guidance and supervision. Working with Dr. Jaeger has undoubtedly prepared me for and opened my eyes to a world of new opportunity in research and development, and I am very grateful to him for that. Construction of the optical voltage transducers presented here was accomplished in collaboration with NxtPhase Corporation. I would like to thank the research and development team at NxtPhase, in particular, Dr. Farnoosh Rahmatian, Director of Research and Development at NxtPhase and my co-supervisor. Dr. Rahmatian's project leadership enabled the speedy development of the optical voltage transducers from a concept to a marketable product. Working closely with Dr. Rahmatian has given me invaluable industrial experience, and I sincerely thank him for it. The NxtPhase team developed the optical electric field sensors with the accompanying analog and digital electronics and the data acquisition software that was used to collect the test data presented in this thesis. Acknowledgement goes out to the Center for Advanced Technology in Microelectronics for developing and supplying the integrated-optic field sensors that were used to construct the xii  first optical voltage transducer based on the quadrature method. I would like to thank Dr. Alina Kulpa for those many hours spent in the laboratory fabricating sensors. This work was financially supported in part by funding from the British Columbia Advanced Systems Institute and the Natural Sciences and Engineering Research Council of Canada.  xiii  Preface  Some of the material presented in this thesis is in publication or is under review for publication. Portions of Chapter 3, Chapter 4, and Section 6.3 have been submitted as three journal papers, respectively, for publication in IEEE Transactions on Power Delivery and are currently under review. These papers are as follows: P. P. Chavez, F. Rahmatian, and N. A. F. Jaeger, "Accurate voltage measurement by the quadrature method," submitted to IEEE Transactions on Power Delivery, November 8, 2000, P. P. Chavez, F. Rahmatian, and N. A. F. Jaeger, "Accurate voltage measurement with electric field sampling using permittivity shielding," submitted to IEEE Transactions on Power Delivery, November 30, 2000, F. Rahmatian, P. P. Chavez, and N. A. F. Jaeger, "230 k V optical voltage transducers using multiple electric field sensors," submitted to IEEE Transactions on Power Delivery, June 15, 2001.  The material of Section 6.2 has been adopted from the following conference paper: P. P. Chavez, N. A. F. Jaeger, F. Rahmatian, and C. Yakymyshyn, "Integrated-optic voltage transducer for high-voltage applications," in Applications of Photonic Technology 4, R. A . Lessard, G. A. Lampropoulos, Editors, Proceedings of SPIE Vol. 4087, 2000, pp. 1229-1237.  Also, Fig. 6.20 and Fig. 6.25 will be published in the following conference paper: F. Rahmatian, P. P. Chavez, and N. A. F. Jaeger, "Resistively shielded optical voltage transducer," to be presented at IEEE/PES Transmission and Distribution Conference, October 28 - November 2, 2001.  xiv  1  Chapter 1 Introduction  1.1  Motivation  Currently, and in the years to come, electric utilities in many countries around the world, including Canada and the United States, are and will be entering a deregulated, competitive business environment. Deregulation of the power industry involves partitioning it among private companies; some of these companies will concentrate on power generation, creating the power supply, while others will concentrate on transmission and distribution, delivering the power to retail (the public) and wholesale (industrial) customers on demand. In this scenario, electric power becomes a tradable commodity, similar to a natural resource like oil or gas. Trading power in a deregulated environment makes the processes of power revenue metering, protection relaying, and power quality monitoring high priorities in both the power generation sector and the transmission and distribution sector. Revenue metering is the quantification of power delivered and emphasizes measurement accuracy. Relaying is the protection of the power system and requires a wide measurement bandwidth to maximize the reliability or robustness of the power circuit. Power quality monitoring is the performance management of the power delivered and also requires a wide measurement bandwidth in order to observe harmonics. Sensor technology, specifically voltage and current transducers, is at the heart of all of these processes. Furthermore, the reliability and familiarity offered by established conventional technologies, on which the industry has been based for the past few decades, is simply not enough. Optical voltage measurement techniques using the electro-optic effect to sense the electric field associated with the voltage being measured have been under investigation in recent  2 years. This thesis advances the current state of the art by introducing and describing a new method and a new mechanism for obtaining an accurate voltage measurement using electric field measurements and demonstrating the practicality of these techniques through the design and construction of high-voltage instrument transformers combining existing electro-optic field sensing technology and these new techniques.  1.2  Conventional Voltage Transformers  Voltage sensors in the power industry convert a high-voltage signal to a scaled-down signal that is suitable for analysis. Conventional voltage sensors are primarily comprised of inductive voltage transformers and, to a lesser degree, capacitive voltage transformers. Inductive voltage transformers (IVTs) consist of an iron core and primary and secondary windings that are linked by the magnetic flux in the core. This technology is well established and offers reliable precision and accuracy as required for metering applications. However, IVTs have several inherent disadvantages. Size and weight are the most apparent shortcomings of this technology. The iron core windings, bushings, and general insulation are necessarily large and heavy components at high voltages. As the rated voltage increases, the size and the weight also increase. This results in expensive construction, installation, and maintenance of high-voltage IVTs, i.e., 69 k V or above. In addition to this, high-voltage IVTs suffer from a high sensitivity to electromagnetic interference. Furthermore, along with these transformers come a high inductance due to the nature of their operation and a nonlinear response due to the flux-voltage relationship in the core. Consequently, for measurement purposes, IVTs are not as transparent to the observed system as would be desired, resulting in a limited bandwidth and a small dynamic range of operation.  3 Capacitive voltage transformers (CVTs) consist of capacitor elements stacked on top of each other to form a voltage divider. Typically, the elements are made of a paper-oil dielectric. High voltage is applied to the top of the stack and the lower voltage existing across one of the capacitor sections is then measured, typically using a small inductive transformer to further step down the voltage (known as capacitively-coupled voltage transformers (CCVTs). CVTs are considerably less expensive than high-voltage inductive transformers. They also do not suffer from the same large weight and size of inductive transformers. However, concerns exist with respect to the stability of CVTs over time and temperature. Temperature- and timedependence of the paper-oil electrical properties and how they relate to drifting outputs and failure of the device has been under investigation [1], [2]. Also, CVTs have poor high-frequency response, making them unsuitable for many relaying applications.  1.3  Optical Voltage Transducers  Although in its early stages of development, optical voltage transducer (OVT) technology offers an attractive alternative to the previously mentioned conventional technologies. Following in the footsteps of their already successful and proven optical current transducer counterparts, OVTs offer promising and superior alternatives over the conventional technologies. Most OVTs rely on electro-optic principles, whereby the indexes of refraction of electro-optic materials are influenced by an electric field. If the effect is linear, i.e., if the change in index of refraction is linearly proportional to the electric field, then it is referred to as the Pockels effect, and several crystals have this property [3]. If the effect is quadratic, then it is referred to as the Kerr effect [3]. OVTs that rely on piezoelectric effects, whereby strains in a crystal are induced by an electric field [3], have also been constructed [4], [5], [6].  4 A voltage, or potential difference, between two points has an electric field associated with it, and sensing the electric field can be used to measure the voltage. Two basic, independent principles are used in making accurate measurements of the voltage by measuring the associated electric field. They are referred to here as "electric field shielding" and "electric field line integration". An O V T uses optical electric field sensors, and its operation relies on at least one of these two principles.  1.3.1  Electric Field Shielding - Existing Technique  For a given system of conductors, all with imposed electric potentials, there exists a unique electric field in the medium surrounding the conductors; this is known as the uniqueness theorem [7]. Qualitatively speaking, although the electric field in the neighborhood of one conductor is more influenced by the voltage on that conductor than the voltages on the other conductors, the electric field changes everywhere for any change in voltage on any conductor in the system. The electric field can also change with changes in the electrical properties, e.g., the conductivities or permittivities, of the media comprising the system. By the equivalence principle [7], the same phenomenon can be expressed in terms of free charges on the surfaces of the conductors that redistribute themselves with any change in voltage on one or more of the conductors or in terms of polarization charges that depend on boundary conditions at media transitions. Since each charge in space interacts with every other charge in space by way of its electric field, as governed by Coulomb's law, it is clear that there exists mutual field coupling among all the conductors and throughout the system. Since the electric field at any point in space is a function of the voltages applied to all the conductors and the electrical properties of the medium, the field measured by an electric field sensor, e.g., an electro-optic sensor, is a function of all these voltages and the medium. For the sensor's output to be a measure of only one voltage difference  5 (between a pair of conductors), the electric field in the region of space where the sensor is located has to be properly shielded from all other voltages on other conductors and the geometry between the measured conductors must be fixed. The electric field around OVTs used in substations is subject to dynamic influences, typically referred to as "stray field effects" in power engineering. Sources of these influences include any changes in the relative location of the O V T with respect to any surrounding objects, e.g., any substation equipment, and changes in the atmospheric environment, e.g., precipitation and pollution deposits. The OVTs that have optical electric field sensors somewhere between two electrodes to determine the voltage between those electrodes [8]-[12], which are typically coupled to a highvoltage conductor and ground, use metallic shields to substantially isolate the region between the electrodes, inside which the sensor(s) is(are) located, from external sources of stray field. Typically, the electrodes themselves are specially shaped to almost completely surround the isolated region (see Fig. 1.1). Because this region is fixed in geometry and is influenced practically only by the voltage difference between the two conductors, the relationship between the electric field, everywhere in the region, and the voltage difference is fixed. Hence, the sensor measurements are simply related to the voltage difference through a constant scaling factor, i.e., through a calibration constant. This technique is referred to as "metallic shielding". While the metallic shields need to substantially surround the isolated region, they also must not form an electrical path between the electrodes, i.e., a short circuit. This means there must be at least one, relatively small insulating gap in the shielding. At high voltages, such a gap poses a problem since high electric field stresses exist across it, and the insulating medium must be able to support these stresses to prevent flashovers and eventual destruction of the structure. So, special insulation, such as SF6 gas, possibly under pressure, is typically required, which complicates the design and may raise environmental concerns. For example, the breakdown  strengths of air and S F gas at atmospheric pressure are typically ~3 kV/mm and ~8 kV/mm, 6  respectively [13].  Electrode 2 Electrically Isolated Region Electric Field Sensor High E-field Stresses V  /  Electrode 1 Fig. 1.1. Example of voltage measurement using metallic shielding.  1.3.2  Electric Field Line Integration - Existing Techniques  In static and quasi-static systems, the voltage difference, V , between two points is equal to ba  the line integral along any path between those two points of the component of the electric field tangent to that path; this is mathematically expressed as  V =-$E-dl ba  where a and b, are the two points, T  ab  [14].  ,  (1.1)  is any path between a and b, and E is the electric field  7 Some OVTs apply this relationship to accurately determine the voltage between two electrodes by using a Pockels-effect crystal spanning a region between the two electrodes [15], [16]. Light is transmitted through the crystal longitudinally from one electrode to the other at least once and is modulated by the longitudinal electric field component along the entire path through the linear electro-optic effect. In this way, the modulation gives a measure of the integral of this electric field component and, therefore, a measure of the voltage between the electrodes. A n O V T that uses piezoelectric crystals between pairs of electrodes and having one arm of an optical fiber interferometer wrapped around them in order to measure the integral piezoelectric effect of the electric field component longitudinal to the crystals has also been developed [6]. This is referred to as a continuous integration method. Such OVTs require a good quality crystal having to ensure a uniform sensitivity along its length. Since the crystal must span the entire distance between the electrodes and the crystal is practically limited in size (typically < 25 cm), the electrodes are necessarily in close proximity, which, as in the case of the small gaps for metallic shielding, requires special insulation to support large electric field stresses at high voltages. Also, since the crystal is in contact with both electrodes, mechanical constraints in the design play a critical role in order to ensure immunity to vibration and safety against "triple point" effects, which occur at high voltages where the crystal, the insulation, and the electrode meet [6], [16]. See Fig. 1.2(a). Other OVTs apply (1.1) by positioning a series of small electric field sensors along a straight path between the electrodes being measured [5], [17]. The sensors are aligned to the path and are equally spaced apart. Light is transmitted through these electric field sensors, and the sensors have equal sensitivities and are oriented in such a way so that the light passing through them is sensitive only to the electric field component tangent to the path of alignment. In this way, the total modulation effectively gives a measure of the Riemann sum approximating the integral in (1.1) and, therefore, also the voltage. This can be expressed mathematically as  8  N  (1.2)  where AJC is the spacing between the sensors, Af is the number of sensors, and E (xi) is the electric x  field component that is tangent to the path, in this case chosen to be along the x-axis, and that is sensed at the ith sensor. This is referred to as a piece-wise integration method. These OVTs require electric field sensors that are virtually identical in their sensitivities to have a good approximation of the voltage. Also, Ax must be chosen so that the variation in E (x) along each interval is small, in which case the Riemann sum accurately gives a measure of x  the voltage despite possible variations in the electric field geometry. Consequently, a large number of electric field sensors must be used for a large separation between the electrodes, or the electrodes must be closer together for fewer sensors, in which case the problem of a small gap requiring special insulation arises again. The mechanical challenge of precisely mounting a large number of sensors also exists. See Fig. 1.2(b).  Electrode 2  Electrode 2  V  2  S\. Distributed £g\ V Linear Electric [ \ \ Field Sensors  Linear Electric Field Sensor High E-field Stresses  Vl Electrode 1 (a)  High E-field Stresses  X V  _  Vi  Electrode 1 (b)  Fig. 1.2. Example of voltage measurement using (a) continuous integration or (b) piece-wise integration.  9 1.3.3 Advantages Over Conventional VTs  In contrast to conventional VTs, OVTs rely on electro-optic or piezoelectric principles. Therefore, O V T technology can have inherent, advantageous features, such as wider bandwidth, larger dynamic range, and lighter weight, that are unobtainable with IVT and C V T technologies. A wider bandwidth and larger dynamic range make it possible for a single O V T to be used for both metering and relaying applications simultaneously, which is not easily achieved with conventional technology. Also, light transmission to and from the region of measurement is typically done using optical fibers. So, the observer is completely electrically isolated from the point of measurement, and the measurement itself is non-intrusive to the power circuit. Also, using optical fibers brings inherent immunity to electromagnetic interference and compatibility with emerging low-power analog and digital equipment used for monitoring and relaying applications. Furthermore, the lighter weight and smaller size of OVTs makes for cheaper installation and better seismic performance than that of conventional VTs. These are the primary advantages of using optical sensors in high-voltage environments over their conventional counterparts.  10  Chapter 2 Tools for Modeling Electrostatic and Quasistatic Fields  2.1  Introduction  In order to study the feasibility of the application of electric field sensors to measuring voltage, finite element analysis tools were developed that could effectively model the electric field around substation-like structures, such as high-voltage standoff insulators and transmission lines. Most importantly, the modeling must be such that it accurately simulates changes in the electric field due to changes in the positions and material properties of such structures, such as the effects of pollution or the installation of new equipment. Numerical methods that have been used for electric field modeling of high-voltage structures are the standard finite element method [18], the finite difference method, the charge simulation method [19], [20], [21], [22], the boundary element method [23], and others [24], [25]. These methods each have their limitations. The standard finite element method and finite difference method can only model bounded domains; the charge simulation methods can only deal with simple geometries and material inhomogeneities before they become impractical, and the computed electric field is inaccurate near the simulated charges themselves; and the boundary element method can also only handle simple material inhomogeneities, and it cannot easily account for charge sources. A specialized finite element processor involving the use of both the boundary integral equation and the finite element method along with appropriate preprocessing (mesh generation) and postprocessing tools have been created in order to offer the efficiency, accuracy, and flexibility that were ultimately needed for developing the quadrature method and permittivity-  11 shielding techniques. These software tools are written in M A T L A B code and are based, in part, on the theoretical methods described in [26] and [27].  2.2  B o u n d a r y Integral/Finite E l e m e n t Analysis  Finite element analysis consists of approximating an unknown function, in this case, the electric potential, by a low order polynomial with unknown coefficients over a subregion, or element in the solution domain. A l l the elements together span the entire finite domain, and qualitatively speaking, each element interacts with its neighboring elements through their common boundaries. The behavior of the function at the edges of those elements that border the solution domain is pre-defined by the solution domain's boundary condition specifications. In solving for the polynomial coefficients by way of a matrix inversion, the boundary conditions are, in effect, propagated throughout the solution domain via element boundary interactions as dictated by the governing system equations, in this case the Poisson equation. Software based on the technique described in [26], that removes the limitation of a finite domain, has been developed. A basic description of this technique, referred to as the boundary integral/finite element (BIFE) method, is given in the following paragraphs, and details on the standard finite element method can be found in various textbooks such as Zienkiewicz [28], Buchanan [29], and Burnett [30]. The finite element method solves differential equations, handling spatial inhomogeneities, and irregularly shaped boundaries with relative ease provided that the solution domain is bounded. For electrostatic field problems (Poisson equation), the unknown electric potential §(x,y,z) is approximated by a summation of shape functions N(x,y,z) = [N\(x,y,z) N2(x,y,z) N3(x,y,z) •••], with unknown coefficients O = [cpi <> |2 <)>3 ...] and is written as T  12  <> | = Nfc  (2.1)  The coefficients represent the value of < > | at a particular point, or node, in.space. Completing the finite element formulation of a differential equation through a variational or weighted residual approach yields a matrix equation which is used to solve for <&:  F = K<3>  The boundary integral equation expresses the function that satisfies the governing differential equation over the entire domain (possibly infinite) in terms of the function over a closed contour (for a 2D-domain) or surface (for a 3D-domain), C, in or around the domain. For the case of source-free electrostatic field problems, the function is (j), and the differential equation is the Laplace equation. The boundary integral equation is given by  where G is the Green's function of the Laplace equation. Substituting (2.1), the boundary integral equation is rewritten:  (2.2)  This expression can be used to generate the following constraint equation:  13 cp = Mcp/ fi  (2.3)  where cp represents the nodal values of the potential field at the boundary B of the boundary B  integral/finite element (BIFE) region, and <•>/ represents the values of points inside the BIFE and FE regions. The BIFE region, between the boundaries B and C, must be source-free, linear, and homogeneous, and the finite element (FE) region, inside C, can be inhomogeneous, anisotropic, active (with sources), and nonlinear. The remaining region, external to B, is referred to as the boundary integral (BI) region and is also source-free and homogeneous. Fig. 2.1 shows the different regions of an unbounded 2D BIFE domain and their governing equations.  Fig. 2.1. A 2D unbounded BIFE domain.  14 For a problem with an open, infinite domain, the original finite element formulation can be rewritten in terms of interior nodes and boundary nodes that span a particular region of interest:  "K  7I  _ B/ K  K  K  / B BB_  Substituting the boundary integral constraint (2.3) results in the following expression, which is the hybrid boundary integral/finite element formulation:  F/=(K +K M)& f l  / B  /  Solving this equation gives the potential field inside the finite region of interest (FE Region). With O/ determined, the constraint equation, (2.3), is evaluated to give the nodal values of the potential at the boundary. Finally, to calculate the potential at any other point in space, i.e., the B l Region, the boundary integral equation, (2.2), is used. If there are many small regions of interest containing nonlinearities, inhomogeneities, sources, and/or other irregularly shaped boundaries with Dirichlet, Neumann, or mixed conditions, each can be contained in a local BIFE/FE region with its own B and C boundaries. In such a scenario, the closed integration path of the boundary integral can be thought of as the combination of all the local C boundaries together with overlapping branches that connect these boundaries to form a single closed path. Since the net contribution of the overlapping branches to the integral is zero, the result is a series of local BIFE/FE regions whose boundary conditions are coupled to each other in the B l region by way of the boundary integral equation. This is a very powerful feature of the BIFE technique and is the main reason for choosing it over other  15 specialized finite element schemes such as infinite elements [31], substructuring [32], and mapping [33] that also remove the finite domain limitation of the finite element method. It allows for local meshes to be created for individual power structures, e.g., power lines or buses, standoffs, corona rings, etc., which can then be easily placed together in an unbounded domain to model various configurations. Following is a description of the mesh generation tools that were developed to exploit this feature.  2.3  Mesh Generation  A preprocessor was developed for the purpose of creating the input data for the finite element processor. Input data includes information regarding boundary conditions, charge distributions, material distributions, and geometry specifications. A semi-automatic mesh generation technique based on [27] and various other tools that make possible an object-oriented approach to creating meshes for relatively complex configurations and that take advantage of the BIFE formulation described above are discussed below. Prior to mesh generation, a minimal amount of input data must be manually entered by the user. The finite element domain must be divided into zones. For two-dimensional domains, zones have four sides with four corner nodes and four midsize nodes. Each side has a parabolic curvature defined by its three nodes. For three-dimensional domains, zones have six faces with eight corner nodes and twelve mid-edge nodes. Each face has a bi-parabolic curvature defined by its eight nodes. Zones are chosen in such a way as to define regions of a particular material, to border boundaries with special conditions and/or complex shapes, or to aid in the grading process of the final mesh (described next). Through the use of isoparametric curvilinear mapping of quadrilaterals (2D) and hexahedra (3D), the coordinates in a normalized domain ranging from -1 to 1 in each dimension  16 can be easily transformed to coordinates in the curved zone as shown in Fig. 2.2. So, for a given zone, which has particular material properties and boundary conditions, a mesh grading in each dimension of the normalized domain is specified. Based on this grading information, nodes are defined in the normalized domain, elements are defined by appropriate grouping of nodes along with the material and relevant boundary conditions of the zone, and the nodal coordinates are then mapped to the curved domain resulting in a finite element mesh. Care must be taken to ensure that adjacent zones have the same grading along their common edges (2D) or faces (3D) as to maintain connectivity of elements across zonal boundaries and to prevent "dangling" nodes  Normalized Space  Curved Space  Fig. 2.2. Isoparametric mapping.  Isoparametric mapping is commonly used in a finite element processor in order to numerically compute spatial integrals over arbitrarily shaped elements. In fact, the zones are actually parabolic serendipity elements, which are discussed in various textbooks on finite element analysis [28], and the mesh generating routine borrows the mapping routines of the finite element processor used here. The two-dimensional mesh generator is capable of generating linear, parabolic Lagrangian, or parabolic serendipity quadrilateral elements and  17 linear or parabolic triangular elements. The three-dimensional mesh generator is capable of generating linear or parabolic serendipity hexahedral elements. When constructing a mesh for complex geometries, it is convenient to build the mesh part-by-part. Taking advantage of the clearly defined grading that is used in the mesh generation scheme described above, a mesh generating tool has been developed that connects finite element mesh sections. In a sense, this allows for an object-oriented approach to building finite element meshes and is analogous to building complex "Lego™" structures from various simpler, smaller blocks that fit together. So, a complex geometry can be attacked in sections, or objects, with the mesh generator, and the mesh tool is then used to connect these objects to give the final mesh. The mesh tool can also shift, rotate, and mirror a mesh object. Individual, local finite element meshes are also treated as objects and can easily be put together in an unbounded boundary integral domain with virtually any orientation and position. The result is great flexibility and time-saving when modeling configurations with complex geometries and/or many individual objects in an unbounded domain.  2.4  Postprocessing  Various postprocessing routines have been developed to manipulate the raw data created by the processor. The raw data is simply the electric potential at each node of the finite element mesh. In order to find the value of the potential at any other points in the problem domain, either interpolation or the boundary integral equation, (2.2), is required. If the point of interest lies in the FE region or BIFE region, the element that contains the point must be found. This is accomplished by mapping the point and each element to an orthonormalized space in which it is easily determined whether the point lies inside the element or not. This mapping, from curved  18 space to orthonormalized space, is the inverse operation of the mapping described earlier for the processor and preprocessor. For all element types other than the linear triangular elements, this inverse mapping is a nonlinear problem. A Newton-Raphson iterative technique has been developed and is used to solve the problem. Once the element containing the point of interest is found, (2.1) is used to find the value of the potential at that point. For points in the B l region, external to the F E and BIFE regions, the boundary integral equation is used. The order of interpolation in the finite element domain is the same order as that of the element. Although a finite element solution using standard isoparametric elements is a function that is continuous across element boundaries, the derivative of the solution is discontinuous from element to element. Since interest lies in the study of the electric field, which is the divergence of the electric potential, both polynomial curve-fitting and linear or cubic spline interpolation are used in order to obtain a smoother representation of the data for analysis. In power systems, primary voltage is either constant or sinusoidal with frequencies of 50 Hz or 60 Hz. Technically, the energy field associated with this system is electromagnetic and is governed by Helmholtz' Wave Equation [7], [14]. However, since power structures are typically much smaller than a wavelength (5000 km at 60 Hz in free space), quasi-static conditions apply [33]. Furthermore, for sinusoidal sources under quasi-static conditions, the resulting electric field is also sinusoidal, and the potential and electric field components can all be represented by magnitudes and phases, or complex numbers. Also, while dielectric materials are modeled with real permittivities, resistive materials are modeled with imaginary permittivities (see Section 4.2).  19  Chapter 3 The Quadrature Method  3.1  Overview  A novel concept that is the basis for designing a new type of voltage transducer (VT) is presented here. This concept allows for VTs that require only a few electric field sensors, each of which measures a component of the electric field, ideally at a point in space. By using alldielectric electric field sensors in regions of high electric field intensity, such a V T is safe, even at very high voltages. Therefore, the all-dielectric designs of small optical electric field sensor technology (see Chapter 5) are particularly well-suited for this purpose. Furthermore, by using optical electrical field sensors in this type of V T , resulting in an O V T , one gets all of the inherent advantages. The design method consists of obtaining a numerical integration formula (Gaussian quadrature) for the V T and for the number of sensors to be used. The resulting numerical integration formula is an approximate electric field line integration using the sensor readings and is such that the variation in the formula due to stray field effects between the electrodes, caused by external voltage sources or conductors or other environmental influences, is minimal for the given number of electric field sensors. In the following sections, the theory behind the quadrature method is explained. Results from numerical simulations demonstrate the effectiveness of the method. Also discussed is how the quadrature method could be directly applied to design high-voltage transducers.  20 3.2  Theory  It is desired to find the voltage difference between two points by measuring the electric field at at least one point surrounding them. The voltage between two points, b and a, is calculated from the surrounding electric field by  (3.1)  where the path of integration r  is any path in space from a to b (see for example [14]). Using a  ab  cartesian coordinate system and letting T be along the x-axis with a at the origin, (3.1) can be ab  written as the line integral  b  ,  V =-JE (x)dx o b  x  (3.2)  where E (x) is the x-component of the electric field, along the x-axis. The problem of x  representing the voltage by a finite number of electric field samples can be approached by approximating the integral in (3.2) with a finite weighted sum  b  /V  V =-JE (x)dx~-^a E (x ) b  x  o  i  x  i  i=i  (3.3)  21 where a, is the weight of the ith sample, x,- is the position of the ith sample, N is the number of samples, and E (xi) is the x-component of the electric field at JC,-. If the voltage difference x  between b and a is fixed, the field along the path of integration may vary depending on conductor geometries and the dielectric medium. So, it is desired to have a weighted sum that remains virtually unaffected, within a given accuracy, for a fixed voltage difference between b and a despite possible variations in E (xi). x  Bohnert and Nehring [5] describe a unique voltage sensor made up of many piezoelectric electric field sensors whose positions are equally spaced by Ax and whose outputs are equally weighted, i.e., a, = Ax. In this way, the integral in (3.3) is actually approximated by a Riemann sum [34]. Using a Riemann sum, the area under E (x), i.e., the integral of E (x), is found by x  x  summing the areas of equally spaced adjacent rectangles along the x-axis. In this case, Ax is chosen so that the variation in E (x) along each width is small, in which case the sum accurately x  gives a measure of the voltage despite possible variations in the electric field geometry, i.e., N  V = -^ b  AxEj (x,-). While being very straightforward, such an approach requires many sample  i=i  points for a small span of integration. Consequently in this approach, Bohnert et. al. use 22 electric field sensors spanning a 15.4 cm length. It will be shown that, using more sophisticated integration techniques, significantly fewer sensors can be used to measure voltage while still maintaining a high degree of accuracy in the presence of variations in the field geometry. Approximating a definite integral by a finite sum is often done using Cotes and Gaussian quadrature formulas. Tables containing the abscissas and weights of such formulas are readily available (see for example [35] and [36]). Cotes formulas use equidistant abscissas and include the well-known Simpson formula and trapezoidal rule. Gaussian quadrature formulas include Gauss-Legendre, Gauss-Laguerre, Gauss-Hermite, and Gauss-Chebyshev formulas, which are  22 widely used to efficiently perform integrations for many numerical differential equation solvers, e.g., [28], [37], and [38]. For a given number of sample points, a Gaussian quadrature formula can be more accurate than a Cotes formula if it is appropriately chosen for the particular definite integral to be approximated. For example, if the definite integral has a finite domain and a smooth, continuous integrand, Gauss-Legendre quadrature should be chosen, whereas if the definite integral has a domain spanning from zero to infinity and a decaying integrand, a Gauss-Laguerre quadrature is more appropriate. Furthermore, an Appoint Gauss-Legendre quadrature formula is exact if the integrand is a polynomial of degree 2N - 1 pr less, and an Af-point Gauss-Laguerre formula is exact if the integrand is the product of the appropriate kernel, in this case the decaying exponential e' , and a polynomial of degree 2N - 1 or less. Gauss-Hermite and Gaussx  Chebyshev quadratures have similar traits [35], [36]. An Appoint Gauss-Legendre formula could be used to give the abscissas and weights in (3.3) provided that the electric field component E , the integrand of the integral, can be x  reasonably represented by a polynomial of degree 2N - 1 or less. Hence, Gauss-Legendre quadarature would likely be the first choice for solving (3.3), however a better approach is to develop a customized Gaussian quadrature formula which accounts for the basic shape of E , x  referred to as E  unp x  (for E unperturbed), along the path of integration. Clearly, for a customized x  Gaussian quadrature formula, the integration will be exact if the integrand is the product of E  unp x  (E  unp x  is now the kernel) and a polynomial of degree 2N - 1 or less. Here, the measured electric  field, E , is expressed as the product of the kernel and a scaling function, p, i.e., x  E (x) = x  p(x)E (x) u np x  23 Techniques for finding such quadrature formulas exist. The technique that is used here requires that E  unp x  be known. E  unp x  is preferably the E that exists in an unperturbed system. E"  np  x  can be  determined analytically or experimentally but, in practice, is most easily determined through the use of field modeling with the aid of a computer and numerical methods (see Chapter 2). Here, we refer to this approach for finding a, and x, in (3.3) as the "quadrature method". The process of determining our quadrature formula (see for example [39]) begins by substituting  E (x) = o(x)E P(x) u n  x  x  and  (3.4)  into (3.3) which gives  b  N  (3.5)  o  Letting p be polynomials of ascending degree, specifically choosing p(x) = 1, x, x ,..., x 2N-1 2  and forcing (3.5) to be exact for these p gives 2N equations:  24 m = p + p + . . . + p, 1  0  2  m  =P x  m  =p x  x  1  1  2  2  1  2  +p x  1  2  v  + ... + $ x N  N  +p x|+... +p ,xJ 2  A  (3.6)  where  u  m = k  ^E {x)x dx u np  k  x  This nonlinear set of equations, (3.6), can be solved by first defining the characteristic polynomial  N  N  P(x) = Y[(x-x ) = Y^C 'k  x  i  k  k=0  i=l  where the x are also the roots of P and the Q are the coefficients of P. Then, the first equation t  in (3.6) is multiplied by Co, the second by C\, and so forth for the first ./V equations, giving  N N £ c , m , = £ p , . P ( * , . ) =0  *=0  .  i=l  Next, skipping the first equation, the next N equations are multiplied by the C* as before, and the results are summed yielding  25 N k=0  This step is repeated until a system of N linear equations results  N ^ k k=0 C  m  k + l = °  where I = 0,1,2,...,N-l. Now, letting CN = 1 and assuming \m | * 0, the remaining C* are found k+l  by solving the linear system. Since the system is real, the Q , and, therefore, also P, are real. Knowing the Ck, the roots of P are computed giving the abscissas x,. Following this, the p\ are found by solving any N equations of (3.6). Finally, the weights oc, can be determined using (3.4). From (3.6), it can be seen that the resulting Gaussian quadrature formula  i=i  which is, in fact, (3.3), is exact if p is any polynomial of degree 2N - 1 or less. As long as E  unp x  is of constant sign, there are N real, distinct zeros of P, i.e., real abscissas  in (3.3), between a and b. The proof for this is available in [39], and a similar proof is given here. Since the characteristic polynomial is zero at its roots and the quadrature formula is exact if p is any polynomial of degree 2N - 1 or less, the following expression is true:  26 b  0  where P is the characteristic polynomial, and Q is any polynomial of degree N -1 or less. Decomposing P into two polynomials gives  P(x) = R{x)l(x)  where R is a real-valued polynomial having M real roots (M <N) inside the interval between a and  and / is a real-valued polynomial with roots that are real and outside the a-b interval or  that are complex conjugate pairs. Letting Q = R gives  b  0  If E"  np  is of constant sign, this expression is a contradiction since the integral cannot be zero  when there are no zero crossings in the integrand. Therefore, P must have N distinct, real roots inside the a-b interval, and the sample positions exist in reality. In the examples given below, the resulting weighted-sum approximation of the voltage is minimally affected by changes in the E {x ) that are caused by unforeseen and/or variable x  t  external influences, e.g., the introduction of a conducting surface in the vicinity of the voltage sensor. The severity of these "stray field effects" is manifested in the shape of p. Again, if p is a polynomial of degree 2N - 1 or less, the formula is exact. So, the accuracy, for a given number of electric field sensors, N, is determined by the shape of p. Similarly, the number of sensors  27 chosen for a particular application will depend on the desired accuracy and on the worst-case shape of p that is to be expected.  3.3  Simulations  In order to demonstrate the effectiveness of the quadrature method, an unperturbed system is modeled and perturbations are simulated using BIFE analysis (see Chapter 2). The unperturbed system consists of two spherical electrodes which have diameters of 200 mm and a center-to-center spacing of 2000 mm and which are suspended in space over a ground plane 1000 mm below the center of the lower electrode. The 2000 mm electrode separation is similar to the line-to-ground separation found in a 230 kV voltage class structure, such as a high-voltage standoff. The upper electrode is at 1 V while the lower electrode is grounded, i.e., at 0 V . Fig. 3.1(a) illustrates this configuration. For this case, E  unp x  is defined to be the x-component of  the electric field along a straight line between points a and b. E  unp x  is found numerically and is  then used to compute the quadrature abscissas and weights for the cases of N = 1,2,3,4. Table 3.1 gives the resulting weights and abscissas for these four cases, and Fig. 3.2 shows E  np x  and  the perturbation p. By definition, p is uniform having a value of 1 for an unperturbed system. Perturbed systems constitute systems with other conductors nearby the electrodes and/or variations in the dielectric medium. The electric field distribution in a perturbed system is different than that in the unperturbed system. While the voltage difference between the two electrodes is fixed at 1 V in the simulations, the value calculated by our formula is slightly different because p cannot necessarily be exactly approximated by a polynomial of degree 2N - 1 or less (even for N as large as 4). The  28  Fig. 3.1. (a) The unperturbed two-electrode system, (b) a perturbed system with an infinite vertical ground plane, (c) and a perturbed system with an electrically floating sphere 50 mm below the most heavily weighted sample point (M.H.W.S.P).  29  0  200  400  600  800  1000  1200  1400  1600  1800  x (mm) (a) 1.4  1.2  /  1.0  0.8 Vertical Ground Plane (d = 200 mm) Floating Sphere (h = 1500 mm, d = 200 mm)  0.6 4  0.4  /  /  /  A  0.2  0.0  200  400  600  800  1000  1200  1400  1600  1800  x (mm) (b) Fig. 3.2. (a) E distributions and (b) the perturbations for unperturbed and perturbed systems. x  30 Table 3.1. Weights and abscissas for different N and for a - 0. N 1 2 3  4  i  1 1 2 1 2 3 1 2 3 4  a, 1228.39 1859.06 394.56 679.77 932.05 215.27 318.80 781.66 541.50 145.70  xi (mm) 1550.28 465.19 1688.30 169.25 1172.16 1731.28 90.21 723.74 1423.14 1751.94  difference between our calculated value and 1 is taken to be the error. A perturbed system with an infinitely extending, vertical ground plane positioned a distance d from the center of the electrodes is simulated (see Fig. 3.1(b)). This ground plane perturbs the field, changing it along the line of integration between points a and b. Table 3.2 gives the errors for d = 2000 mm to d - 200 mm. Another perturbed system, with an electrically floating sphere positioned 50 mm below the most heavily weighted sample point (see Fig. 3.1(c) and Table 3.1), giving a large local disturbance, and a horizontal distance d from the electrodes is also simulated. Table 3.3 gives the error of the quadrature for d = 1000 mm to d = 200 mm. Generally, as the sources of electric field disturbance are brought closer to the integration path, the less smooth the corresponding p becomes. Fig. 3.2 shows E and p for the case of the x  vertical ground plane 200 mm away from the x-axis and for the case of an electrically floating sphere at a distance 200 mm horizontally and 50 mm vertically from the sensor in the N = 1 case.  31 Table 3.2. Percent error for perturbations caused by a vertical ground plane. d 2000 1500 1000 800 600 400 200  N= 1 -0.79 -1.35 -2.52 -3.29 -4.21 -4.22 8.48  N=2 0.00 0.02 0.11 0.14 0.01 -0.99 -3.87  N=3 0.02 0.02 0.03 0.04 0.07 0.16 -1.22  N=4 0.01 0.01 0.01 0.01 0.00 0.01 -0.06  Table 3.3. Percent error for perturbations caused by an electrically floating sphere 50 mm below the most heavily weighted sample point.  d 1000 800 600 400 200  N= 1 -0.04 -0.05 -0.06 0.42 4.04  N=2 0.00 0.01 0.03 0.13 1.04  N=3 0.00 0.00 0.01 0.09 1.21  # =4 0.00 0.00 0.01 0.04 0.48  The shape of p determines the number of sensors required for a minimum desired accuracy. In other words, the more nonlinear that the shape of p is, the higher the degree of polynomial needed to accurately approximate p, and the larger the value of N must be for a given accuracy. TV-point quadrature formulas achieve near-perfect accuracy if N is high enough. This can be seen for the floating-sphere cases of d = 1000 mm and d = 800 mm, for which nearperfect accuracy (< 0.01% error) is achieved for N > 1 and N>2, respectively. For the purpose of comparison, Table 3.4 gives the errors for approximating the voltage with a Riemann sum, i.e., equally weighted and equally spaced samples, using 100 electric field samples, or N = 100.  32 Table 3.4. Percent error of the Riemann sum for perturbations caused by a vertical ground plane for N =100.  100 0.0871 0.1180 0.2017 0.2824 0.4426 0.8194 2.0917  d  N =  2000 1500 1000 800 600 400 200  3.4  E r r o r in an Applied Quadrature Formula  In practice, when applying a quadrature formula using electric field sensors in an OVT, the output,  V ut, 0  is some measure of the voltage in digital or analog form and is expressed as  N  V =-Y <*i ( i) E  out  J  x  >  + 0  X  ;=l  (-> 3  7  where cc, and x, are the ideal quadrature weights and positions, E (xi) is a measure of the electric x  field at JC„ and O is an offset from the ideal numerical integration formula (quadrature formula). O can originate from uncertainty in positioning, uncertainty in weighting (calibration of sensor outputs), and "blurring" of point measurements. The determined applied voltage, Vb ', is de  a  related to the V  out  through a calibration factor C:  V  aa ba  = CV  l  (3.8)  33 C is determined by measuring V for a known applied voltage, typically the rated voltage,  V d,  out  mte  and for an unperturbed OVT:  .  C = V IV r rated  ou  (3.9)  C is analogous to the turns ratio between primary and secondary windings in conventional VTs. The error, e, in  Vb ' de  a  is given by  Vu  -Vu  et  *_ba  _ba_  v  e=  (  3  1  0  )  Vta  The error is zero for an unperturbed O V T at rated voltage. Also, if O varies linearly with applied voltage (this will be shown to be the case), the error is constant for any system (zero for the unperturbed system) regardless of the voltage applied. Equations (3.7), (3.8), and (3.9) are substituted into (3.10) to give an expression for the error in going from the unperturbed system to a perturbed system:  N  A  + AO 1=1 rrUUp  out  v  where  (3.11)  34 N  N  N  = -Y a E (x ) j  i  x  nP  x  (3.12)  x  +  i=i  i=i  1=1  Y,*i " ( i) E  i  and  AO =  (3.13)  0-O  unp  From (3.11), it can be seen that the error in Vba" is proportional to the error built-in to the quadrature formula (given by (3.12)) and to the change in O in going from an unperturbed to a perturbed system (given by (3.13)). Breaking down the error into its different parts can be expressed as  e = e +e +e +e , x  a  b  q  (3.14)  where e is the error caused by positioning error, e is the error caused by weighting error, e is x  a  b  the error caused by "blurring" of point measurements, and e is the error built-in to the q  quadrature formula.  3.4.1  Positioning  Error  Here, an expression for e is derived. To do this, an expression for the offset in the ideal x  quadrature due to a positioning error is sought. If the positioning of the mth electric field sensor is slightly in error, or has an uncertainty associated with it, represented by a small distance Ax  35 from the ideal position x , a two-term truncated Taylor series expansion of E around x could be m  x  m  used to represent the actual sample in terms of the ideal sample:  Replacing E (x ) in the ideal quadrature formula with this expression gives x  m  N  Vout = - X  * (*i  E  )  ~  ( X  m  A x E  x  im ) x  (=1  By comparing this expression with (3.7), the offset is given by  0 = -a AxE (x ) m  x  .  m  Since Ax is small,  A x * / ( x ) = A x ^ f e J * jE {x lSx)-E {x )) m  x  m+  x  dx  m  Ax  and substituting this expression into that of O gives  0~-a AE {x ) m  x  m  ,  =  ^  ^  )  36 which shows that the offset is approximately proportional to the difference in E between x and x  m  x +Ax. m  So, the positioning error, e , is given by x  unp out  r  \junp "out  (3.15)  L  i.e., the error due to an error in positioning of a sample point is approximately, linearly proportional to the change in the difference of E between the ideal and actual sample locations x  in going from the unperturbed to the perturbed system.  3.4.2 Weighting Error  If the weight of the mth electric field sensor is in error, oc can be substituted by a +Aa, m  m  in the ideal quadrature formula to give  N i=l  Comparing this expression with (3.7) gives an expression for the offset:  0 =  -Aa E (x ) m  x  m  which shows that the offset is proportional to the electric field at the sample point and to the error in the weight of that sample.  37 It follows that the error, e , is given by a  AO a  unp out  T  Aocm  17 unp 'out  E  p x  (x ) m  E  (3.16)  {x )  x  m  which shows that the error due to an error in the weighting of a sample point is linearly proportional to the error in weighting and the change in E at the sample location in going from x  the unperturbed to the perturbed system.  3.4.3  Blurring Error  In practice, the output of each electric field sensor may effectively be a measure of the average intensity of E over some finite length inside the sensor-head (see Section 5.3). This x  averaging effect is loosely described as a "blurring" effect on the measurement. The error resulting from averaging E over a finite path length instead of measuring E at a point, i.e., at the x  x  center of the path, is a function of the curvature of E along the path. A four term Taylor series x  expansion of E along the x-axis and around each sample point, x„ is used to demonstrate this: x  1  f  x ()  E  x  = x E  ( i )+( ~ x  x  i )x  x  E  (i ) + - ( - i ) x  x  x  2 x E  / r  (• [x^-ix-x^E,  6  (*,-)  .  (3.17)  The average of E along a length /,• centered at xi is expressed as x  (3.18)  E x, blurred x,-/,./2  38  Substituting (3.17) into (3.18) and simplifying gives  1  "  x,blurred { i ) = x ( i ) + ~^ i *  E  x  E  x  l  )  E  Using this expression to substitute for E (x ) in the ideal quadrature formula gives x  t  I  N  i=l  N  /=!  Comparing this expression with (3.7) gives an expression for the offset:  N  0 =  -j^£cLihE "(x ) x  i  1=1  which shows that the offset caused by blurring is proportional to the curvatures of the electric field at the sample points, to the lengths of the sensors, and to the weights of that samples. It follows that the error, e , is given by t  b =  AO  N  unp out  i<j unp out 1=1  e  v y  V  1Z,V  '  E  u np x  (x )-E t  x  fe)  (3.19)  Also, expressions similar to (3.19) for blurring along any other axis can be derived by using a Taylor series expansion of E along that axis. x  39 So, the voltage measurement error caused by blurring of the sample points is proportional to the lengths along which the field measurements take place and the change in curvature of E at x  each sample point in going from the unperturbed system to the perturbed system. It should be noted that (3.19) includes the error contributions from all the sensors in the summation, as opposed to (3.15) and (3.16), which focus on a single sensor.  3.4.4  Quadrature Error  A straightforward way of understanding the error built-in to a Gaussian quadrature formula (see (3.12) is to represent p by a power series expansion. Since the quadrature represents the exact value of the integrals (with E  unp x  as the kernel) of the first 2N-1  terms, the  error is represented by the sum of the integrals of the remaining terms in the power series expansion. From Taylor's theorem, these remaining terms can be represented by a remainder term (see [34]), and integrating this term gives an expression for the error  a  and using the mean value theorem gives  (3.20) a  where X and Y are some values between a and b.  40 A similar expression, having a much more insightful derivation (found in [35]), of the error built-in to the quadrature formula involves determining an expression for the 2N-  1-degree  interpolating polynomial that approximates p(jc). To begin this approach, the interpolating polynomial, /(JC), is assumed to be Hermitian, i.e.,  N  I(x) = X [p(*. )A- M + P'fo )Bi (x)] ,  (3.21)  where A,(JC) and Bfa) are polynomials that satisfy the following constraints:  A"( m) x  =  0  \  m  A/(*J =O Bi{x ) m  i  B  (3.22)  = 0 = &im  { m) x  where  M^hl  1  l U = m  It is clear from (3.21) and (3.22) that / and its derivative match p and its derivative, respectively, at the N sampling points. To determine A, and 5 „ the characteristic polynomial, P, and related functions, P, and L „ are used:  41 N  P(x) = Y[(x-x ) m  m=l  N  Pi(x) = Y[(x-x ) m  m=l  Next, by setting A, and 5 , each equal to the product of L and another polynomial, 2  A {x) = C {x)L {xf i  i  i  and  B {x) = D {x)L {x)  2  i  i  i  ,  all but the following four constraints in (3.22) are already met:  A-'(* -)=o (  *,•(*,• )=o Bi\x ) = l t  42  Having two constraint equations, (3.28) or (3.29), for each independent equation to be solved, (3.26) or (3.27), implies that C, and D, are each a first degree polynomial, i.e., a line function. From (3.25), (3.26), and (3.28),  C,.(x .) = l |  and  A/ (X) = c[ (x)l, {x)  + 2 Q (x)Z, (X)L/ (x)  2  A. ( .) = o = Ct {xi )L {xi ) + 2Ci {xi % (x,- )L- (X,- ) 2  x  t  = C,.'(x,.) + 2 L ' ( x ) /  Ci'{xi) =  /  -2Li'(xi)  so  C,(x) = 2L,'(x,)(x,--x) + l  From (3.25), (3.27), and (3.29),  D (x,) = 0 /  and  .  (3.30)  43 B {x) = D (*)Z,(x)\2 + 2D {x)L (x)L  (x)  2  t  i  i  B- {x ) = l = D {  i  i  )L {x f + 2D,- { )L, (x, )L- (x )  f  t  t  t  Xi  t  D-{ ) = l Xi  so  D (x) = x-x i  .  i  (3.31)  So, substituting (3.26), (3.27), (3.30), and (3.31) into (3.21) gives  X  'M = Z U i ( i x  (i  2L  ~)  x  x  +  K M*  + P'( i x  )( ~ i ) i x  x  L  M*  (3.32)  From (3.32), it can be seen that the interpolating polynomial, /, is a polynomial of degree 2N — 1. Next, the interpolating error, E, is defined as  E{x) = p(x)-l{x)  .  (3.33)  E is zero at N distinct points, x = x , x ,..., x , in the domain of p. In order to find an x  2  N  expression for E, another function, F, is formulated:  F(JC) = P ( J C ) - / ( J C ) - C P ( J C )  2  .  The constant C is such that F is zero at a distinct point, XQ, in addition to the N zeros of P:  (3.34)  44  p(x )-/(x ) 0  0  By Rolle's theorem, F' has A zeros, each one between a pair of the N + 1 zeros of F, or 7  x = x , x , x , • • •, x 0  x  2  N  . By definition (see (3.23), (3.32), and (3.33)), F' also has an additional N  zeros at x - x , x ,..., x . x  2  step it is clear that  N  Again by Rolle's theorem, F" has 2N-1  zeros, and by repeating this  has one zero at some point X giving  F( Xx) 2N  = p( Xx)-C{2N).=  0  2N  p( Hx) 2N  C = , P  Note that X is a function of xo and a<X<b.  \ ' 2iV!  .  (3.36)  Substituting (3.35) into (3.36) and rearranging  gives  p(-o)-/(-o) + ^ ^ ^ o )  2  This equation is still correct if XQ is replaced by x. Making this substitution and substituting the result into (3.33) gives an expression for the interpolating error function:  (3.37)  45 From (3.32), (3.33), and (3.37), an expression for p is derived giving  P  w=/(*)+E( X )=^  (*.• i i - *)+*K ( f+p'fe  [>(* i t  x  2L  x  - i )i () x  L  x  2  p  ]+ ^ f ^ p (  w  (3.38)  Integrating p with E  UNP X  as the integral's kernel function using (3.38) gives  ]p(x)ET (x)dx = Yp f o ) J W ^ ' f o X * -x) + \Y (xfdxMx )\E? l*)(x-M*?dx p  P  l  i  i  +lEr(x)^y^P(xfdx  (3.39)  In order to have the sum depend only on p(jc,-) and not p'(*,•) as would be the case for a Gaussian quadrature formula, the following conditions must be met:  u  J E" (x){x - x )L,- {xf dx = 0 NP  {  Rewriting this using (3.23)-(3.25) gives  ^  fE  U NP X  {x)P{x)L {x)dx = 0 i  (3.40)  This equation has implied in it the defining equations (see (3.6)) for explicitly finding Gaussian quadrature formulas, i.e., a Gaussian quadrature's abscissas and weights are such that they  46 satisfy the above expression. In other words, P is orthogonal to polynomials of degree N - 1 or less. From (3.39), the quadrature error e is given by q  a  and using the mean value theorem gives  a  where Y is some number between a and b. Clearly, from (3.20) and (3.41), it can be seen that if p is any polynomial of degree 2N1 or less, the quadrature error is zero. Due to the fact that p is derived from physical quantities, it is believed that it will always have decaying higher order Taylor series terms. More specifically, the fact that the Laplacian operator of the Poisson equation is essentially an averaging operator, p will always be a smooth, polynomial-like function in practice. This is one of the key reasons why the quadrature method is so effective!  3.4.5  Quasi-Quadrature for Known Position Offsets  In practice, the positioning of a field sensor may not be as accurate as the measurement of that sensor's position. If sensor positions are known to a very high accuracy, and if these  47 positions are different than the ideal positions, an adjusted quadrature formula can be calculated. This is done simply by calculating the weights using the first N defining equations (see (3.6)) and using the values of the unperturbed electric field at the known positions rather than at the positions of the characteristic polynomial. Since the positions are not freely chosen, the result is not, strictly speaking, a quadrature formula. However, the resulting formula can be thought of as a "quasi-quadrature" formula. Finding a numerical integration formula with pre-assigned weights is equivalent to approximating p with a Lagrangian interpolation formula, J, where  N  (3.42)  Using a similar method to that described in Section 3.4.4, it can be shown (see [35]) that  (3.43)  and that the built-in error of the quasi-quadrature formula, e , is given by q  (3.44) a  where X is a function of JC and  a<X<b.  48 If the chosen x,- happen to be quadrature sample points, then the first N terms of the power series expansion of  O^ \X)/N\ N  do not contribute to e . This is due to the fact that P is q  orthogonal to polynomials of degree N- 1 or less (see (3.40)). Taking this into account, if the x,are close to being quadrature points, then the first N terms of the power series expansion of contribute very little to e . This is so because the P associated with these points is q  nearly orthogonal to polynomials of degree N - 1 or less. So, a quasi-quadrature formula is expected to be nearly as accurate as the quadrature formula, depending on how close the sample points are to the quadrature points. Adjustment of the original quadrature formula due to known small offsets in the actual sampling positions from the ideal sampling positions will still result in a very accurate numerical integration of the electric field.  3.5  Special Case - Sensor-Dependent Field  In deriving the quadrature formula to this point, it was assumed that the electric field sensors themselves do not affect the electric field in which they are placed, i.e., E  unp x  is not  influenced by the presence of the sensors. In reality, the sensors are made of some material and may have to be mounted on a flange or other small section of either insulating or conducting material, which in turn is then mounted inside an insulator during construction. So, the sensors and their mounting supports may significantly alter the electric field from that which would otherwise exist inside the insulator, thereby making the problem of finding the quadrature points a more complex one. Specifically, the problem becomes  49 f = "J T E  P  (X, X , X X  2  X )p(X)dx = -  n  where the kernel, E  unp x  WiP(Xi )  n  (3.45)  1=1  is now also a function of the location of the sensors as compared to the  original problem (see (3.5)). It is believed that this equation is generally solvable by an iterative technique, such as one in which the derivation process described in Section 3.2 is repeated over and over again until a consistent solution is reached. In order to demonstrate that (3.45) is solvable for a specific case, the two-sphere system used in Section 3.3 is modified by incorporating sensors for the case of N = 3, and the quadrature formula for this modified system is found. Each sensor simply consists of two, small, conducting spheres having 15 mm diameters and separated (edge-to-edge) by 20 mm. Initially, the vertical centers of the sensors (halfway between the sensor electrodes) are located at the x,calculated in Section 3.3, for which the sensors' presence is not taken into account. The solution process consists of calculating E  unp x  for this system, calculating the new x, and oc„ checking the  change in x„ and repeating the process if the change exceeds some tolerance or stopping the process if the change is less than the tolerance. The resulting x, and a, satisfy (3.45). The convergence of the solution is demonstrated in Fig. 3.3, which shows the change in x, as a function of the iteration number.  50 .2-1  iteration number Fig. 3.3. Convergence of an iterative solution process.  3.6  Application  The previous examples show that the quadrature method is very accurate for relatively large perturbations. With regards to revenue metering, voltage transducers can be required to have accuracies of 0.3% and 0.2% or better in North America and Europe, respectively. The simulated vertical ground plane and floating sphere perturbations are examples of overly extreme situations that do not happen in practice. The minimum clearance between high-voltage conductors in a substation is governed by safety standards, e.g., ~2 m for 230 k V structures [40]. The examples indicate that with the quadrature method only two electric field sensors (N - 2) may be sufficient for measuring voltage to metering accuracies at high voltages, e.g., 230 kV.  51 A feasible design of a high-voltage voltage sensor, e.g., for measuring the line-to-ground voltage on a bus in a substation, consists of a standard high-voltage standoff with a number of electric field sensors mounted inside. Fig. 3.4 depicts such a design. The standoff would have a conductive cap attached to a bus, an insulating section, and a conductive stand. As in the examples, N electric field sensors would be placed along the center of the insulating section, and b and a would be centrally located at the top of the stand and at the bottom of the cap, respectively. E  unp x  would be chosen to be the x-component of the field between points a and b in  the absence of other sources or electric field disturbances and would be calculated numerically. A customized weighted sum based on E  unp x  would be derived and would dictate the exact  positions and weights of the electric field sensors and their outputs. Preferably, these electric field sensors would be small, all-dielectric, optical field sensors, resulting in an OVT. A processing unit, e.g., a computer, would be used to calculate the formula and give instantaneous readings of the line-to-ground voltage on the bus. Alternatively, and depending on the type of electric field sensor, the weighting and summing could be performed physically by having the sensors in series and having their assigned weights built-in to their transfer functions. In the presence of stray field effects due to other electric sources, such as neighboring phases, or other field disturbances, such as rain, the measured electric field components would change slightly, but the measured voltage would remain relatively unchanged, as was demonstrated in the previous examples. Also, these effects would be limited because their sources would be kept at a minimum distance from each sensor by the standoff.  52 HVbus Corona ring Insulating off-the-shelf standoff Electric field sensors  Metallic stand  Fig. 3.4. A high-voltage transducer.  3.7  Conclusion  A new concept for obtaining voltage from discrete electric field measurements has been introduced. Simulations demonstrated that, using the quadrature method, as few as two electric field measurements may be sufficient for designing accurate high-voltage transducers. It was shown how this method can be used as a tool for designing voltage transducers requiring only a small number of electric field measurements. Measuring electric fields at points can be accomplished with the optical electric field sensor technology available today (see Chapter 5), resulting in an O V T . High-voltage transducers based on this concept would not require the special insulation or electric shielding that is required in most conventional and optical voltage transducer technologies of today.  53  Chapter 4 Permittivity-Shielding  4.1  Overview  The "permittivity-shielding" approach to making an accurate voltage measurement with electric field sampling is presented. Permittivity-shielding between two electrodes is used to reduce stray field effects inside the shielded region. This allows for a few electric field sensors to be used to accurately measure the voltage difference between the two electrodes, even in the presence of severe stray field effects, such as those due to pollution. Also, permittivity-shielding does not introduce large electric field stresses and, in fact, moderates the electric field. It removes the need for expensive insulation to support otherwise high field stresses. In the next section, permittivity-shielding is explained and demonstrated using computer simulation. Following this, further computer modeling demonstrates how permittivity-shielding can be used to make a voltage sensor, out of an "off-the-shelf standoff insulator, that can measure voltage accurately even in the presence of severe stray field effects.  4.2  Description  In recent years dielectric materials with high dielectric constants and semi-conductive materials have been under investigation for their use in cable terminations [41]-[43]. Their ability to structure an electric field is useful for moderating the electric field intensities at cable terminations, which would otherwise have to be done using a combination of specially designed conducting structures and insulation to avoid breakdown in air. Here, it is proposed that the field structuring properties of such materials can also be used to shield the electric field in a region  54 between two electrodes from external sources of electric field disturbance, such as changes in geometry of other conductors and of the external medium, commonly referred to as stray field effects. The use of these materials for this purpose is referred to as "permittivity-shielding''. With very high shielding located between two electrodes, a nearly fixed one-to-one relationship between the electric field internal to the shielding and the voltage difference between the electrodes can be achieved, and an electric field sensor placed somewhere in this region will give a measurement proportional to the voltage between the electrodes. In this way, a voltage could be measured with a single electric field sensor. With lower levels of shielding, stray field effects will not be completely eliminated, but multiple sensors together with the quadrature method can be used to measure voltage accurately with fewer sensors than would otherwise be required without the shielding. To illustrate the concepts of permittivity-shielding, a simple structure is studied. This structure consists of two identical, disc-shaped electrodes having diameters of 200 mm and separated from each other by 2000 mm. The lower electrode is grounded, and the upper electrode is energized. Between the electrodes, there is a tube made of a uniform material with permittivity, e , and having a thickness of 10 mm and a variable inner diameter, DD. The r  dimensions of this structure are similar to the basic dimensions of 230 k V standoff insulators found in high-voltage substations. The structure is suspended in air 1000 mm above a ground plane. See Fig. 4.1.  55  Energized Electrode  Shield, e  tr  1000  mm  -200  mm  10  1  r  mm  Grounded Electrode  Fig. 4.1. Two-electrode structure (unperturbed system).  A sinusoidal voltage of magnitude 1 V and frequency 60 Hz is applied to the upper electrode, and the electric field surrounding the structure is calculated using BIFF analysis. Since the medium is linear, the electric field is also sinusoidal with the same frequency and has both a magnitude distribution and phase distribution. The vertical component of the electric field along the x-axis of the structure, or E , is plotted in Fig. 4.2 in terms of its magnitude and phase x  for tubes with ID = 20 mm and varying relative complex permittivities; e = Ei - JE2 [14]. From r  these plots, it is apparent that increasing the tube's permittivity significantly moderates the field.  56 In addition, Fig. 4.2(b) and (d) shows that, when the medium between the electrodes is predominantly either capacitive or conductive, the phase distribution for E is uniform and that x  the phase of E is the same as that of the applied voltage, i.e., has value zero. For the cases of x  dielectric shielding, i.e., £2 = 0, the medium (aside from the perfectly-conducting electrodes and ground plane) between the electrodes is purely capacitive, and, as a result, the electric field in this medium has a constant phase distribution having a value of zero. With the introduction of a slightly conductive material, i.e., £2 * 0, the medium becomes resistive-capacitive, and the electric field has a nonuniform phase distribution. For sufficiently large conductivities the phase distribution effectively returns to zero.  0.005  0.004 A £,fore = 1 r  s .s  /v, for >: = 10 r  0.003 A  fore,-=.100 /'.;fori: - 1 . 0 0 0 r  T3  3  0.002 A  •  ^fors^  1 0 ; 0 0 0  A  E  1 0 0 , 0 0 0  x  for  s= r  0:001 A % 6 A 6 A 6 A A 6 6 6 6 6 6 6 6 d_A-A-» 0 . 0 0 0 0  500  1000 x (mm) (a)  1500  2000  57  x for v. - 1  E  r  E f6re x  = 10  r  E fors .=  100  ^fors =  1,000  x  i  r  •  E fat:e = 10,000  A  /: forr =  x  r  x  v  100,000  1000  500  1500  2000  x (mm) (b)  0.005  o;oo4 = I]  -j for t: = I - J10 £ .fore = ;I - jl'00 E for s = I - j 1.000 EJove  r  s S >  •.•  4-*  "§b  0.002  r  x  r  x  r  x  0:003  3  E  •  E .fme = I' - j 10.000  A  £ .fore =  x  A  r  r  -j 100,000  0.001  0 0 0 0 6 6 0  0 0 6  o o a a a a  O O O jO_0  0.000 1000  500  x (mm) (c)  1500  2000  58 60 Ej0TE = 1 - j r  l\Jorv, = r  40  1 -jlO  /<; for e = 1 -jlOO EJoTs =l - j 1,000 ^fore,.- 1 -j 10,000 4 f o r e : = 1 -j 100,000 r  —  •  20 o.  A  r  /  r  l-c  00 -A -A _A_0_ ft.  T3..  R  R  7  *  £  A  A  V\  -20  /  -40  /  /  • • A  A  /  -60 0  500  1000  1500  2000  x (mm) (d) Fig. 4.2. (a) Magnitudes and (b) phases of E for dielectric tubes and (c) magnitudes and (d) phases of E for resistive tubes and for 1 V applied. x  x  Next, various stray field effects are simulated by introducing a vertical ground plane, a grounded conducting sphere, and a resistive outer tube surrounding the structure. The systems involving stray field effects will be referred to as perturbed systems, and the structure depicted in Fig. 4.1 will be referred to as the unperturbed system. The simulated perturbed systems are shown in Fig. 4.3. The first and second perturbed systems (see Fig. 4.3(a) and (b)) consist of a semi-infinite vertical ground plane and a grounded sphere with a 200 mm diameter and that is vertically centered, respectively, 200 mm away from the axis of the structure. The third perturbed system (see Fig. 4.3(c)) consists of a 10 mm thick layer of resistive material in the form of a tube with an inner diameter of 180 mm. The tube has either a high resistivity, specifically, p = 10,000 £2-m, or a low resistivity, specifically, p = 10 Q-m. Resistivity is inversely proportional to the  59  Fig. 4.3. Perturbed systems involving (a) a ground plane, (b) a grounded sphere, and (c) a uniform resistive layer with an air-gap at the top.  60 imaginary permittivity and frequency; p = ( C O E ^ ) " [14], where e is the free-space permittivity. 1  0  The shielding tube of the structure has ID = 20 mm. The electric field distributions in the perturbed systems are different than that in the unperturbed system. However, with shielding, the field inside the tube changes less than would otherwise be the case. Fig. 4.4 shows how E changes in going from two unperturbed systems, x  i.e., no shielding (e = 1) and with shielding (e = 1 -7'10,000), to two perturbed systems both r  r  consisting of a low-resistivity (p = 10 £lm at 60 Hz) outer tube with an air-gap between the top of the resistive tube and the top electrode. The gap is approximately 67 mm in height. Along with the significant field structuring, these plots make it apparent how significantly the shield "screens" E from the external disturbance. x  i i  0.010 ^  0.008  r  x  6 E  ••E - perturbed, e = 1  — 0:006  T3  i  E - unperturbed, e = 1 x  i  r  K - unperturbed. r. = 1 - j 10,000  i  r  E - perturbed, e = 1 -j 10,000  *  x  r  i  3  00  0.004  0.002  A—A—A—A—A—A—A—A—A—A—A—A—A—A—A—A—A—A—A—A—A—A—A—A,A«r-&  *A  fj  A  A  A  0:000 500  1000  1500  2000  x (mm) Fig. 4.4. Unperturbed and perturbed electric field magnitude distributions with and without shielding for an applied I V .  61 Next, this shielding phenomenon is used to improve the accuracy in determining the voltage, Vdet, between the electrodes from N electric field samples according to the quadrature method in the presence of the stray field effects just described. The expression for the voltage determination is given in Chapter 3 and is repeated here for the reader's benefit:  N  (4.1)  where N is the number of sensors, x,- are the sample locations, and oc, are the sample weights. Both the sample locations x,- and weights a, are determined by the quadrature method, and the case of two sensors, i.e., N=2,is  used.  Since the frequency is fixed, at a single value of 60 Hz, for all the simulated cases, the applied voltage Vcan be represented by its magnitude and phase as follows:  v =  \v\em+v  where oo is the radian frequency, |V| is the voltage magnitude, and (|> is the phase. 0 has a value of zero for the simulations. Vdet is sinusoidal with the same frequency (60 Hz in the simulations) and can be described by a magnitude and a phase as well. For the following cases, the difference between V  det  and V i s expressed in terms of a magnitude error and a phase error. In Table 4.1,  normalized magnitude errors are given in percent and are relative to, i.e., a percentage of, the magnitude of the applied 60 Hz voltage signal;  62 (kdetl-M)  magnitude error = - — ' V  xl00%,  The phase error of Vdet is relative to the phase of V, i.e.,  phase error = (j)  det  The second column to the fifth column in Table 4.1 demonstrates the effectiveness of the shielding as regards reducing the error in Vdet for the perturbed systems considered. Two additional perturbed systems, one consisting of a resistive outer layer and an air-gap as described before but with the air-gap positioned at the vertical center of the structure and the other consisting of the same resistive water layer without an air-gap, are also included in Table 4.1 (last four columns). Higher phase errors occur with the introduction of resistive material, i.e., £2 ^ 0, between the electrodes. However for sufficiently large shielding-tube conductivities, the phase error decreases again and can be made to be very small. For the case of dielectric shielding, Table 4.1 indicates that the levels of real relative permittivity needed to achieve reasonable shielding against stray field effects for two sensors is well out of the range of what is presently available [42], [43]. In contrast, virtually any level of imaginary permittivity is attainable using existing, resistive materials. For this reason, resistive shielding seems to be the more promising method of shielding and will be focused on for the remainder of this discussion. Power is transmitted through the structure when using a resistive inner tube, an issue that warrants consideration. The lower the resistance of the shielding, the higher the power loss. For example, 230 kV transmission lines typically can deliver a rated power of about half a gigaWatt. The power dissipated, Pi, by a 100 MQ shield (corresponding to £2 = 6,500 at 60 Hz in the  63 O  £  >v  00  CO  00  cS  CD >  x: CX  .—i  I—.  O  00  CM  T-H  CM  CM  T—t  CT5  CT)  CT)  i—I '—i  00  CD  i—i  o  CT)  o  O  LO CT)  CM  o  §  o  o o  O  CO  «2  c  CO  to cu  s  i—l  CM  i—"  T—<  CM  x;  CJ  LO  o  CO  o  CT)  00 CD i—l CM  T-H  CT)  O  CM CT)  ,—i  O  CT)  O O  oo o  oo  CM  00  a OH  o u  '>  .22 "55  CJ  ex  OJ  CO  'S so  CT) O00 CT) CT) 00 C D CT) CT) CT) LO  CO Cv] CO CM  CT)  CO O  o  00  CT)  CM'  CD CO  CO CO  o o  O  < ]> 00 1-. 00 00  00 CJ  cu  CD LO  *"*  00 00  LO  LO  LO CM  X! 00  CO cq  00  CT) CO  CM' CM  CO  CT)  oo  o o  o  o  §1  CJ  ex  >  CJ  oo  a.  3 •4—> I  LO  ro  CO CM  CO  o  o o  LO  o  o  a  LO  i—I CM  ocs  o  O  C  to  "3 x:  cx cx  00  bO C  00  '>> u-  oo  >  X, 00  CO  CD CO  o  o o  LO CM  O  — t CO  CT)  CM  O  O o  00  e  •a CU  CJ  -a >(D-  -t—» oo >. 00 T3  CM  CT) CM  CO  CT)  CO  CM  O  o o  CM  CD  O  CM  T-H  -*'  CJ  00 o  o o  X) '—  3  t!  •a  CJ  cx  G 3  O  J-H  —  o  ro  fc! . CJ  1>CT) CT) co  o 0J 3  CM  co o  CO o  o o  CO  CT)  CO  1—I  LO LO  o o  CO  FT  1  CL)  CJ  >  00 *•>  c  is I f § *-*—•  CJ  1  "33  "  '3  T)  cx S  O  o o  o o o  o o o o  o o o o o  o o  o o o  o o o o  o o o o o  5051.0  -137.9  -0.2 -21.3  o o o o o o o o o o o o  -0.9  o o  o o  o o  o o  o o  o o  -5.5  o  o  o  o  o o  "P  o  O  o  ci  o o -137.0 -13.7  -133.0 -13.4  o  o  o o  o  o  1-j 100000  o  1-j 10000  84.7  75.6  21.8  cvi  1-j 1000  -809.5  93.4  -18.4 O o  194.1  -777.5  -500.6  -447.7  od  1-jlOO  -3.6 -0.4  -3.5 -0.4  186.3  77.4  5576.9  -783.5  -718.7  T—H cvi  -3.9  "a*  -30.4  -237.4  o O  -307.6  o o  -3936.0  CV)  -4073.3  o  115.5  C)  8761.5  1—1  31.4  o  -35.2  o  -4718.6  23.5  CO  100000  -7.7  159.7  o o  10000  -44.5  393.8  o  12.1  -0.1  -65.7 32.5  4662.9  o  1000  -0.1 -66.7  3495.0  5026.1  o  65.1  low resistivity  high resistivity  low resistivity  high resistivity  Uniform Resistive Layer  low resistivity  Middle Air-Gap  CVl O  high resistivity  Upper Air-Gap  o  70.2  Grounded Sphere  o  112.6  Relative Permittivity of Vertical Ground Plane Shielding (ID = 20 mm)  |  64  CO  65 structure) is given by P, = (230kv/V3) /lOOMQ = 176 W . So, the power loss in the resistive shield 2  is less than 0.00005% of the rated power. This power loss is in the form of heat generated by the shielding tube. If the resistance is too small, too much heat will be generated, and the tube may overheat. Nevertheless, dissipating 176 W in such a large structure should not pose any problems. For a fixed shielding resistance, the shield is preferably closer to the region of measurement, i.e., the axis of the structure. Table 4.2 gives voltage measurement errors for a 100 M£2 tube with varying ID and using two sensors.  4.3  Application  It is proposed that compact electric field sensors can be placed inside an off-the-shelf standoff insulator in order to measure the line-to-ground voltage of a power structure such as a bus in a substation. Electro-optic field sensors are particularly well-suited for this application, and they bring with them all the inherent advantages of optical sensor technology. The performance of such a voltage sensor is simulated. The voltage sensor consists of, from top to bottom, a conducting cap with a corona ring, an insulating column having height 2245 mm and outer diameter 312 mm (these dimensions are those of a particular 230 kV composite insulator), and a conducting stand having height 2000 mm and diameter 400 mm. The cap is connected to the high-voltage line, and the stand is grounded. The insulating column is hollow with fiberglass walls having silicone rubber sheds attached. A n electroded tube, supplying the shielding and having inner diameter 198 mm and wall thickness 4 mm, is placed inside the insulating column (see Fig. 4.5).  66 '5-  Ui  cu  c  yS  jrT> CO  co  c u u.  ve  G  'oo  o o  TP  LO  CO  o  O  o  o  o  1  &  o  fo o  00  o o  00  o o  t  1  CT)  O O  O —  <u  H  H co  Re:  "c/i  Si  c  >  u  00 'oo  G  g  c2 C D  (D  -o  cu  high i  <U  CO  00  o  CO  CO  CJ)  CD  CD  o  o  o  cd  o  o  O  O  CM  rl CO  o  oo CD  I O LO  O  o  1  o  1  O  'S >  die Air- ap  CD  res  "3  o c  o  CO  oo CL)  r_|  CO CM  TP  O  o  o  CD CM  LO  t-»:  r—1  t-;  CT) CD  & >•,  resisti  '>  oo  a  CT) CM r-H  r—1 o CM  CT) o CD CO  J3  LO  "T  oo  CT) t»LO  r-H  CM CM CO  CM CM LO  CV]  1  SP  o o T3  CD  X  00 CO  00  cu  ap  T3 C  o  CO  L  u  >•  CL,  >  cu  OH  CO >  00 00  cu  JS  u,  » 00  "O  >.  cu  T3  ai u,  oun phe  00  T3 ca  u,  -e ^  O  ^  o  CD 1—< o  CM CO  LO LO  O  o  00 CO CM  o  o  LO  p LO  r—1  T-H  o  CT)  CM TP  CT) CT)  oo CT)  o  CM 00  r-H  CO CO  CD CM  TP LO  TP 1  ui  FA  2  CL)  £  o o  :ica:[Gr  O  ca <D &o cu 00  CO  O  -S  r—i O  o  O  CM  CM  1—H o  o  r—, o  r—, o  CM o  CM  o  o  CO  CO  TP  LO  CD  CO  o  o  o  O  o  o  o  o  o  o  o  O  cu r-H  J O  EC  o  O  o  o  o  O  o  o  O  o  o  CO  00  o  o  O  o  t-i  > a S  00 M hieldi  CO r=H  co  n O  -a c  8.1  CO  O  o  HS  sCO  CN Tr'  o  CT)  SP  oo  CO  LO LO  4=  '%  E  CT) CM  r-H  ow  u,  LO  ^  CO  Q i—i  CM  TP  o  o  o  CM r—1  TP  r—1  CO 1—1  2  o  a £  3  2 —< on  u  2  15.8  -390.2 -532.3 -758.3 -1175.4 -2177.7 150.1  -367.9 -487.1 -654.7 -888.4 -1134.0 63.0  OO  od  -3.1 -62.9 -529.3 56.7  -5.0 -10.7 -31.0 -108.9 -429.5 -34.0  -1.2 LO  oq  rq  cq  ro  CO  od  o  00  O  o  CO  ro  O  r— o  o  oo  -0.9  -0.7  -0.5  -3.6  -3.5  -290.9  -279.0  -5.0  low resistiv  LO  o  -3.4  high resistivity -4.9  low resistiv -214.9  high resistivity  Uniform Re:sistive Layer  -208.3  low resistiv  Middle 1Dryband  oo  -0.3  a  ai  -0.6  high resistivity  Upper L)ryband >^  cq  -3.0  Grounded Sphere  >>  oq  -0.2  Vertical Ground Plane  oq oo  oq  CD id od  12.0  67  ro od  CD  o o — i <  oo •  OO  id  LO  oo id  o  o  o  •<3< i  CO  68  In the simulations, the sensors are assumed to be positioned along the column axis with locations dictated by the quadrature method, and the electric field sampling occurs at these locations. Polluted water having varying conductivities is placed on the sheds, and the voltage measurement errors induced by the resulting stray field effects are computed for a shield resistance of 100 MQ and for various values of N. In the first set of simulations, a uniform layer of water is introduced on all but the top 250 mm (vertically down from the cap) of shedding, constituting a "dryband" condition just under the cap (upper dryband). The water layer has a thickness of 0.25 mm. Two cases are considered, one having a high level of pollution and one having a low level of pollution. In one case the water has impurity concentrations resulting in a water resistivity of 100 k£2cm and in the other case resulting in a resistivity of 100 Q c m , referred to as the low pollution and high  V  Fig. 4.5. Voltage sensing standoff with shielding.  69 pollution cases, respectively. The errors of the measured voltage, in terms of amplitude error in percent and phase error in minutes, are determined and are shown in Table 4.3. Similarly, in the second set of simulations, another dryband condition is modeled by introducing the same water layer everywhere but on a section in a lower region of the insulator, specifically, between 620 mm and 870 mm above the bottom of the base of the lower flange (lower dryband), and results are also shown in Table 4.3. It should be noted that such dryband conditions with a uniform contaminated water layer of 100 Q c m and a thickness of 0.25 mm are very extreme cases, intended to simulate extreme stray field effects. It is apparent that the more highly polluted water generally results in larger errors in the measured voltage. However, it should also be noted that pollution levels of resistivities below 100 Q,cm do not significantly increase amplitude errors and, in fact, improve phase errors. Table 4.3 indicates that with only six sensors, a voltage sensor using permittivityshielding can achieve IEC 0.2 class metering accuracies (< 0.2% magnitude error and < 10 minutes phase error) and ANSI/IEEE 0.3 class metering accuracies (< 0.3% magnitude error and < 8 minutes phase error) even under heavily polluted conditions. Table 4.2 suggests that the same accuracies can be achieved with even fewer sensors by using a shielding tube with a smaller diameter. This kind of voltage sensor does not require special insulation due to the wide spacing between the tube electrodes. Also, existing optical field sensor technology is particularly wellsuited for making the electric field measurements inside the structure.  71 4.4  Effect of Frequency Dependence of Resistive Shielding on the Quadrature Method  As mentioned before, resistivity is inversely proportional to the imaginary permittivity and frequency; p =  (CO8082)"  1  [14], where e is the free-space permittivity. It follows that the 0  imaginary permittivity of a shield with fixed resistance is inversely proportional to the frequency. So, since the quasi-static electric field resulting from a sinusoidal source depends on the complex permittivity of the medium, it also depends on the frequency of the source. Therefore, the unperturbed field for the purpose of the quadrature method is only "correct" at one frequency. At any other frequency, the field typically appears to be perturbed. For example, referring back to the two-electrode structure of Fig. 4.1, if the shield is said to be resistive, then for some base frequency it will have a corresponding complex permittivity of e = 1 - 100,000j, which results in an E that is uniform and at zero phase as shown in Fig. 4.2(c) r  x  and (d), which is repeated below as Fig. 4.6 for the reader's convenience. If this field is chosen as the unperturbed field, then at a frequency 10 times higher than the base frequency the shield will have a complex permittivity of e = 1 - 10,000j resulting in a field that is also shown in the r  figure. Increasing the frequency by factors of 100, 1000, 1000, 10000, and 100000 similarly results in the other fields shown in the figure. Since E"  np  is a constant, the shapes of these fields  are also the shapes of the associated p(;c)'s. As a result, there is a different numerical integration error associated with each frequency. Generally, real signals have a wide frequency spectrum. Under nominal operating conditions, a typical power voltage signal will have a large, main frequency component at the rated frequency (50 Hz or 60 Hz) with smaller harmonic components across the rest of the spectrum. For a time period during which there is a transient, such as a switching impulse or a lightning impulse, the corresponding spectrum of the signal will have a distribution of contiguous frequency components over a wide range of the spectrum. In either case, measuring  72 the voltage signal in real-time using any field sampling technique, e.g., the quadrature method, can be thought of as effectively computing each of the measured signal's frequency components, each of which will have a different error associated with it. The result is a determined signal that is a slightly distorted version of the measured signal, with the level of its distortion depending on the range of the measured signal's frequency spectrum, the change in E with respect to x  frequency, and the accuracy of the quadrature formula with respect to this change.  73 60 E fors -  H 1 - J10 £,fore = 1-J100 ••- £ fors = 1 - j 1,000 • E for s - 1 - j 10,000 A £ for r = 1 - j 100,000 x  r  t: forc ~ x  40 A  r  r  x  20 %-  r  x  r  x  v  <D  A  A  A  A  TV  A  A ^ A —  <L>  CO  O.  -20 4  /  \ /  /  -40 -60 1000  500  1500  2000  x (mm) (b) Fig. 4.6. (a) Magnitudes and (b) phases of E for resistive tubes and for 1 V applied. x  4.5  Summary  Permittivity-shielding for use with discrete electric field measurements to measure voltage has been described. It has been shown to be an effective tool for significantly reducing stray field effects, thereby allowing accurate voltage measurement with a small number of electric field sensors, even when the standoff is heavily polluted. Important advantages of this type of shielding, over shielding based on specially designed metal-insulation structures, is that it was shown to reduce field stresses, and consequently does not require special, expensive insulation such as SF6 or oil, and is safer. It was pointed out that emerging electro-optic field sensors are particularly well-suited to measuring the electric field at points determined by the quadrature method. Numerical simulations were used to show that a voltage transducer  74 consisting of an off-the-shelf standoff insulator, a small number of electric field sensors, and an electroded tube providing the permittivity-shielding can meet TEC 0.2 class metering accuracies and ANSI/IEEE 0.3 class metering accuracies. Also, the frequency-dependence in determining voltage using field sampling with resistive shielding was discussed.  75  Chapter 5 Optical Electric Field Sensors  5.1  Introduction  In order to use a quadrature formula, point samples of one component of the electric field must be measured with electric field sensors. Small sensors based on electro-optic principles are ideally suited to do this. Such sensors are passive and typically made up of all-dielectric components, which makes them safe in high-voltage environments. Also, since their measurements are carried in light, the observer can be safely insulated from the high-voltage environment, e.g., through the use of optical fiber. Here, immersion-type integrated-optic Pockels cells (IOPCs) and miniature bulk-optic Pockels cells designed for this type of field sensing are described.  5.2  Integrated-Optic Pockels Cell  5.2.1  Description  The integrated-optic Pockels cell (IOPC) [44] consists of an X-cut or F-cut lithium niobate (LiNbC^) substrate having a thickness of approximately 0.5 mm. A long, narrow region just below the surface of the substrate with a cross section of a few microns in each transverse dimension is doped with titanium forming an optical waveguide that is parallel to the Z-axis of the lithium niobate. The waveguide is such that it can support only the fundamental transverseelectric-like (TE-like) mode and the fundamental transverse-magnetic-like (TM-like) mode, which correspond to two orthogonal polarizations. Coherent light (wavelength X = 1.3 pm) is  76 transmitted to the waveguide, or the sensor-head, through an optical fiber that is directly bonded to the substrate. The incoming light is linearly polarized, e.g., with a polarizing fiber (PZ), with its polarization ideally oriented at 45° to the polarizations of both modes. Upon entering the substrate, the light is coupled into the fundamental modes of the waveguide. Ideally, the input power is evenly split between the two modes. The two modes each have slightly different effective indices, and, consequently, they have different phase velocities (the TM-like mode is slightly faster than the TE-like mode). After travelling the length of the waveguide, the light in the two modes emerges and recombines resulting in elliptic polarization states. This light enters an interrogating, polarization maintaining optical fiber guiding it to photodetectors in the electronics. The presence of an electric field in the waveguide anisotropically varies the refractive index of the substrate. These variations change the phase velocities of the two modes, and, consequently, the polarization states of the light at the output also change. Fig. 5.1 conceptually illustrates the operation of an IOPC.  Fig. 5.1. An IOPC sensing system.  77 For an electric field component parallel to the crystallographic 7-axis of the substrate, the change in the two modal effective indices are equal and opposite; specifically An = ± n r 2 £ y 2 , 3  0  2  where An is the change in the effective index, n is the ordinary refractive index of the LiNbC-3 0  substrate, r?i is the appropriate electro-optic coefficient, E is the average intensity of the electric Y  field component parallel to the F-axis along the length of the waveguide, and the plus-or-minus sign refers to the TE-like mode or TM-like mode, respectively. When measuring E , the IOPC is x  oriented so that its y-axis is aligned with the x-axis. The linear electro-optic effect is known as the Pockels effect. Ideally, the length of the sensor head is chosen so that the linearly polarized (ideally 45° to the crystallographic axes) light at the input is circularly polarized at the output. So, the choice of the length depends on the modal birefringence of the waveguide. Introduction of an electric field along the T-axis shifts the polarization state at the output from circular to elliptical, where the axis of the axes of the ellipse are ideally fixed. The fast and slow axes of a polarization maintaining (PM) fiber at the output are aligned with the major axis and minor axis of this ellipse. Using a polarizing beam splitter and photodetectors at the other end of the P M fiber, the power in each axis can be isolated and measured, and these powers are given by  P:OL  $1,2 -  1± ycos  (5.1)  where P, is the total input power, 0.1,2 are loss factors each depending on the modal power losses, E„is the half-wave electric field, y is dependent on fabrication imperfections and is typically near 1 (> 0.99), and <> j ,- is the built-in phase difference between the two modes at the output and is  78 referred to as the bias. As mentioned before, the polarization state of the light at the output is ideally circular, which corresponds to a bias of ()), = 90°. An important and very useful aspect of the IOPC is that it has two outputs, i.e, the two powers measured along the major and minor axes of the polarization ellipse at the output of the sensor head. These are used to serve two purposes: normalization of the transfer function (output powers as a function of the intensity of Ey) and bias tracking. For normalization, the two powers are balanced with respect to one another by using the fact that their A C components, caused by a sinusoidal E (at 50 Hz or 60 Hz in power delivery y  applications), must be equal and opposite to one another, effectively correcting for any difference between cti and 0C2. Then, the balanced powers are added to give the total output power P . Dividing (5.1) by P gives normalized outputs: 0  a  #12  =-  (5.2)  1± ycos  Normalization significantly reduces the effects of variations in the source power, of losses in the input fiber, and of vibrations. Having (j), = 90°, having E small compared to E , and filtering out the D C component in Y  n  (5.2) lead to a small signal expression for the output:  (5.3)  n ^ ^ L 2E  n  In practice, the bias may not be exactly 90° and may slowly drift, particularly due to temperature changes. In this case, the small signal expression in (5.3) loses accuracy. Assuming  79 that the bias varies much more slowly than the modulating electric field, a solution to this problem is to track the bias by monitoring the intensities of the two normalized output powers on a slow time scale. This is done by periodically measuring the D C components of the normalized outputs (see (5.2)), and the bias is calculated using (5.2) with E set to zero. With knowledge of Y  the bias, the small signal approximation in (5.3) can be dynamically corrected during operation, and the dynamic range of the IOPC can be greatly increased. Furthermore, if the bias has been characterized with respect to temperature before operation, the temperature internal to the sensor-head can be measured during operation. Fig. 5.2 shows the basic structure of an immersion-type IOPC. It consists of a lithiumniobate substrate with an embedded waveguide (substrate doped with titanium), a thin protective layer of silicon dioxide, a lithium-niobate cover piece, and two fibers (PZ and PM) bonded to both ends of the substrate. The substrate is preferably a section of an X-cut wafer of lithium niobate, with the electric field oriented along the crystallographic T-axis. The thin layer of silicon dioxide offers protection to the surface of the substrate and a clean boundary at the waveguide's top. The cover piece aids in the direct bonding of the fibers to the substrate by providing a large flat surface area to which to bond the fibers. Cover Piece  Ti:LiNb03  Lithium-Niobate Substrate Fig. 5.2. A n immersion-type IOPC  80 5.2.2  Importance of IOPC Shape  There are several reasons for choosing an Z-cut or a F-cut IOPC rather than a Z-cut IOPC. The TM-like and TE-like fundamental mode profiles for Z-propagating waveguides are very similar, so the polarization state at the end of the waveguide is quite uniform, which improves the on-off ratio of the device. This is not the case in a Z-cut substrate with X- or Ypropagating waveguides due to materials issues dealing with the significant difference between the extraordinary and ordinary refractive indices (transverse to the direction of propagation), n  e  and n , of lithium niobate and how they vary differently as a function of the titanium a  concentration. Also, this difference in indices makes it practically impossible to control the bias by choosing the length of the device since the large birefringence results in a bias that is very sensitive to length [44]. Finally, for X- or F-propagating waveguides, n and n vary differently e  Q  with respect to temperature, resulting in excessive bias drifts. There is also an impetus for using an X-cut substrate instead of a F-cut substrate if the IOPC is to be used as an immersion-type sensor. As an immersion-type device, the IOPC's sensitivity in measuring a component of the electric field in which it is immersed depends, in part, on the electric field that "penetrates" the substrate. The shape of the IOPC significantly affects the amount of field penetration. Ideally, an IOPC is long in the direction of the measured electric field component with its dimensions transverse to this direction being small relative to its length. In this way, the measured electric field component substantially penetrates the substrate and since the permittivity of lithium niobate is very high compared to that of air (approximately 70:1), the transverse components of the field are significantly reduced as dictated by the electric flux boundary conditions [14]. Also, it is necessary that the field component being measured, e.g., E , is parallel to the crystallographic F-axis, since this is the component that gives the x  desired "push-pull" effect that modulates the transmitted light and that is detected at the output.  81 Furthermore, having an electric field component parallel to the X-axis is not desirable. A l l these things considered, for an IOPC measuring the component of the surrounding electric field parallel to its F-axis, the ideal cross-sectional shape of the IOPC has a large height-to-width (Fto-X dimensions) ratio with the waveguide positioned at the vertical center (of the F-dimension). In practice, such a cross-sectional shape can be more readily achieved with an X-cut substrate as opposed to a F-cut substrate. As mentioned earlier, the length (Z-dimension) of the IOPC is chosen to give the proper bias point. A simple IOPC structure having a fixed length of 20 mm and a varying height-to-width ratios (with the shortest side having a 1 mm length in all cases) and having an anisotropic permittivity, where Exx = 70, E Y Y = 70, and Ezz = 40, was modeled. The IOPC was positioned halfway between the two-sphere structure of Chapter 3 and was oriented 45° to the x-axis. Since the electric field along the x-axis is also tangent to it, the IOPC's X and F crystallographic axes are also oriented 45° to the field at the IOPC's location. The electric field to be measured is referred to as the external electric field, and this electric field has X and F components, E  ext x  and  Ey \ that are equal in magnitude at the IOPC's location. Once the IOPC is placed in the field, x  the electric field is distorted in and around it. The average value of the electric field components, E and Ey, at the transverse center and along the length of the IOPC are calculated. These are the x  components of the electric field that would exist at the indiffused waveguide in an actual sensor. Fig. 5.3 shows the value of these components normalized to Ex and Ey , respectively, as ext  ext  functions of the height-to-width ratio. As the height-to-width ratio increases, the IOPC becomes more sensitive to Ey and less sensitive to Ex as is desired. Loosely speaking, Ey penetrates ext  the IOPC more while E  ext  ext x  ext  penetrates the IOPC less for a higher height-to-width ratio. So, the  normalized electric field components in the plot can be thought of as penetration factors. It should be noted that the penetration factors ideally are fixed with respect to field orientation. By rotating the simulated IOPC from 0° to 90° and observing the penetration  82 factors, it was found that its penetration factors depend on the height-to-width ratio, h/w, and were sufficiently insensitive to field orientation for h/w = 0.2 to h/w = 5. 0.12  h/w Fig. 5.3. Penetration factors versus height-to-width ratio, h/w.  5.3  Miniature Bulk-Optic Pockels Cell  For making near-point measurements of the electric field, sensors that are essentially miniature bulk-optic Pockels cells can also be used. An example of such a sensor consists of a cylindrical bismuth germanate (BGO) electro-optic crystal sensor head with a quarter-waveplate and a polarization beam splitter (see Fig. 5.4) and is referred to as a BGOPC. The operation of the BGOPC is similar to that of the IOPC, except that the B G O P C s quarter-waveplate takes the place of the IOPC's built-in modal birefringence and the polarization beam splitter is needed to interrogate the major and minor axes of the elliptically polarized light at the output. Also, graded  83 refractive index (GRIN) lenses are required to collimate light at the inputs and outputs of the sensor head. Electro-optic crystal (BGO) GRIN Ferrule Lens Waveplate  Singlemode tiber  Polarizing beamsplitter  GRIN  Lens,  P o l a r i z e r  Ferrule  Multimode fiber  Fig. 5.4. BGOPC electric field sensor.  The output of each BGOPC is effectively a measure of the average intensity of the longitudinal component of the electric field, or E , inside the sensor head and, due to the B G O ' x  crystal symmetry, is insensitive to the transverse electric field component. The error between averaging E over a finite path length and measuring E at a point, i.e., at the center of the path, x  x  a function of the curvature of E along the path. It was found that using the averages of x  E  is  np x  along 2-cm-long paths centered at the x, in place of the point values of E at the x, introduces less x  than a 0.01% error in (3.1) (see Section 3.4.3). The BGOPC is well-suited as a field sensor. A l l materials making up the sensor are insulating, which is desirable in H V environments. Also, the cylindrical shape of the B G O crystal and its relative permittivity of 16 make for good field penetration of the field parallel to its axis. For example, a B G O crystal having a length of 2 cm and a diameter of 5 mm has a field penetration of about 60%, which is practically fixed with respect to field orientation. This results in a sensor with a good sensitivity.  84  Chapter 6 Development of Novel High-Voltage Optical Transducers for Metering and Relaying Applications  6.1  Introduction  Novel O V T prototypes measuring voltage at 138 kV, 230 kV, and 345 k V levels based on the quadrature method and permittivity-shielding have been constructed and tested. The development of these OVTs begins with the first prototype, the integrated-optic voltage transducer (IOVT), which uses IOPCs for field sensing and which demonstrates the effectiveness of the numerical integration of the electric field using the quadrature method in the presence of stray field effects similar to those caused by installation or movement of nearby substation equipment. The IOVT is succeeded by a more advanced O V T prototype that uses miniature BGOPCs to sense the field and that meets the requirements of the highest accuracy classes of DEC and ANSI/IEEE standards for instrument transformers. Subsequent to the development of this O V T , the impact of severe stray field effects caused by pollution and ice deposits on an OVT's exterior surface are invesitgated. As a result, the final stages of development constitute the design and testing of OVTs using resistive shielding in addition to the quadrature method. These OVTs meet the highest IEC [45], [46] and ANSI/IEEE [47] accuracy standards for instrument transformers, and they also measure voltage accurately even in the presence of severe pollution. The following sections describe these stages of the O V T development in detail.  85 6.2  Integrated-optic voltage transducer - Proof of Concept  Here, a new type of O V T design that does not rely on special electrode structures or special insulation and that uses immersion-type IOPC technology (see Chapter 5) is introduced. This integrated-optic voltage transducer (IOVT) uses very few electric field sensors, which make point-like electric field measurements, and the quadrature method (see Chapter 3) for obtaining the voltage between two electrodes. To do this the IOPCs must be located at specific positions and assigned specific weights depending on the chosen electrode geometry. This allows the design of the IOVT to be very flexible.  6.2.1  Description  The IOVT's physical design primarily consists of an off-the-shelf insulator, a metallic stand, and a metal cap. The insulator is comprised of rubber sheds, inner and outer fiberglass tubes, and flanges on both ends. The diameter of the insulator is about 60 cm and its height is about 2 m, characteristic of a 230 kV class post insulator. In addition, a corona ring surrounds the top flange protecting it from excessive field stresses. Fig. 6.1(a) shows this structure sitting on a ground plane. Fig. 6.1(b) shows the same configuration as in Fig. 6.1(a) with an additional semi-infinite vertical ground plane 1.5 m away from the axis of the insulator. In practice, placing large objects, particularly ones that are conducting, this close to a high-voltage conductor is not allowed in substations for safety reasons [40]. Thus, the nearby vertical ground plane represents an overly extreme case of an external influence. Using finite element modeling, the electric field distribution is computed for 100 k V applied across the standoff, and a plot of the  86 axial component (the x-component) of the electric field, along the axis of the standoff from the bottom electrode to the top electrode (a to b), for both configurations, is shown in Fig. 6.2. Vertical ground plane  Off-the-shelf standoff insulator  Metallic stand \  1.5mGround  (a)  (b)  Fig. 6.1. (a) A an insulating structure and (b) the same structure with a vertical ground plane 1.5 m away. 160  n  1  -  140  120  100  80 no vertical ground plane with vertical ground plane  20  h 200  400  600  800  1000 1200 % (nun)  1400  1600  1800  2000  Figure 6.2. E along the insulator axis for 100 kV applied across the standoff insulator with and without a ground plane present. x  87 The chosen number of sensors is determined by the desired accuracy of the voltage measurement and the most severe electric field distribution changes caused by the environment. The desired accuracy is related to the accuracy class of the instrument transformer, e.g., metering or relaying classes. The most severe electric field perturbations due to the surrounding environment are dominated by surface contamination, worst-case weather conditions, and substation design standards, e.g., the clearances between conductors. Choosing the unperturbed E to be the E for the case of the standoff insulator with no x  x  external influences makes p a nonuniform function for variations in the geometry (or a constant for the case of no geometric change). Fig. 6.3 shows p for the case of the vertical ground plane 1.5 m away from the standoff insulator. For this case and for practical geometric changes in general, p is a smooth function.  TT  r  n  r  r  11 •  Fig. 6.3. The scaling function p representing the introduction of a vertical ground plane 1.5 m away from the standoff insulator.  88 The structure shown in Fig. 6.1(a) is transformed into a voltage transducer by mounting IOPC electric field sensors inside the inner, hollow, fiberglass tube. The positions and weights for the sensors are determined by the calculated unperturbed electric field and the number of sensors to be used. The number of sensors is predicated on the worst-case shape of p that can occur in practice. The shape of p is important because it determines the lowest degree of the approximating polynomial for which the numerical integration is exact. The field along the axis between the electrodes of the structure, as shown in Fig. 6.1(a), is used for the unperturbed electric field. Here, the vertical ground plane case (see 6.1(b)) is representative of an overly extreme change of geometry, and the p associated with it is used to determine N. Three sensors are chosen since p (see Fig. 6.3) can be accurately approximated with a 5 degree polynomial. th  Fig. 6.4 shows the difference between F, the best (in the least-squares sense) 5 degree th  polynomial fit to p, and p.  0  200  400  600  =800  1000  1200  1400  1600:  .1800  2000  x (nun)  Fig. 6.4. The difference between F, the best (in the least-squares sense) 5 degree polynomial fit to p, and p.  89  The resulting positions (with respect to point a) and weights (normalized to the weight of the middle sensor) of the sample points are given in Table 6.1.  Table 6.1. Formula sample positions with respect to a and weights normalized to the middle sensor's (Sample #2) weight.  Sample Number 1 2 3  6.2.2  (mm) 1868.8 1083.7 180.1  a, 0.4894 1.0000 0.5944  Xi  Simulation and Test Results  The accuracy of the numerical integration was tested by using it to integrate E for the x  simulated model of the insulating structure with a vertical ground plane at a distance of 1.5 m (see Fig 6.1(b)). Also, the percentage changes in the values of E at the sample points, from x  those values of E when there is no vertical ground plane present, are given in Table 6.2. x  Using the principles described, an integrated-optic voltage transducer, with three immersion-type IOPCs (see Chapter 5) mounted internally along the column axis, was constructed and tested in a high-voltage laboratory. The IOPCs were positioned at the sample point positions and were oriented so as to measure the vertical electric field component. Optical fibers, bonded directly to the IOPCs, transmitted light to and from the sensor-heads. The outputs of the sensors were weighted and summed by an A / D converter and a digital signal processor. The electronics had an absolute accuracy of ± 1 % (however measurements are relative).  90 Table 6.2. Simulation percentage errors in voltage measurement (AV) and percentage changes in local electric field measurements (AE (xi)) for a vertical ground plane 1.5 meters away. x  Variables  AV L\E (X!) x  AE (x ) x  2  L\E (X ) x  3  % Changes Due to a Vertical Ground Plane -0.12 8.1 2.4 -18.7  Introduction of the IOPCs and their holders along the column axis influence the field locally, but this effect is expected to introduce a very small error to the final measurement. This is due to the fact that the IOPC has dimensions on the order of millimeters (compared to the 2 m path length between electrodes) and each sensor holder is made of an insulating material with a low dielectric permittivity. The IOVT was tested for accuracy for the case of a large, grounded screen standing vertically -1.5 m away from the center of the column. Water, ice, and any other deposits resting on the surface of the standoff sheds also constitute a change in the external geometry and may influence the electric field internal to the standoff. In particular, impure water or deposits that are conductive and that cover large regions of the shed surface may substantially affect E . Thus, X  the IOVT was also tested for accuracy using ANSI/IEEE [48] and IEC [49] wet testing specifications. The laboratory test results are given in Table 6.3, and the IOVT with the vertical grounded screen is shown in Fig. 6.5.  91 Table 6.3. Laboratory test percentage errors in voltage measurement (AV) and percentage changes in local electric field measurements (AEv(x,)) for several external conditions.  Variables  AV AE (xj) AE (x ) AE (x ) x  x  2  x  3  % Changes Due to a Vertical Grounded Screen 0.6 19.1 -4.6 -22.9  % Changes for EEC Wet Test Conditions  % Changes for ANSI Wet Test Conditions  0.0 -0.1 0.2 0.1  -0.1 0.4 0.0 -0.4  % Changes for Wet Insulator Surface with No Water Flow 0.1 0.6 0.1 -0.3  The changes in the E (xi) for the vertical ground plane simulation and for the vertical x  grounded screen laboratory test follow the same trend, i.e., the field increases near the top electrode and decreases near the bottom electrode. Differences between the simulated values and test values may be able to be attributed to the fact that the actual geometry of the test setup is not accounted for in the simulation, which assumes that only a semi-infinite ground plane is present. In contrast to the case of the vertical grounded screen, the wet tests have a very small effect on the E (xi). x  Voltage transducers for metering applications are required to meet particular accuracy class specifications. According to IEC standards [45], [46], these range from 0.1% class to 3% class, for which the corresponding magnitude error bands range from ±0.1% to ±3%, respectively. Metering class specifications also impose limits on phase angle errors for each accuracy class. The voltage magnitude errors in Table 6.2 and Table 6.3 show that the IOVT can meet 1% class metering accuracy for magnitude and may, in practice, be able to meet 0.2% class metering accuracy for magnitude. The IOVT provides a collection of advantages over previous, conventional and optical VTs. It does not require special insulation since no particularly high electric field stresses are present. Its mechanical design is primarily that of the standoff insulator, which is available offthe-shelf. The insulator serves to mechanically support the sensors and to protect them from  92 harsh weather conditions. It also prevents electric field perturbations from occurring close to the sensors, which is important since such changes could introduce excessively large errors into the integration. It should also be noted that the VT's insulating structure could also be placed between phases to give a measurement of the phase-to-phase voltage in a three-phase system. Here, IOPCs were used, but other small electric field sensors can serve the same purpose. During and shortly after the time that the IOVT was built, both IOPC technology and BGOPC technology (see Section 5.3) for field sensing were being developed in parallel. At the time, IOPC technology had a higher cost associated with it than that of B G O P C technology. Since the development of OVTs was industry-driven with the purpose of introducing them to the instrument-transformer market in a timely and cost-effective fashion, it was decided to use BGOPCs, instead of IOPCs, in subsequent O V T prototypes.  Fig. 6.5. Integrated-optic voltage transducer with a vertical grounded screen 1.5 m away.  93 6.2.3  Summary and Conclusion  The first voltage transducer based on the quadrature method was built. A n IOVT, consisting of three IOPCs mounted inside an off-the-shelf standoff insulator, was shown to be accurate even with significant changes in the electric field distribution. In conclusion, an IOPCbased voltage transducer can achieve DSC 1% class metering accuracy and may, in practice, be able to meet DEC 0.2% class metering accuracy. A key feature of this design concept is that it does not require special electrode structures and/or insulation, making it safe and environmentally friendly.  6.3  Optical Voltage Transducer  6.3.1  Introduction  A 230 k V O V T designed to meet DEC 0.2% class requirements [45] was built and tested. The 230 k V O V T uses three small BGOPCs and the quadrature method to determine voltage. The sensors are housed inside a hollow off-the-shelf H V insulator (as with its predecessor, the IOVT) that is filled with N2, providing a secure environment to the sensors and making the O V T structure mechanically and electrically very robust. In the next section, the basic operation of the O V T is described. Then, an overview of standard H V laboratory tests and of additional tests for the purpose of confirming the accuracy of the O V T is given, and results are reported.  94 6.3.2  Principles of Design and Operation  The OVT structure consists of an off-the-shelf 230 kV insulator supported by a stand that is sitting on a ground plane, as it would be in a substation. The insulator consists of a fiberglass tube having an inner diameter of 35.8 cm and thickness of 8 mm, with silicone rubber shedding on the outside of it, and provides a structurally robust design. A corona ring is also positioned around the top of the insulator. Inside the insulator tube, there is a smaller fiberglass tube having an inner diameter of 19.8 cm and a thickness of 4 mm, and extending from the top flange to the bottom flange, to support field sensors. At the ends of this inner tube are two electrodes separated by a distance of approximately 2.16 m. This configuration is defined to be the unperturbed system (see Fig. 6.6). The path of numerical integration is along the tube axis, or xaxis, from the bottom electrode, at point a, to the top electrode, at point b. E  unp x  is E along this x  path and is calculated for an applied rated voltage of 140 kV and for a stand having a height of 2.5 m (see Fig. 6.7).  b Corona ring  Electrodes  Insulator  —a  Stand .  Fig. 6.6. The unperturbed O V T system.  95  >  200  2 5  o o4)  'HS  52  Fig. 6.7. Axial electric field along the path of integration for an unperturbed O V T and a perturbed O V T with 140 kV applied.  The O V T is specified to maintain good accuracy, i.e., to meet IEC 0.2% class specifications, in the presence of the kinds of changes in external geometry that it may experience upon its installation or the installations of neighboring equipment during its operation in a 230 k V substation. Substation design safety standards specify minimum clearance distances between any two pieces of H V equipment installed on two different phases or on a phase and ground depending on their voltage class (typically about 3 m or 2 m, respectively, for 230 kV), thereby limiting the severity of the field perturbations and, consequently, the required number of sensors, N. Fig. 6.7 shows the axial electric field for a perturbation consisting of a vertical ground plane 1.6 m away from the jc-axis, and Fig. 6.8 shows the p that corresponds to that perturbation.  96  1  n o -, 1.05 -  ^  -  1.00 -  0.95 -  a  0.90 -  /  0.85 -  0.80 -  /  /  0.75 -  0.70  -I  1  0  500  1  1000  n  1500  1  '  2000  .v (mm) Fig. 6.8. p for a semi-infinite vertical ground plane 1.6 m away from the OVT's axis.  Various insulators with various perturbations were simulated. Examples of perturbations simulated include neighboring-phase buses, a nearby, vertical ground plane, and a nearby conducting sphere. For each of these cases, sample positions and weights for various numbers of sensors were computed using the quadrature method, and the accuracy of the resulting weighted sums were observed. It was found that three sensors are sufficient to accurately measure voltage (< 0.2% error) for these types of perturbations and for insulators with internal electrodes spaced apart by approximately 2 m (length of the integration path). Table 6.4 shows simulated errors for computer models of perturbed systems involving either a ground plane or a grounded sphere near our OVT. The distance of a perturbing object refers to the distance between the edge of the object and the jc-axis, and a sphere's height is approximately the height of its center with respect to the bottom of the insulator, e.g., "Low" is at the height of the bottom of the insulator and  97 "High" is at the height of the top of the insulator. Table 6.5 shows the changes in E at the three x  sample locations inside the O V T for two of these simulated cases.  Table 6.4. Simulated magnitude errors due to extreme perturbations.  Perturbation Type  Distance  Height  % Magnitude Error  None  NA  NA  0.00  Ground Plane  1.6 m  NA  -0.03  Ground Plane  2.2 m  NA  -0.02  Grounded Sphere  1.2 m  Low  -0.05  Grounded Sphere  1.6 m  Low  -0.03  Grounded Sphere  1.2 m  Middle  -0.02  Grounded Sphere  1.6 m  Middle  -0.02  Grounded Sphere  1.2 m  High  -0.01  Grounded Sphere  1.6 m  High  -0.01  Table 6.5. Simulated changes in E (xi) due to perturbations. x  Perturbation Type  Distance  AEx(x{) (bottom)  (middle)  (top)  Ground Plane  1.6 m  -22.2%  0.9%  5.6%  Middle Grounded Sphere  1.6 m  -3.0%  0.4%  0.5%  AE (x )  L\E (X ) x  x  2  3  So, having established the need for three electric field sensors, three positions x, and three weights cc, were determined using the quadrature method with the calculated E  unp x  of the O V T  system to be built. Then, electric field sensors were mounted inside the inner tube at these x , t  and their outputs were weighted with these a, and summed in real-time using digital electronics to give the output of the OVT. The sensors measured the jc-component of the electric field in which they were immersed. Table 6.6 gives the x, and the a, that were used in the OVT.  98 Table 6.6. Formula sample positions, x„ with respect to a and weights, a „ normalized to the middle sensor's (#2's) calculated weight. Sample Number 1 2 3  Xi  (mm)  231.04 1229.69 1997.65  a, 0.60048 1.00000 0.41681  Light was transmitted to and from each sensor using optical fibers. The outputs of the sensors were then digitally processed giving measures of the electric field, and further digital signal processing performed the numerical integration in real-time.  6.3.3  Laboratory Test Results  Four identical three-sensor 230 kV OVTs, as described earlier and illustrated in Fig. 6.6, were constructed and tested in an H V laboratory (see Fig. 6.9). The OVTs were filled with N  2  gas at 170 kPa. Each O V T weighs approximately 220 kg. During the testing, they were supported by a grounded platform about 2.5 m high (see Fig. 6.9). The electric field sensors were positioned and their outputs were weighted according to the the JC, and a, given in Table 6.6. Cables supported the fibers that transmit light to and from each O V T , and the analog and digital electronics resided in the control room, where digital data acquisition took place. The output of the digital electronics was passed through a D/A converter and was amplified to give an analog voltage output, which is required for standard testing. For an applied rated voltage of 140 kV, the OVTs read 2 V , corresponding to a voltage transformation ratio of 70,000:1.  99  5t  ass  Fig. 6.9. High-voltage test set-up.  Various tests were performed on the OVTs (at least one O V T per test) in accordance with IEC standards [45], [46], [48], and they included error testing, lightning impulse testing, wet testing, power-frequency withstand testing, partial discharge testing, chopped impulse testing, and mechanical testing. Special tests were also performed to evaluate the accuracy of the O V T in the presence of "substation-like" changes in local conductor geometry. Since the O V T design is essentially that of a standard standoff insulator with a few extra internal dielectric components, it inherits the advantageous mechanical and electrical properties of the insulator, particularly with respect to H V withstand and seismic withstand. The O V T successfully passed all of the standard withstand tests. Additionally, it withstood negativepolarity, full-wave impulses down to -1211 kV, which exceeds the standard full-wave impulse test voltage magnitude of 1050 kV, with no sign of disruptive discharge or insulation failure.  100 Linearity tests were done on three of the OVTs, one at a time, by observing the ratio of their voltage measurements to the voltage output of the reference. The OVTs were tested for DEC 0.2% accuracy class [45], [46], which means that the magnitude error should not exceed ±0.2% and the phase displacement error should not exceed ±10 minutes at 80%, 100%, and 120% of rated voltage. The OVTs demonstrated magnitude errors of less than 0.1% and phase displacement errors of 1 minute or less, meeting the DEC 0.2% standards. A l l the OVTs had a rated phase delay of 0.95°. Future versions of the O V T will have user-defined rated phase delays to be set to any value depending on the requirements of the application, e.g., 0° to meet the standard requirement in [46]. Additionally, switching impulses were applied to an O V T , and the OVT's output faithfully traced the applied waveforms demonstrating the high bandwidth and dynamic range of the OVTs. Errors were recorded at various other voltages outside of the standard range. Fig. 6.10 shows magnitude and phase errors for voltages from 3 kV to 350 kV. The magnitude errors do not exceed 0.2% and phase errors do not exceed 1 minute through this entire range. It should be noted that the upper range of the digital measurements was limited only by the test equipment. 0.30 0.25  0.2% Class  0.20 0.15  9T o fc CO  <u -o 3  A n a l o g Output  O  Digital Output  0.10 0.05 0.00  o  -0.05  CO CO  •  -0.10  a  O  o  -0.15 -0.20 -0.25  80%  100% 120%  Percent of Rated Voltage  -0.30 100  200  applied voltage (kV) (a)  300  400  101  0  100  200  300  400  applied voltage (kV) (b) Fig. 6.10. OVT (a) magnitude errors and (b) phase errors.  Perturbation tests were also conducted in which the accuracy of an O V T was tested at rated voltage with various objects placed in its vicinity. A suspended, vertical, grounded metallic screen at various distances from the O V T (see Fig. 6.11), a suspended, grounded metallic sphere (1 m in diameter) at various heights and distances with respect to the O V T (see Fig. 6.12), a grounded truck near the O V T (see Fig. 6.13), and a neighboring energized O V T 120° out of phase with and 3 m away from (< minimum phase spacing) the tested O V T were used. The magnitude errors for the cases of the grounded sphere and grounded screen at safe distances from the O V T and for the cases of the truck and the neighboring phase are less than 0.1%. The magnitude errors for the unsafe cases do not exceed 0.3%. Table 6.7 shows the errors for some of the safe cases, and Table 6.8 shows the errors for some of the unsafe cases.  102 Here, the distance of a perturbing object is defined as the distance between the edge of the object and the OVT's central axis. The minimum safe distance for a grounded object is defined as approximately 2 m or more (the exact number depends on the local substation safety regulations). Heights of perturbing objects are with respect to the bottom of the insulator. The phase displacement errors are less than 1 minute for all of the perturbed cases.  Fig. 6.11. Grounded screen perturbation 1.6 m away.  Fig. 6.13. Truck perturbation.  104  Table 6.7. Magnitude errors due to safe perturbations. Perturbation Type None Grounded Screen Grounded Sphere Grounded Sphere Neighboring Phase  Distance NA 2.2 m 2.2 m 2.2 m 3 m  Height NA NA Low Middle NA  % Magnitude Error 0.00 0.04 -0.01 0.03 0.04  Table 6.8. Magnitude errors due to unsafe perturbations. Perturbation Type None Grounded Screen Grounded Sphere Grounded Sphere Grounded Sphere Grounded Sphere Grounded Sphere Grounded Truck  Distance NA 1.6 m 1.6 m 1.6 m 1.2 m 1.6 m 1.2m 1 m  Height NA NA High Middle Middle Low Low NA  % Magnitude Error 0.00 0.14 0.06 0.07 0.25 -0.02 -0.05 -0.07  Demonstrating the effectiveness of the weighted sum used in the O V T design to determine voltage accurately, Table 6.9 shows the percentage changes in the measured electric field at the sensor locations in the perturbed cases with respect to the unperturbed case. Table 6.9 also shows phase changes in the measured electric field for the case of the neighboring phase. It should be noted that for the cases of the suspended sphere, the sphere was vertically supported by a grounded cable, which results in a more severe perturbation compared to the simulated grounded-sphere perturbations. Three of the tested OVTs have been installed as a three-phase voltage measurement system at B C Hydro's Ingledow substation in Surrey, B C , Canada. They have been in operation since June 2000 and are being evaluated by B C Hydro (see Fig. 6.14).  105 Table 6.9. Changes in E (x ) due to perturbations. x  t  (bottom)  (middle)  A£*(x ) (top)  1.6 m  -28.9%  1.9%  14.0%  Middle Grounded Sphere  1.6 m  -11.5%  0.85%  5.3%  Grounded Truck  1m  -5.0%  1.4%  1.5%  Neighboring Phase at 120°  3m  -14.7% (-9.95°)  0.4% (0.15°)  8.9% (4.15°)  Perturbation Type  Distance  Grounded Screen  3  Fig. 6.14. OVTs at the Ingledow substation, Surrey, B C , Canada.  6.3.4  Conclusion  Four 230 kV OVTs were constructed and tested. One or more of the OVTs were subjected to various IEC electrical and mechanical withstand tests, and all tests were passed successfully. They demonstrated 0.2% class accuracies in accordance with [45], i.e., magnitude errors not exceeding ±0.2% and phase displacement errors not exceeding ±10 minutes. The  106 OVTs have a fixed rated phase delay of nearly 1°, which will be set by the user in future versions of the O V T to meet the requirements of specific applications, e.g., 0° to meet the requirement of [46]. Further tests demonstrated that the OVTs' voltage measurements are accurate to within 0.1% and 1 minute for various cases of electric field perturbations caused by nearby metallic objects and a neighboring OVT. The OVTs were also shown to be very accurate (within 0.3%) even when the metallic objects were placed dangerously close to the OVT. Significant measured changes in the local electric field measurements at the field sensor locations due to perturbations were also shown, displaying the effectiveness of the implementation of the quadrature method.  6.4  Study of Pollution Effects  6.4.1  Introduction  The O V T discussed in Section 6.3 above was designed to maintain accuracy in the presence of substation-like changes in geometry, such as the installation of neighboring equipment. These types of changes in geometry are limited in severity by substation safety standards, e.g., minimum phase-to-ground and minimum phase-to-phase spacings. It was demonstrated that for these kinds of perturbations, only three sensors are sufficient to measure voltage with IEC 0.2% class accuracies. However, a V T operating in a substation may experience more severe perturbations caused by exposure to pollution and weather. For example, the O V T was tested for accuracy under severe pollution conditions and showed an error of almost 30%. As a result, shielding and/or more sensors must be used to maintain accuracy under such conditions. Here, the effects of pollution on accuracy are investigated by modeling and laboratory testing.  107 6.4.2  Laboratory Pollution Tests  Tests were conducted so that a concrete connection, or correlation, between simulated results and laboratory results could be determined and so that the performance (accuracy) of an O V T design, specifically with regard to the number of sensors needed and their positions, in the presence of severe pollution conditions could be better predicted using more accurate modeling. Additionally, a better understanding of dynamic stray field effects, e.g., due to drybanding and arcing, was accomplished. These phenomena are practically impossible to deterministically model using presently available simulation tools.  6.4.2.1 Approach  The output of the O V T is a combination of the outputs of the electric field sensors mounted inside of it. Because the accuracy of the combined signal depends on how the vertical component of the electric field along the axial center of the column changes in the presence of changes in the medium, the conductor geometry, and electrical sources external to the insulating column, a laboratory test is set up to measure this electric field using electric field sensors uniformly distributed along the center of a standoff insulator in the presence of controlled field disturbances, which are expected to be comparable to levels of pollution substation equipment may experience during its operation in the field,. These perturbed systems are also simulated with a computer and the electric field is computed. The measured and computed perturbed electric field distributions are then compared with each other. Also, a laboratory test is set up at a high-voltage laboratory in which the insulator is painted with a conductive substance, that is typically used for pollution withstand tests of insulating columns. The change in the axial  108 electric field component is measured with the electric field sensors uniformly distributed along the center of the column in the presence of this disturbance with and without exposure to fog.  6.4.2.2 Test Setup and Results  A 138 kV off-the-shelf, fiberglass-rubber-sheds insulator (inner diameter = 300 mm; height = 1525 mm) with nine equally-spaced BGOPC field sensors (see Section 5.3) mounted between internal electrodes was built. The internal electrode-to-electrode spacing is 1182.1 mm and the column is filled with air and is not pressurized. See Fig. 6.15.  i.  •-  Fig. 6.15. Al-foil test setup (prior to foil application).  Fig. 6.16 shows the expected E  mp x  (see Chapter 3) for the column and the corresponding  p, which is uniform with value 1 (see Chapter 3). In the testing, the experimental p for each  109 perturbed system is investigated instead of the actual E of that system. This is done for two x  reasons. The first reason is that the electric field sensors were not calibrated (with respect to electric field), so neither an absolute nor relative (sensor-to-sensor) measurement of E was x  possible. The second reason is that p is directly related to the accuracy of an iV-sensor OVT. An JV-sensor O V T has perfect accuracy if p can be exactly approximated by a 2N - 1 degree polynomial.  400 -|  1  300 200 -  I  100 0 -  (a)  Q.  1  200  400  600  x  800  1000  (mm)  (b) Fig. 6.16. (a) Simulated E  u n p x  for the rated applied voltage of 80.5 k V and (b) unperturbed p.  110 6.4.2.2.1  Aluminum Foil Tests  Aluminum foil is wrapped around the entire insulator except for a small section referred to as the "dryband" (see Fig. 6.17). Pollution tends to deposit on the surface of a standoff insulator installed at a substation. When moisture appears on the surface, e.g., by way of precipitation and condensation, it dissolves the pollution deposits, and conductive regions form on the insulator's surface. Drybands refer to regions of lower conductivity than that of the surrounding regions, and they typically severely distort the electric field along and near the insulator's surface. Aluminum foil wrapped around the insulator is used to model the highly conductive regions, while the absence of aluminum is used to model the drybanding. The effect of this drybanding at various positions along the height of the column was examined for applied voltages of up to 25 kV and at 60 Hz.  Fig. 6.17. Insulator with A l foil and a middle dryband.  Ill Aluminum foil testing for two cases of single dryband conditions was performed, and the results were correlated with those of computer simulations of these cases. Fig. 6.18 shows experimental and simulated results for the case of a three-shed dryband (approximate height of 160 mm) at the top of the column. Fig. 6.19 shows experimental and simulated results for the case of the same-size dryband at the middle three sheds of the column.  20 -I OX)  CD  -20  c/3  -40  si OH  *  0  a <+-( o  cu  • •  «  —  w  w  w  measured simulated  -60 w 1  200  i  i  600  400  x (mm) (b)  1  800  1  1000  i p  112  CL  measured simulated  O •*-» O  J3 «  CL  c  200  400  600  800  1000  x (mm) (c) Fig. 6.18. (a) The magnitude, (b) the phase, and (c) a snapshot (at the peak of the voltage waveform) of p for top dryband.  •  -  •  measured simulated  o 0  1  1  i  200  400  600 x (mm) (b)  800  1000  113  10  0  200  400  600  800  1000  x (mm) (c) Fig. 6.19.  (a) The magnitude, (b) the phase, and (c) a snapshot (at the peak of the voltage waveform) of p for middle dryband.  In both cases, there is very good agreement between the simulated and experimental results. There are large phase differences in areas where the field magnitude becomes very small. This is due to the fact that for very small signals, the accuracy of the phase calculation decreases (it is hard to see the signal in the noise). In any case, observation of the snapshots of p in time shows good agreement between the experimental and simulated results. It is expected that the top sensor deviates from the simulated p, in part, because it is positioned partially inside a hole in the top electrode (half inside and half outside) and does not exactly measure E at the x  surface of the ideal electrode as is done in the simulations. Single middle dryband is shown to be a severe kind of perturbation introducing large errors in the voltage readings. Using the plots, it can be shown that the middle dryband p requires a high-degree polynomial (>20 degree) to approximate it "reasonably" accurately, th  which directly corresponds to needing at least 10 sensors to "reasonably" accurately determine the voltage for this case (-1%)  .  114 When applying 50 k V (the maximum for the setup), there was significant buzzing due to partial discharge for the cases of top and middle dryband. 80.5 kV is the rated line-to-ground voltage for 138 kV VTs, and it is expected that flashover would occur at the rated voltage for these kinds of drybands.  6.4.2.2.2  Pollution Tests  In a high-voltage laboratory, clay, water, and salt were mixed together to form a conductive paste that was painted on the entire shed surface (see Fig. 6.20). Since the silicone rubber shedding of the insulator is highly hydrophobic (to prevent drybanding and extend its lifespan), isopropyl alcohol was applied to its surface to significantly, temporarily destroy the hydrophobicity in order so that the mixture would more easily adhere to it. Obviously in doing so, the rubber material could potentially be irreversibly damaged. The particular mixture used is according to the Level 4 insulating standoff pollution standard [50], i.e., considered medium pollution. The insulator was placed inside a fog chamber. Measurements were made just after application of the mixture, after it has dried, in the presence of fog, and after removal of most of the mixture from the shed surface.  115  Fig. 6.20. Polluted insulator.  Fig. 6.21, Fig. 6.22, and Fig. 6.23 show p for the cases of wet pollution, dry pollution, and fog pollution, respectively. It is noted cubic spline interpolation is used to interpolate the sample points in these figures. In the wet pollution case, the conductive paste described earlier is applied, and the column is energized soon afterward while the paste is still wet. Significant arcing (visible discharge) took place when initially bringing the voltage to 80 kV, so it was dropped to 20 kV during most of the data acquisition for this case. In the dry pollution case, the conductive paste is dried (see Fig. 6.20), and in the fog pollution case, fog was applied for approximately 80 minutes. Most of the data acquisition for these cases was done at 80 kV, and only small traces of arcing were noticed during them. Clean fog tests were also conducted prior to the pollution tests, but field changes were very small by comparison.  116  a  tw  o cu -a3 w  •a oo  B  00 CD  T3  Q. tw O  1)  cn oi •w  CL,  (b)  Q. CH  O -w O J3  cw CL,  cj C  Fig. 6.21. (a) The magnitude, (b) the phase, and (c) a snapshot (at the peak of the voltage waveform) of p for wet pollution.  117  (a)  (b)  (c) Fig. 6.22. (a) The magnitude, (b) the phase, and (c) a snapshot (at the peak of the voltage waveform) of p for dry pollution.  118  o CD - O  -t-»  '3  oo S  oo T3 C Cj-I  O  <L> (/I  03  O.  (b)  (c)  Fig. 6.23. (a) The magnitude, (b) the phase, and (c) a snapshot (at the peak of the voltage waveform) of p for fog-pollution.  119 Using the simulated E  unp  x  shown in Fig. 6.16(a) and the experimental p's, these given by  the sample points and cubic spline interpolation, along with the relationship E = pE x  unp  x  (see  Chapter 3), the experimental field distributions for the perturbations can be computed. Fig. 6.24 shows these E for some of the tested cases with the severest perturbations. Note that these plots x  are for a rated applied voltage of 80.5 k V ; the plots scale linearly proportionally to the applied voltage. It can be seen from the regions of higher field intensity (the humps in the plots) that the distribution of the pollution was such that there were two dryband-like regions at the top and near the bottom of the insulator. Fig. 6.18, Fig. 6.19, Fig. 6.21, Fig. 6.22, and Fig. 6.23 indicate that the Al-foil single dryband scenarios create perturbations that are at least as bad as the pollution perturbation, with respect to the quadrature method. So, Al-foil testing can serve as a practical and efficient way to test an OVT's accuracy performance under severe pollution conditions without damaging the insulator surface. 1000  800  A E - top dryband E - middle dryband £ - wet pollution x  600  B B  x  > ^*  400  200  200  400  600  x (mm) Fig. 6.24. Experimental E s for some of the tested cases using the experimental p and simulated E for the rated applied voltage of 80.5 kV. x  x  np  120 6.4.3 Simulated Accuracies for Unshielded and Resistively Shielded OVTs in Dryband Conditions  The laboratory testing discussed above indicates the need for considerably more than three field sensors or some permittivity-shielding or both. The 138 kV insulator was modeled, and magnitude and phase errors were calculated for perturbed systems of each of two cases: no shielding and center-tube shielding. The center-tube shielding consists of a single, solid, nearly cylindrical resistive tube extending from one electrode to the other along the insulator axis that has a diameter of approximately 60 mm. The shielding has a total resistance of approximately 75 M£2. The errors are calculated for N = 1 to 22 sensors positioned and weighted according to the quadrature method. The perturbed systems involve a 0.25 mm-thick resistive layer resting on the shed surface in four different pollution cases: top three-shed dryband, middle three-shed dryband, and two cases of uniformly spaced multiple drybands (two and four). The three-shed dryband is a dryband having a height of about 160 mm. The multiple-dryband cases consist of drybands whose heights and spacing are equal. For example, the two-dryband case has two drybands located at the top and bottom of the insulator and having heights that are each about one third of the insulator height and a spacing that is also one third of the insulator height. The resistive layer also has varying resistivities: 100 £2cm, 1 k£2-cm, 10 k£2cm, and 100 k£2-cm. Table 6.10 shows errors for resistivities of 100 Q,cm and 1 k£2cm and for N = 10 to 22 in the case of no shielding. Table 6.11 shows errors for resistivities of 100 Q-cm and 1 k£2-cm and for N = 1 to 10 in the case of center-tube shielding. Of the four resistivities, 100 £2-cm and 1 k£2cm are the two severest perturbed systems, having the highest magnitude or phase errors. After analysis of the data, it is expected that at lower resistivities of the pollution layer, the errors should, at worst, only marginally increase, and that, at very low resistivities, the phase errors should significantly decrease (see Chapter 4).  II  2  o  o  CO  CD  t—  cn  r-  LO  czi  O  o  c—  CO CD O  czi  czi  o  |  |  -0.016  -0.014  0.065 0.081  -0.024  -0.019 o  o  o  czi  czi  -0.020  czi  -0.022  czi  0.259  — t  o  00 CO CO  CZi  CO  0.047  0.305  CO OO CNJ  -a*  CO  2  CZi  o o  CD  LO  o  CO CO CNl  o o p  CNl  II  II  2;  2; LO  II  2; -<r  II  2 CO  II  2  czi  czi  rCD CNl  CD LO  CZi  czi  o p -0.006  CO  CO  czi  t—  oczi  -0.009  2:  ii  CD UO CO  II  LO CNl  oczi  -0.024  II  2  o  oo  t— LO  czi  0.001  LO  CD  II  2  CD CD CNl  0.312  2  CO CD  cn  czi  o  czi  czi  -0.162  II  cn  2  CO  o  o  -0.435  CO  II  t—  CNJ  LO CNl  -0.346  2;  CD OO  o  CNJ  czi  cn f—  o  Phase Erro r in M inutes  ll  o  0.295  2;  c3 O  00  O  2  r—  0.166  II  ^t-  IS  O  o  0.372  oo  C KS CO  c is  CO  CO  cn  II  0.114  -2  LO  ISO  o ex.  CO LO CD  0.155  c o  CNl CZ)  0.270  2;  Magrlitude Erroi i n %  -5  o  0.045  II  I'-  CO  o  0.032  cn  CZ)  o  o  -0.267  cs  oo CO  CNl  -1.262  2;  o o  t— CNJ CNJ  CO  cn  CO  II  CNl  -0.161  T3  d  cn  0.899  II  o  2  TP OO  CN]  B  czi  CNl  o  C  o  -0.040  d  o  o  2  o  CD CO CNl  -0.295  a  CNl CNl  oo co o  2  II  LO  0.699  II  CO CO  0.557  CNJ CNJ  -0.106  |  121  00  CO  CZi  czi  o o  o o  00  0.009  -0.001  II  2  -0.121  LO CNJ  CNl  -0.026  o  -0.156  II  2  -0.003  CN]  s-  0.345  c  0.033  I  O  o  o  CO t-—  o  o  CD  O  o  czi  II  2  -0.021  -0.018  II  2  -0.010  2  czi  4 (uniform dist.)  co  CD  CO CO  -0.025  oo r— ll  o  oo  2 (uniform dist.)  o  u  cn  0.508  &, -a  1  CNJ  -0.216  2  '  o  middle 3-shed dryband  LO  0.306  II  CO oo C3  top 3-shed dryband  co  -0.549  O  CNJ  Dryband  'uniform dist.)  [uniform dist.)  NO  Dryband  cd  top 3-shed dryband  •4—*  "S  midd'le 3-shed dryband  T3  2: II  b C  o cra  Q  o ll 2; CO  rcCM  o  .—i  o  CO LO  o o o o  2i  II  T  2: II  CO  o d II 00  2;  ca  D  2;  ai  CM  o  II  oo p  ll 2:  o  2: II  o cra  -1.046  LO  -0.462  II  -0.766  2;  -1.723  CD  -0.350  II  -1.767  2;  -0.125  r-  -12.518  p  CO*  c~  a>  oo  oq  CD CO  o  CO  O p  CM  LO CM  r-  O  -1.589  -0.987  -0.956  0.201  -2.962 4.631  -1.433 | -1.147 |  |  |  |  CO CM LO  oo  00  CM  oo 00  o CM  CM o  -2.113 -1.540  4.719  0.228  |  o co c-  CD CO CO  -1.990  -6.344  2.075  -0.315  -0.014 |  CO CM  CD O O  16.407  II  18.989  2:  -9.007  oo  -11.165  II  -14.366  2;  -0.590  CM CO  2;  4 (uniform dist.)  cco o o  II  -1.665  00  CM  0.091  0.297  o  -11.335  0.360  fO  4.369  0.158  0.113  O  2;  -2.381  0.287  0.258  o  -0.162  O  2 (uniform dist.)  0.160  0.043  -0.039  o  Phase Error in M inutes  0.048  0.273  -0.927  CO o  II  28.499  o  CD CO  o  CM  -9.969  o  CM CO  CM  CD  middle 3-shed dryband  LO CM  oo  II  -18.202  ro  oo  2;  10.528  2  CO fCD O  CM CM  top 3-shed dryband  II  0.152  CVl  0.381  2;  0.086  II  CD CM  0.031  LO  4 (uniform dist.)  2; o  0.184  2;  0.400  II  cn  0.349  t-~  o  -0.927  ll 2:  CO LO  0.313  00  2 (uniform dist.)  II  2; o  -1.373  II  0.503  2;  o  -0.105  o>  0.291  II  0.310  CM  -0.065  o  0.181  II  -0.145  CM  middle 3-shed dryband  II  0.406  II  -0.273  CVI CM  top 3-shed dryband  91 = AI  Magrlitude Erroi- in %  122  CVl  ^  p  CO  CD  o  123 o II  B o d  fe  M  CD  -—\  II  .0  fe  T3  c  o o p  o o o  o o o  o o o  o  o  o  o o p  o o p o  o o o o o" o  O  oo CV]  r—  CT>\  o  oo  oo O O  Cvi  CTJ  cm  1  oo II  a o o  O o  o o o" o  o o o o  o  o  Cv]  CV]  fe  o o  t>-  oo  o  o  00  SI  oo  II  _>  fe  "•4—» CO  o p p  TOO"  ites  03  p  o p o  cv) CO  o p o  cn LO  CO  <s  <D  OS  C  >~> 03  s-.  Er  CO II II  •~  tion  1-  <U  o  1—1  fe  o o o  LO  00  o  o o o  o  o o o  +-J 3  o  o o o  SS  o o  w  CO CO  co •"tf  03 TD 3 HH  "o  II  c  OJO  IH  £  CO  fe  o o p  co  o o p  o o o o  o o  co o co oo o CD  cu CO  CO  JS PH  CO  03 "—'  II  gca  CO  fe  e  LO  o  o o  Cvi  *—1  o  o o  o  o o  CV]  < —N c J  o  o o  00  Cv] LO LO  o  1  1)  co  JS  II  CO  be  i  f—  o o o  CJ5  o o o  CO  o p o  1  3  1  ^H  co  o o o  oo CO  (U  CO II  CU o  II  fe  o  ciS  •^r o o o o  CO CVl  co  o O  o o o  1—1 Cv]  00  cn cd  1  t-H  +-» C  o  cn  oo co  LO CD  CO O  <U  II  cO  fe  < u CO JS  c—  CD  oo o  p  o  o  CV]  o  o o  o  CO  CO 00 CD LO Cv]  O-  -d  <u 03  H  JH  o. o  S  TD  c  CO  175  Cv]  TD C  CO JD  -Q  -a  niforn  3-she dryban  oo  -a  idd  VO  TD -H  •shed ryband  00  TD  Dryba  .. Ma nitude a  c  TD  TD  c  a a  CO JD  Q  TD TD  cu JS co i  CO OJ  CU  JS to i  CO CU TD  I  0.085 0.015  3.676 -0.092 0.100  2 (uniform dist.)  4 (uniform dist.)  2.373  16.226  II  4 (uniform dist.)  II  14.764  CN]  -17.703  II  2 (uniform dist.)  oo  -0.995  -4.118  2.990  0.209  -0.686 -0.993  1.169  0.268  -0.208 -0.695  0.298  0.117  0.121  0.005  0.016  -0.116  0.904 -0.801 0.815  -2.021  -0.090 0.178  II  -31.259  II  131.418  un  middle 3-shed dryband  II  0.068  C O  0.056  II  0.486  II  -0.116  oo  0.004  ll  1.487  cn  -1.120  0.001 0.000  -0.002  0.000  -0.008  0.000  0.001  0.000  -0.009  0.041  0.117  0.140  -0.070  II  -28.804  0.001  -0.001  0.002  Phase Error in Minutes  0.004  -0.007  0.026  -0.010  o  top 3-shed dryband  -0.007  II  0.001  II  -0.003  un  -0.021  II  0.019  CJO  -0.057  ll  0.012  I'-  Dryband  0.108  -0.983  II  middle 3-shed dryband  2:  -0.002  II  -0.006  oo  0.006  ll  -0.011  cn  0.019  II  -0.026  0.024  II  0.103  II  -1.551  CO  top 3-shed dryband  Magnitude Error in % o  000 0  Dryband  124  125 The tables indicate that if no shielding is used, 22 sensors may be enough to meet the accuracy requirements of IEC 0.2 class metering accuracies (< 0.2% magnitude error and < 10 minutes phase error) and ANSI/IEEE 0.3 class metering accuracies (< 0.3% magnitude error and < 8 minutes phase error). With center-tube shielding (75 MQ), 3 sensors are enough to meet the same requirements even in the presence of severe pollution perturbations.  6.4.4  Summary and Conclusion  Artificial drybanding on a 138 k V insulator was modeled using aluminum foil on the insulator surface, and the measured changes in the electric field along the axis of an insulator due to the drybands were compared with simulated results. Good agreement between the simulations and the experimental tests was demonstrated. Additionally, changes in the field due to the presence of a pollution-like mixture on the insulator surface were studied. From these changes and the use of interpolation, an experimental p can be approximated. Comparing the pollution p's with those of the aluminum-foil drybanding cases indicates that single drybands are the more severe perturbations with regards to the quadrature method. As a result, using aluminum-foil drybanding experiments could probably serve as a worst-case benchmark for the accuracy performance of an O V T exposed to pollution. Modeling showed that if the tested 138 k V insulator was fitted with about 22 sensors with no shielding or 3 sensors with 75 M£2 center-tube shielding, it could measure voltage accurately even under severe pollution conditions.  126 6.5  Resistively Shielded Optical Voltage Transducers  One 345 kV and two 138 kV resistively shielded OVTs were constructed and tested. They underwent IEC standard testing as well as various perturbation tests. The results are discussed in the following sections.  6.5.1  OVT Structures  One of the 138 kV OVTs is made up of the same off-the-shelf fiberglass insulator described in Section 6.4.2.2, having a height of 1525 mm and an inner diameter of 300 mm, and a three-stage resistor. Similarly, the 345 k V O V T is also made up of an off-the-shelf insulator having a height of 2685 mm and an inner diameter of 510 mm that is internally fitted with a high-voltage electrode at the top and a grounded electrode at the bottom. Both columns are fitted with shielding between the two electrodes that consists of a three-stage resistor configuration. The three-stage resistor is made up of three identical resistors that are connected to each other through electrically floating electrodes. Inside and at the center of each resistor stage is a B G O P C electric field sensor. The three sensors are weighted equally. In so doing, the field is assumed to be uniform along the length of each resistor, and each resistor stage is treated as an independent voltage stage. So, according to the quadrature method, each sensor must be placed at the center of each stage and gives a measure of the voltage across that stage, and since the resistances are equal, the ideal, uniform electric field along each resistor is the same, and the weights are equal as well. Each resistor in the 138 kV O V T has a height of 460 mm, outer and inner diameters of 33 mm and 25 mm, respectively, and a resistance of 30 MQ, while each resistor in the 345 kV O V T has a height of 770 mm, outer and inner diameters of 48 mm and 40 mm, respectively, and a resistance of 50 MQ,. The resistors are a proprietary technology and  127 are available off-the-shelf, and they basically consist of a hollow alumina tube coated with a thin metal-oxide layer. The other 138 kV O V T is also made up of a fiberglass tube having a height of 1372 mm and an inner diameter of 300 mm and is internally fitted with two electrodes at both ends. A single resistor is mounted between these electrodes, and three electric field sensors are mounted inside of it. The resistor has a height of 990 mm, outer and inner diameters of 48 mm and 40 mm, respectively, and a resistance of 100 MQ. Inside the resistor are mounted three B G O P C electric field sensors. They are positioned according to Gauss-Legendre quadrature, which assumes that the electric field is uniform along the length of the resistor and which is a specific case in the quadrature method. The positions are 105.9 mm, 469.65 mm (resistor center), and 833.4 mm measured from the bottom electrode, and the weights are 0.625, 1, and 0.625, respectively. A l l OVTs were filled with pressurized dry nitrogen gas. The 138 k V OVTs were pressurized at 83 kPa to 110 kPa above the atmospheric pressure, while the 345 k V O V T was pressurized at 170 kPa above atmospheric pressure. It was found that the higher the pressure is, the higher the breakdown threshold of the nitrogen insulation is. This was expected as per the abundant literature available on the topic of breakdown strengths of gaseous dielectrics [13]. For example, at 170 kPa above atmospheric pressure the nitrogen gas threshold is 2.66 times higher than at atmospheric pressure. The use of pressurized nitrogen gas provides a dry environment to the sensors and increases the safety margin with regards to the OVT's H V withstand.  6.5.2  Tests  A l l the OVTs were subjected to standard IEC linearity and withstand tests [45] and passed them successfully. In addition to those tests, aluminum foil tests, fog-pollution tests, and  128 ice tests were also conducted to verify the effectiveness of the resistive shielding and the quadrature method.  6.5.2.1 Temperature Rise due to Resistors  A l l of the OVTs experienced a temperature rise caused by the heat dissipation of the internal resistors. Using the shift in D C bias of the B G O P C field sensors, it was found that the sensors themselves rose in temperature by up to 35° C when the OVTs were operating at rated voltage and with an ambient temperature of 20° C. The temperature rise in each sensor depended on its position in the O V T and on the resistance and geometry of the O V T itself. The OVTs' good electrical insulation tends to also serve as good heat insulation causing the significant temperature rise internally. Since the OVTs are required to operate in a wide range of temperatures (-50° C to +50° C), this is an issue of considerable importance, and work is currently in progress to deal with it. Solutions may include improving the internal-to-external heat transfer by some means, such as better heat sinking and internal ventilation, and/or increasing the number of sensors and reducing the levels of shielding to reduce the heat dissipation but maintain accuracy performance.  6.5.2.2 Aluminum-Foil Test Results  Top, middle, and bottom aluminum-foil drybanding tests, for which a dryband is located at the top, middle, and bottom of the insulator, respectively, were performed (see Section 6.4.2.2.1). The drybands used here have heights that are approximately one quarter of the height of the respective insulator. It was found that at heights less than this, flashover across the dryband is very likely to occur at rated voltage. Accuracies for these three cases and for all  129 the OVTs are given in Table 6.12. Table 6.13 shows the relative changes in the electric field at the sensor locations for the dryband that caused the greatest error for each OVT.  Table 6.12. Magnitude and phase errors due to quarter-height dryband using aluminum foil of (a) the three-stage 138 kV OVT, (b) the single-stage 138 kV O V T , and (c) the 345 k V OVT. (a) Dryband (30 cm) Top Middle Bottom  Magnitude Error (%) 0.12 0.18 0.12  Phase Error (minutes) 1.9 0.5 1.5  (b) Dryband (30 cm) Top Middle Bottom  Magnitude Error (%) 0.32 0.14 -0.13  Phase Error (minutes) -4.1 7.5 2.8  (c) Dryband (60 cm) Top Middle Bottom  Phase Error (minutes) 14.6 2.1 11.5  Magnitude Error (%) 0.09 0.07 0.27  Table 6.13. Relative magnitude and phase changes in the electric field at the sensor locations. Dryband (60 cm) Three-Stage 138 kV OVT with Middle Dryband Single-Stage 138 kV OVT with Top Dryband Three-Stage 345 kV OVT with Bottom Dryband  Top Magnitude Change (%)  Middle Magnitude Change (%)  Bottom Magnitude Change (%)  Top Magnitude Change (minutes)  Middle Magnitude Change (minutes)  Bottom Magnitude Change (minutes)  -2.15  2.84  -0.22  -307.2  266.4  40.8  6.30  -0.60  -1.51  513.6  -52.3  -449.5  -9.52  -0.34  31.03  -1838.0  -211.4  1579.4  130 6.5.2.3 Pollution Test Results  The three-stage 138 kV O V T and the three-stage 345 kV O V T underwent fog-pollution tests (see Section 6.4.2.2.2). Fog-pollution testing consists of applying a salt-clay-water mixture to the insulator surface, allowing it to dry, and then subjecting the O V T to artificial fog conditions while measuring its accuracy at rated line-to-ground voltage (80.5 k V for the 138 kV and 207 kV for the 345 k V OVT). As the polluted surface becomes wet, the salt dissolves and the result is an uneven, dynamic conductive (pollution-like) substance. The electric field in its vicinity is significantly and unpredictably distorted, which is a source of error in the OVT's voltage determination. These kinds of local field disturbances are considered to be some of the worst, practical perturbations that an O V T can experience during its operation in a substation. It should again be noted that the inherent hydrophobic properties of the silicone rubber shedding must first be temporarily destroyed using alcohol in order so that the salt-clay mixture, when applied, can adhere to the surface while drying. Table 6.14 gives the errors in the 138 kV OVT's measurements during a fog-pollution test. Table 6.15 gives the errors in the 345 k V OVT's measurements before, during, and after a fog-pollution test. Some manual temperature compensation was used to give the corrected values in the tables. This manual correction is based on the characterization of each OVT's drift in ratio and phase versus time while the rated line-to-ground voltage is applied. The drift is caused by heat dissipation of the resistor internal to the O V T and a resulting temperature rise in the sensors. It should be noted that this compensation is less than ideal, and some of the error appearing in the fog-on pollution values may be caused by heating or cooling as opposed to the pollution itself. Also, some of the error may be attributed to the linearity error in going from 240 k V to 70 k V during the test. Furthermore, the overall accuracy of the testing instrumentation that was used is approximately 0.05% and 3 minutes.  131 These results indicate that, even under pollution and deliberately destroyed hydrophobic properties of the silicone sheds, the resistively shielded OVTs can meet fEC 0.2% accuracy class requirements.  Table 6.14. Magnitude and phase errors due to pollution on the shed surface of the three-stage 138 kV OVT. Time fog on (minutes)  Magnitude error (%)  Phase error (minutes)  0 5 10 15 20 30 40 50 60 70 80 86  -0.11 -0.11 -0.09 -0.04 -0.02 -0.02 -0.02 -0.04 -0.04 -0.05 -0.07 -0.09  -6.6 -6.0 -5.4 -4.2 -3.0 -1.2 0 0.6 1.2 1.2 1.2 1.2  Table 6.15. Magnitude and phase errors due to pollution on the shed surface of the 345 k V OVT. Perturbation Fresh Pollution (70 kV) Dried Pollution Fog on - Pollution (0 min) Fog on - Pollution (14 min) Fog on - Pollution (22 min, 138 kV) Fog on - Pollution (25 min, 138 kV) Fog on - Pollution (65 min, 138 kV) Fog on - Pollution (69 min, 240 kV) After Fog-Pollution Test  Magnitude Error (%) 0.06 0.03 0.14 0.22 0.21 0.16 0.04 0.04 0.03  Phase Error (minutes) 1.7 3.3 6.4 7.3 14.1 14.9 12.9 8.7 1.1  132 6.5.2.4 Ice Test Results  Electric field disturbances due to melting ice on a polluted insulator can also result in severe field perturbations and may occur in substations. To simulate this condition, the threestage 138 kV O V T was rinsed with water (some pollution still remaining on the insulators and the hydrophobicity of the silicone sheds still absent) and placed inside an environmental chamber. Then, the temperature was reduced to approximately -20° C overnight. On the morning after, the insulator was covered with ice over a 3-hour period by spraying the insulator with freezing tap water (one layer sprayed and built every 10-15 minutes). Fig. 6.25 shows the insulator covered with ice.  Fig. 6.25. O V T covered with ice at -18° C.  133 Next, the chamber temperature was increased to above zero, allowing the ice to melt over a 3-hour period. Fig. 6.26 shows the results of the test as a function of time. Due to a lack of availability of a temperature- and condensation-compliant H V bushing for the environmental chamber, only about 30 k V was applied to the O V T during this test. This lower voltage may create a worse scenario (as concerns the accuracy of the OVT) as compared to that when the rated voltage is applied because a lower voltage may allow for the formation of smaller drybands and, therefore, relatively more severe localized field disturbances. As shown in Fig. 6.26, the O V T stays within a 0.3% accuracy band. As expected, the field disturbances at the locations of the optical electric field sensors are significantly moderated, but not eliminated, through the use of the resistive shield. Fig. 6.27 shows the changes in the field at the sensor locations. The largest changes, or perturbations, occur while the ice is melting since this is when conductive regions form on the insulator. 0.4 0.3 A  •0.2 A | » Ice Melting 0.3 0  20  40  60  80  100  time (minutes) (a)  120  140  160  180  time (tributes) (a)  135  15  10  H  Top Sensor Middle Sensor Bottom Sensor  CD CD U,  00  cu  T3 CO  OS  CL  _2 "cu  _c  CU  00  -5  -10  A Ice Melting  -15 0  20  40  60  80  100  120  140  160  180  time (minutes) (b) Fig. 6.27. Field changes at sensor locations under melting ice: a) magnitude and b) phase displacement.  6.5.3  Conclusion  The resistively shielded OVTs presented here demonstrate excellent linearity and accuracy suitable for stringent revenue metering applications. They maintain high accuracy even in the presence of sources of extreme field perturbations, such as pollution and melting ice, while offering a safe, environmentally friendly, insulation system. They provide all the advantages of optical technology including wide bandwidth and galvanic isolation of high-voltage and ground conductors.  136 6.6  Summary  The development of OVTs based on the quadrature method and permittivity-shielding has been described. The first O V T developed, the IOVT, used the quadrature method and three IOPCs mounted inside an off-the-shelf insulator and demonstrated the quadrature method's effectiveness at dealing with stray field effects. This stage in development was followed by an O V T that instead uses three BGOPCs and that meets IEC 0.2% class accuracy requirements even in the presence of substation-like changes in external geometry. Subsequently, the severe stray field effects caused by pollution deposited on the insulator surface were investigated. As a result, permittivity-shielding in the form of off-the-shelf, thin, hollow resistors with the sensors mounted inside of them was used to construct OVTs that would remain accurate under severe stray field conditions.  137  Chapter 7 Summary and Conclusion  There is a growing need in the power industry, in part due to the industry's looming widespread deregulation, for cheaper, safer, smaller, and more flexible instrumentation systems than those which currently exist for the observation and control of electrical power. Optical voltage sensor technologies promise such benefits and are aimed at replacing the conventional capacitive voltage transformers and inductive voltage transformers. Existing OVTs offer the accuracies needed for power revenue metering and the wide bandwidths needed for protection relaying and power quality monitoring, and they are relatively compact, cheap, and lightweight. They also offer immunity to electromagnetic interference, isolation of the observer from the high-voltage environment, and sophisticated interfacibility when used with fiber-optic transmission to and from the sensor heads. Prior to the work presented in this thesis, all OVTs, as well as conventional voltage instrument transformers, existing in industry had one aspect of their design in common, and that is that high-voltage and grounded electrodes are in close proximity with one another. In OVTs, this was necessary for one of two basic reasons. One is to offer proper metallic shielding to one or more optical field sensors from external changes in geometry to give a sufficiently fixed relationship between the field measurements and the applied voltage difference. The other is to force the entire voltage across an optical field sensor (single or distributed), which is practically limited in size, in order to obtain a measure of the integral of the electric field from one electrode to the other, which is a measure of the voltage. This aspect of their design results in the need for special insulation systems and/or specially designed internal electrode structures. Special insulation systems typically involve using environmentally unfriendly materials, such as SF6 and  138 oil, while specially designed internal electrode structures add complexity and overall cost in their designs. Using extensive finite element modeling, the quadrature method and permittivityshielding have been developed and have been presented here. The quadrature method is a technique that determines the necessary number of sensors that make point-like measurements of the field and the best positioning and weighting of those sensors for any chosen electrode geometry, for a given worst-case field perturbation due to external changes in the field (stray field effects), and for a particular minimum accuracy requirement. The weighting and the summing of the sensor measurements based on the quadrature method is effectively an efficient numerical integration of the field giving the voltage between the electrodes. This numerical integration is optimal in the sense that it minimizes the error in voltage measurement with respect to field perturbations that can be represented by a set of functions. Specifically, these functions are the products of the unperturbed field and polynomials of order 2N — 1 or less. As a result, electrodes can be positioned much farther apart for a given number of sensors than was previously known to be possible. This, in turn, results in lower field stresses in and around the electrodes, which allows for less extreme insulation requirements. Permittivity-shielding is a technique that offers an alternative to metallic shielding for the purpose of isolating electric field measurements from stray field effects. Materials with high dielectric constants, high-enough conductivities, or both supply the shielding by filling a space spanning from one electrode to the other and shield the field inside and nearby that space. While materials with high-enough dielectric constants for shielding purposes are not currently available, it was shown that existing materials that are considered to be highly resistive can be used to supply ample shielding. Resistive shielding is limited only by the amount of heat dissipation that can be tolerated in the OVT. With permittivity-shielding, electrodes are not  139 required to be in close proximity with one another, avoiding the need for any special insulation. This kind of shielding also has the added benefit in that it moderates the field between the electrodes, removing zones of extremely high field stresses that are typically the source of flashover. With respect to the quadrature method, the use of permittivity-shielding has the effect of smoothing perturbations caused by external changes in geometry. As a result, more shielding reduces the number of required sensors, and, conversely, more sensors reduce the required amount of shielding to maintain a particular accuracy in the presence of stray fields. 230 k V OVTs basically consisting of and off-the-shelf standoff insulator and using three field sensors according to the quadrature method were constructed and tested. They meet the highest IEC and ANSI/IEEE accuracy requirements for instrument transformers, i.e., ±0.2% magnitude error and ±10 minutes phase error and ±0.3% magnitude error and ±8 minutes phase error, respectively, even in the presence of substation-like changes in local geometry, such as the movement or installation of neighboring equipment. Following this, 138 kV and 345 k V OVTs that use off-the-shelf resistors to supply the resistive shielding and that use three sensors according to the quadrature method were constructed and tested. These OVTs also met the highest requirements set forth by EEC and IEEE/ANSI standards for instrument transformers. However, the use of shielding enables these OVTs to maintain accuracy even in the presence of severe stray field effects, which are caused by the mixture of moisture and pollution on an OVT's surface. The OVTs presented here offer all the inherent advantages of optical sensor technologies but are simpler in design than other types of OVTs. The bulk of their structure is off-the-shelf, i.e., the insulator and the resistors. Any electric field sensors that can measure a component of the electric field at nearly a point and that are small enough to be relatively non-intrusive can be used. Electrode structures are simple and are located internally near the ends of the insulator.  140 This provides a safe separation between high-voltage and ground with no need for special insulation. As a result, the OVTs presented here have improved safety, robustness, and environmental friendliness over existing OVTs. The quadrature method and permittivity shielding, as well as OVTs operating on these principles, have been patented [51].  141  Bibliography  [I]  J. Rungis, "Investigation of capacitor voltage transformers," 1990 IEEE International Symposium on Electrical Insulation, Toronto, Canada, lune 1990, pp. 388-391.  [2]  A . Bergman, "In-situ calibration of voltage transformers on the Swedish national grid," Metering and Tariffs for Energy Supply, Conference Publication no. 426, July 1996, pp. 227-239.  [3]  J. F. Nye, Physical Properties of Crystals, New York, N Y : Oxford University Press, 1985.  [4]  K. M . Bohnert and J. Nehring, "Fiber-optic Sensing of electric field components," Applied Optics, Vol. 27, No. 23, pp. 4814-4818, December 1988.  [5]  K. Bohnert and J. Nehring, "Fiber-optic sensing of voltages by line integration of the electric field," Optics Letters, Vol. 14, No. 5, pp. 290-292, March 1989.  [6]  K. Bohnert, J. Kostovic, and P. Pequignot, "Fiber optic voltage sensor for 420 kV electric power systems," Optical Engineering, vol. 39, no. 11, November 2000, pp. 3060-3067.  [7]  J. A. Kong, Electromagnetic Wave Theory, New York, N Y : John Wiley and Sons, 2nd Ed. 1990.  [8]  N. A . F. Jaeger and F. Rahmatian, "Integrated optics pockels cell high-voltage sensor," IEEE Transactions on Power Delivery, vol. 10, no. 1, January 1995, pp. 127-134.  [9]  T. Sawa, K. Kurosawa, T. Kaminishi, and T. Yokota, "Development of optical instrument transformers," IEEE Transactions on Power Delivery, vol. 5, no. 2, April 1990, pp. 884891.  [10]  L. H. Christensen, "Design, construction, and test of a passive optical prototype high voltage instrument transformer," IEEE Transactions on Power Delivery, vol. 10, no. 3, July 1995, pp.1332-1337.  [II]  A. Bosco, T. Hertig, and A. Kaczkowski, "Measuring device for a metal-enclosed, gasinsulated high-voltage installation," US Patent 5,917,316, June 29, 1999.  [12]  R. Baumgartner, K. Y . Haffner, H. Hageli, and A. Kaczkowski, "Current and voltage transformer for a metal encapsulated, gas-insulated high-voltage installation," US Patent 5,272,460, December 21,1993.  [13]  L. G. Christophorou and L. A. Pinnaduwage, "Basic Physics of Gaseous Dielectrics," IEEE Transactions on Electrical Insulation, Vol. 25, No. 1, pp. 55-74, February 1990.  142 [14]  S. V . Marshall and G. G. Skitek, Electromagnetic Concepts and Applications, 3 Ed., Englewood Cliffs, NJ: Prentice-Hall, 1990.  [15]  S. Weikel and G. Stranovsky , "Application of an electro optic voltage transducer at 345 kV," EPRI Optical Sensors for Utility T & D Applications Workshop, Portland, Oregon, July 20-21, 1995.  [16]  C. P. Yakymyshyn, M . Brubaker, P. Johnston, and C. Reinhold, "Manufacturing challenges of optical current and voltage sensors for utility applications," SPIE Conference on Sensors and Controls for Advanced Manufacturing, October 14-17, 1997.  [17]  J. C. Santos, M . C. Taplamacioglu, and K. Hidaka, "Pockels high-voltage measurement system," IEEE Transactions on Power Delivery, Vol. 15, No. 1, January 2000, pp. 8-13.  [18]  N. Alame and S. Shihab, " A model for calculating the field distribution of polluted post insulators during flashovers," Proceedings of IEEE Conference on Electrical Insulation and Dielectric Phenomena, pp. 817-828, October 1992.  [19]  A . Skopec, J. G. Wankowicz, B. Sikorski, "Electric field calculation for an axiallysymmetric insulator with surface contamination," IEEE Transactions on Dielectrics and Electrical Insulation, Vol. 1, No. 2, pp. 332-339, April 1994.  [20]  H. El-Kishky and R. S. Gorur, "Electric field computation on an insulating surface with discrete water droplets," IEEE Transactions on Dielectrics and Electrical Insulation, Vol. 3, No. 3, pp. 450-456, June 1996.  [21]  H. El-Kishky and R. S. Gorur, "Electric field and energy computation on wet insulating surfaces," IEEE Transactions on Dielectrics and Electrical Insulation, Vol. 3, No. 4, pp. 587-593, August 1996.  [22]  J. Weifang, W. Huiming, and E. Kuffel, "Application of the modified surface charge simulation method for solving axial symmetric electrostatic problems with floating electrodes," Proceedings of the 4 International Conference on Properties and Applications of Dielectric Materials, Paper 4208, pp. 28-30, July 3-8, 1994.  rd  th  [23]  J. L. Rasolonjanahary, L. Kraehenbuehl, and A. Nicolas, "Computation of electric fields and potential on polluted insulators using a boundary element method," IEEE Transactions on Magnetics, Vol. 28, No. 2, pp. 1473-1476, March 1992.  [24]  A. S. A. Farag, F. M . Zedan, and T. C. Cheng, "Analytical studies of H V insulators in saudi arabia," IEEE Transactions on Electrical Insulation, Vol. 28, No. 3, pp. 379-391, June 1993.  [25]  U. van Rienen, M . Clemens, and T. Weiland, "Simulation of low-frequency fields on high-voltage insulators with light contaminations," IEEE Transactions on Magnetics, Vol. 32, No. 3, pp. 816-819, May 1996.  143 [26]  B. H. McDonald and A . Wexler, "Finite-element solution of unbounded field problems," IEEE Transactions on Microwave Theory and Techniques, Vol. 20, No. 12, pp. 841-847, December 1972.  [27]  O. C. Zienkiewicz and D. V. Phillips, " A n automatic mesh generation scheme for plane and curved surfaces by 'isoparametric' co-ordinates," International Journal for Numerical Methods in Engineering. Vol. 3, pp. 519-528, 1971.  [28]  O. C. Zienkiewicz, The Finite Element Method, London: McGraw-Hill, 1977.  [29]  G. R. Buchanan, Schaum's Outline Theory and Problems of Finite Element Analysis, New York:McGraw-Hill, 1995.  [30]  D. S. Burnett, Finite Element Analysis, Reading, Massachusetts: Addison-Wesley, 1987.  [31]  P. Bettess, "Infinite elements", International Journal for Numerical Methods in Engineering, Vol. 11, pp. 53-64, 1977.  [32]  P. P. Silvester. D. A. Lowther, C. J. Carpenter, and E. A . Wyatt, "Exterior finite elements for 2-dimensional field problems with open boundaries," Proceedings of the IEEE, Vol. 124, No. 12.  [33]  C. W. Steele, Numerical Computation of Electric and Magnetic Fields, New York, N Y : Chapman and Hall, 2nd. Ed., 1997.  [34]  R. A. Adams, Single-Variable Calculus, Don Mills, O N : Addison-Wesley Publishers, 1986, pp. 289-300, 504-505.  [35]  C. E. Froberg, Introduction to Numerical Analysis, Reading, M A : Addison-Wesley, pp. 146-148, 183-184,187-194, 1965.  [36]  M . Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Dover Publications, 1977, pp. 172-201.  [37]  G. E. Howard, J. J. Yang, and Y . L. Chow, " A multipipe model of general strip transmission lines for rapid convergence of integral equation singularities," IEEE Trans, on Microwave Theory and Techniques, vol. 40, no. 4, April 1992, pp. 628-636.  [38]  N. A. F. Jaeger and B. P. C. Tsou, "Calculation of the fundamental mode sizes in optical channel waveguides using Gaussian quadrature," IEEE Trans, on Microwave Theory and Techniques, vol. 41, no. 11, November 1993, pp. 1907-1912.  [39]  R. W. Hamming, Numerical Methods for Scientists and Engineers, New York, N Y : McGraw-Hill, 1962, pp. 130-132.  [40]  International Standard IEC 60071-2 (1996), "Insulation Co-ordination - Part 2 Application Guide," International Electrotechnical Commission (TEC), Geneva, Switzerland.  144 [41]  C. Brackeniers and A. Cheenne-Astorino, " A new family of M V terminations," Third International Conference on Power Cables and Accessories 10kV-500kV, 1993, pp. 135139.  [42]  S. V. Nikolajevic, N. M . Pekaric-Nadj, and R. M . Dimitrijevic, "Optimization of cable terminations," IEEE Transactions on Power Delivery, vol. 12, no. 2, April 1997, pp. 527532.  [43]  J. Pingkai, X . Chuanxiang, L. Fuyi, and W. Shoutai, "Study of dielectric properties of a new high permittivity material," Proceedings of Electrical/Electronics Insulation Conference, 1995.  [44]  N. A. F. Jaeger and F. Rahmatian , "Integrated electro-optic high-voltage sensors," EPRI Optical Sensors for Utility T & D Applications Workshop, Portland, Oregon, July 20-21, 1995.  [45]  International Standard IEC 60044-7 FDIS "Instrument transformers - Part 7: Electronic voltage transformers," International Electrotechnical Commission (IEC), Geneva, Switzerland.  [46]  International Standard DEC 60044-2 (1997), "Instrument Transformers - Part 2: Inductive voltage transformers," International Electrotechnical Commission (DEC), Geneva, Switzerland.  [47]  DEEE Standard C57.13-1993, "DEEE Standard Requirements for Instrument Transformers," 1993.  [48]  International Standard DEC 60060-1, "High-voltage test techniques - Part 1: General definitions and test requirements," International Electrotechnical Commission (DEC), Geneva, Switzerland.  [49]  DEEE Standard 4, "DEEE Standard Techniques for High-Voltage Testing," 1993.  [50]  International Standard DEC 60507 (1991), "Artificial pollution tests on high-voltage insulators to be used on A.C. systems," International Electrotechnical Commission (DEC), Geneva, Switzerland.  [51]  N. A. F. Jaeger, F. Rahmatian, P. P. Chavez, S. E. Cherukupalli, and G. S. Polovick, "Method and apparatus for measuring voltage using electric field sensors," US Patent 6,252,388, June 26, 2001.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0065370/manifest

Comment

Related Items