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Spatial scales of geomagnetically induced currents in B.C. Hydro's power transmission system Butler, David Buchanan 1990

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Spatial Scales of Geomagnetically Induced Currents in B. G. Hydro's Power Transmission System By David Buchanan Butler B.Sc, Queen's University, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF MASTER OF SCIENCE in T H E FACULTY OF GRADUATE STUDIES (Department of Geophysics and Astronomy) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1990 © David Buchanan Butler, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada Date fO. DE-6 (2/88) Abstract Geomagnetically Induced Currents (GIC's) in B. C. Hydro's 500 kV trans-mission system have in the past been responsible for the generation of harmonics of 60 Hz, system voltage drops, and misoperation of relay units. Characterization of the associated magnetic storms' spatial scales would further the understanding of GIC generation in the area, and allow advanced warning of potential problems in future power transmission projects. Data collected in 1984 at four substations were analysed to determine lateral variations in geomagnetically induced earth surface electric fields. Inversion techniques were employed to find a variety of solutions that would reproduce the data. Results suggested that the magnetic storms were larger than the area moni-tored, and that resultant electric fields seen by a large portion of the transmission grid were uniform. Departures from this uniformity in other portions of the elec-tric field models were felt either to be due to earth induction effects, or in some cases, to be artifacts of the data analysis techniques. An experiment designed to determine the controlling factors behind GIC's is oudined. Considerable effort would be required to explicidy determine all electric fields affecting the transmission system ii Table of Contents Abstract ii List of Figures v Acknowledgements xiii Chapter 1 A Review of the GIC Project 1 The GIC Phenomenon 1 Long Term Goals 7 Overview of Previous Work 7 Initial Goals of the Thesis 10 Chapter 2 Data and Initial Processing 11 Results and Discussion 17 Chapter 3 A Critical Evaluation of the Experiment 20 Section 1 Resolution of the Experiment 23 Discussion 38 Section 2 Smallest Model Inversion 38 Results and Discussion 42 Section 3 Smallest Deviatoric Model 44 Results 47 Discussion 55 Chapter 4 A Redesigned Experiment 76 Placement of HET's 77 Summary 82 iii Conclusions 86 List of References 88 Appendix A Monitored Transformer Percentages 90 Appendix B The Least-Squares Uniform Electric Field 93 Appendix C Handling of Errors in Inversion 96 Appendix D Computer Code 103 Appendix E Data 104 Appendix F Smallest Deviatoric Inversion Results 119 iv List of Figures 1.1: Deviation of earth's magnetic field from dipolar configuration, due to so-lar wind. The solar wind speed is, in the presence of the earth's mag-netic field, supersonic. Therefore a shock front exists solarward of the earth. Cusps exist where lines of force are swept aft, deep into the tail. Solar particles are thought to enter at these cusps, and into the neutral sheet portion of the tail, a planar surface separating lines of force directed towards or away from the earth. (After Merrill and McElhinny, 1983.) 2 1.2: Potential Differences at the earth's surface cause electric currents in three-phase power transmission lines that are grounded at two or more points by wye-connected transformers. (After Albertson and Van Baelen, 1970.) 5 1.3: GIC's bias the core of a transformer in one direction, so that dur-ing alternate half-cycles of the applied 60 Hz voltage, the exciting current reaches very high peaks as determined by the saturated por-tion of the curve. The dashed curve shows the excitation current during normal operation. (After Kappenman and Albertson, 1990.) 6 1.4: B.C. Hydro 500 kV system. Series capacitors are installed in the lines from the Gordon M Shrum generating station to the Ingledow substation. They are shown by the bars perpendicular to the power lines. (After Lundby et al, 1985.) 9 V Fig 2.1: Major B.C. Hydro transmission lines. GIC's on the 500 kV transmission lines in central B.C. are isolated from the rest of the province by capacitors at McLeese (MLS) and Kennedy (KDY) substations. The 230 kV lines were presumed to be of comparatively high resistance, and were therefore neglected in the analysis 12 Fig 2.2: During 1984, 138 kV transmission lines in the area ended in delta-wound transformers. The lower voltage lines are typically wye-connected, but no GIC path exists through the transformer 14 Fig 2.3: D.C. approximation of power transmission system. Knowing the currents grounded at the four substations, one can immediately calculate the earth surface potentials 15 Fig 2.4: Electric field results using model in figure 2.3. The solid, dotted, and dashed curves represent E i , E2 , and E3, respectively. The long dashed curve repre-sents both the fields from Williston to Gordon M . Shrum and from Williston to Kelly Lake (this curve was not considered correct, but had no effect on the calculation of the other three). Notice the very strong fields east of Telkwa that are not seen to the west 18 Fig 3.1: Stations north of Kelly Lake that were capable of grounding GIC's. Capacitors were shunted at Kennedy and McLeese 21 Fig 3.2: Electric fields responsible for the GIC's measured at the four sta-tions. Lines 5L2, 5L12, 5L61, 5L62 and 5L63 are 500 kV lines; 2L96, 2L97, 2L99, 2L101 and 2L353 are 230 kV lines. 22 Fig 3.3: A completely determined problem uniquely links one point in data space with vi one point in model space (part a). An underdetermined problem (not enough data) maps a single point in data space to a region in model space, as the path is not fully known. Resolution is a measure of how well the linkage can be narrowed, and how small the range of solutions can be. One attempts to narrow the path linkage and minimize the range in model space wherein the true solution lies 27 Fig 3.4: A perfect resolution matrix. Y-axis values show which model parameter is being resolved by each vector. Vertical lines represent the magnitude of each element in the vector (90% of the distance between horizontal lines corresponds to magnitude 1.0). This function would perfectly resolve the true model, as only the main diagonal has non-zero elements. Therefore (m) would equal m 33 Fig 3.5: Resolution of the ten electric fields from figure 3.2 using only data from Skeena, Telkwa, Glenannan, and Williston. The unimodular constraint was not used. E3, E 4 , and E 6 are the best resolved, but are smeared with noise from neighbouring fields ; 34 Fig 3.6: The best resolution possible if the sum of the elements of each vector equals unity. The extra constraint disfigures the function, adding strong off-diagonal components 36 Fig 3.7: The resolution matrix if the Backus-Gilbert spread condition is used. Again, strong off-diagonal components appear. The unimodular delta condition and the spread condition are therefore not considered to give appropriate resolution functions 37 Fig 3.8: Results from the smallest model inversion using data from event 3. Sim-ply trying to minimize field amplitudes results in field polarities being Vll split at Telkwa (for the east-west portion of the grid) and at Williston (for the north-south portion). Notice that E i , E2 , and E3 oppose E4, E5, and E6, and that E7 and Es oppose E9 and E i 0 . This type of result led to the conclusion that the smallest model was inappropriate. 43 Figures 3.9, 3.10, 3.13, 3.14, 3.17, and 3.18: Smallest deviatoric inversion results. These are the closest to uniform results that fit the data. Part a is the northerly component of E 4 , and b, c, and d are the easterly components of E 3 , E4, and E6. The solid curves are the solutions, and the dashed curves represent the corresponding error bounds. Note that these do not represent the total possible range of electric fields that fit the data; rather, the errors show the range possible for results of this inversion — the smallest deviation from a uniform field. The improvement over the smallest model results is that the field directions are not always split at Telkwa and Williston. Figures 3.11, 3.12, 3.15, 3.16, and 3.19: Pictorial representations (without errors) of the strength and direction of the total fields E i , E 3 , E 4 , E 5 , and E6. The x-axis represents time, with the base of each arrow showing the time of occurrence for that particular field orientation. The y-axis shows distance east of Prince Rupert; the y-coordinate of each trace distance east from Prince Rupert to the centre of the power line concerned. North is to the left, parallel to the x-axis. The length of each vector gives the field strength, using the provided scales. Fig 3.9: Event 1 49 Fig 3.10: Event 7 51 Fig 3.11: Event 5 smallest electric fields 53 viii Fig 3.12: Event 5 smallest deviatoric electric fields 54 Fig 3.13: Event 9 5 Fig 3.13.2:Micropulsation seen at beginning of event 3. This short period fluctuation differs from the rest of in that E3 is stronger than E 4 and E6. The data have been subjected to a trapezoidal bandpass frequecy filter (corners at 0.01, 0.012, 0.054, 0.056 Hz) 58 Fig 3.14: Event 10 .' 60 Fig 3.15: Event 2 64 Fig 3.16: Event 4 65 Fig 3.17: Event 2 67 Fig 3.18: Event 4 69 Fig 3.19: Event 8 71 Fig 3.20: Auroral electrojets that form during the growth and expansion phases of a magnetic storm. The vortices have opposing senses of rotation, forming a zone of confusion where they meet between 9:00 and 10:00 pm local time. (After Iijima and Nagata, 1972.) 73 Fig 3.21: Schematic representation of conductive structure (named the Canadian Cordilleran Regional conductor) in the Intermontane and Omenica tectonic provinces, with locations of power lines superimposed. Depth to the conduc-tor's upper surface is estimated at 15 to 30 km, and its depth extent is believed to be 30 to 70 km. The conductor may be the cause of the reduced amplitude of E 6 with respect to E 4 . (After Gough, 1986.) 75 Fig 4.1: Possible GIC paths. A 138 kV system, installed from Skeena (SKA) to Stewart (STW), has experienced GIC related problems from since 1984. . 79 ix Fig 4.2: Updated schematic of power grid; electric field and data subscripts were changed to account for effects of all stations. Notice that it is not necessary to monitor Williston to obtain correct values for E12. 80 Fig 4.3: The resolution matrix of the experiment if all 12 data are collected: all fields are perfectly determined 81 Fig 4.4: Resolution for a poor choice of placement of 11 HET's. Neglecting to monitor Kitimat (datum 6) smears images of E7, Es, E9, E10, and E12. 83 Fig 4.5: A good choice for placement of 11 HET's. Neglecting Falls River does little damage to resolution of electric fields 84 Fig 4.6: Resolution of 12 fields using nine instruments. The best place-ment would neglect Falls River, Green River, and Diana Lake. 85 Fig A . l : Operating one line diagram of Glenannan. The station is connected to Telkwa and Williston by lines 5L62 and 5L61, respectively. Transformers TI and T2 step the voltage down from 500 kV to 230 kV. Transformers T5 and TI 1 then reduce it to 138 kV. From there, the voltage is reduced to 60 kV by T3 and T4. All six transformers are wye connected, and can therefore ground GIC's, as can shunt reactors SRX5 and SRX5N 91 Fig A.2: The transformers and shunt reactors from the one line diagram are shown here by their d.c. resistances. The HET monitored TIL, which passed 40% of the total current grounded by the station 92 X Fig B . l : A uniform electric field would induce currents at Skeena, Telkwa, Glenannan and Williston, whose relative amplitudes are controlled only by the loop resistances in each cell 95 Fig E . l : Transformer current for event 1. Date of storm: 1984-07-14 105 Fig E.2: Transformer current for event 2. Date of storm: 1984—07-14 106 Fig E.3: Transformer current for event 3. Date of storm: 1984-07-15 107 Fig E.4: Transformer current for event 4. Date of storm: 1984—07-17 108 Fig E.5: Transformer current for event 5. Date of storm: 1984-07-17 109 Fig E.6: Transformer current for event 7. Date of storm: 1984-08-01 110 Fig E.7: Transformer current for event 8. Date of storm: 1984-08-01 I l l Fig E.8: Transformer current for event 9. Date of storm: 1984—08-02 112 Fig E.9: Transformer current for event 10. Date of storm: 1984-08-02 113 Fig E.10: Magnetic field variation at Smithers for event 1 114 Fig E . 11: Magnetic field variation at Smithers for event 2 114 Fig E.12: Magnetic field variation at Smithers for event 3 115 Fig E.13: Magnetic field variation at Smithers for event 4 115 Fig E.14: Magnetic field variation at Smithers for event 5 116 Fig E.15: Magnetic field variation at Smithers for event 7 116 Fig E. 16: Magnetic field variation at Smithers for event 8 117 Fig E.17: Magnetic field variation at Smithers for event 9 117 Fig E.18: Magnetic field variation at Smithers for event 10 118 Fig F . l : Smallest deviatoric fields for event 1 120 Fig F.2: Smallest deviatoric fields for event 2 122 Fig F.3: Smallest deviatoric fields for event 3 124 xi Fig F.4: Smallest deviatoric fields for event 4 126 Fig F.5: Smallest deviatoric fields for event 5 128 Fig F.6: Smallest deviatoric fields for event 7 130 Fig F.7: Smallest deviatoric fields for event 8 132 Fig F.8: Smallest deviatoric fields for event 9 134 Fig F.9: Smallest deviatoric fields for event 10 136 Fig F.10: Vector plot of smallest deviatoric fields from event 1 138 Fig F . l l : Vector plot of smallest deviatoric fields from event 2 139 Fig F.12: Vector plot of smallest deviatoric fields from event 3 140 Fig F.13: Vector plot of smallest deviatoric fields from event 4 141 Fig F.14: Vector plot of smallest deviatoric fields from event 5 142 Fig F.15: Vector plot of smallest deviatoric fields from event 7 143 Fig F.16: Vector plot of smallest deviatoric fields from event 8 144 Fig F.17: Vector plot of smallest deviatoric fields from event 9 145 Fig F.18: Vector plot of smallest deviatoric fields from event 10 146 xii Acknowledgements I would like to express my thanks to a number of people who have played integral parts in helping me complete this thesis. Dr. Tomiya Watanabe is first among these, as his advice, encouragement, and expertise proved invaluable throughout the course of this project. I would like to express my sincere thanks as well to Drs. Bob Ellis and Don Russell for their guidance in the completion of the work, and their much-appreciated comments on the manuscript. The students in the Department of Geophysics and Astronomy were essential for numerous discussions, insights, and the social relief so necessary during a long study. Most importantly, I would like to thank my wife Sandy Stewart for her constant support, and for her ability to put up with me, and often without me, over the entire two years. xiii Chapter 1 A Review of the GIG Project The first chapter serves as an introduction to the project, giving both the background to the field, and the history of the work done here at U.B.C. It has four main sections: an outline of the phenomenon of geomagnetically induced currents; a description of the long term goals of the Aeronomy group with respect to this problem; accomplishments here at U . B . C , and work done by other researchers; and a statement of the initial goals of this thesis. The GIC Phenomenon A geomagnetically induced current (GIC) in a power line is a result of the interaction of the earth's magnetic field with solar particles. The sun constantly emits in all directions an electrically neutral stream of charged particles, known as the solar wind, consisting mostly of equal numbers of protons and electrons. At the distance of the earth's orbit, the particles have speeds ranging from 300 km/s to 800 km/s, and the plasma density varies normally from three to eight particles per cubic metre. Such a conducting fluid has currents induced within it as it encounters the earth's magnetic field. Conversely, the solar wind is repelled by the Lorentz force that acts on the currents. The stream envelopes and contains the earth's field, so that on the windward side of the planet, the magnetic field lines are compressed by the plasma until the perpendicular magnetic field pressure balances the kinetic pressure of the incoming particles. In the lee, the field lines extend deep into space in a turbulent shadow. The result, shown in figure 1.1, is that the earth, with its field lines, can be viewed as a magnetic comet travelling through the solar plasma. 1 / / / , Plasma ' / / / shtel Fig 1.1: Deviation of earth's magnetic field from dipolar configuration, due to solar wind. The solar wind speed is, in the presence of the earth's magnetic field, supersonic. Therefore a shock front exists solarward of the earth. Cusps exist where lines of force are swept aft, deep into the tail. Solar particles are thought to enter at these cusps, and into the neutral sheet portion of the tail, a planar surface separating lines of force directed towards or away from the earth. (After Merrill and McElhinny, 1983.) 2 Any changes in the solar wind's particle density or velocity will force departures from the stable configuration of the magnetosphere — that portion of space containing the earth and its field. Strong departures are termed magnetic storms: rapid variations in the strength of the magnetic field that can be of the order of thousands of gammas at the earth's surface. In general, magnetic storms are of two types: sporadic and recurrent. Re-current storms are a result of mildly varying conditions on the sun's surface that are functions of solar longitude. Solar wind variations caused by such conditions are regularly seen by the earth, and have a period of one solar day (28 earth days). Sporadic magnetic storms occur following drastic temporal changes on the sun's surface. Solar wind flux is affected by solar flares, coronal holes, and disappearing filaments. If the flux increases, so does the kinetic pressure exerted on the magnetopause (the boundary of the magnetosphere), which contracts until the pressure is balanced as a result of an increased magnetic flux density in the interior. It is the sporadic changes, rather than the recurrent, that cause the largest magnetic storms. The magnetic variations are not solely a result of the changing shape of the magnetic field. During the storms, complex current systems termed auroral electrojets are induced in the ionosphere, and follow roughly circular paths around the magnetic poles, located approximately at an altitude of 100 km and a geomagnetic latitude of 65°. The morphology of the currents, which are estimated to be on the order of 106 A , is not precisely known (for a detailed discussion, see Rostoker (1972)), but it is obvious that such currents can have a significant effect on the magnetic field seen at the earth's surface. 3 A magnetic storm of any type will induce electric fields at the earth's surface. During storm conditions the fields are typically of the order of 0.1 V/km, but have been known to exceed 5 V/km. Any conductor that connects different points on the earth's surface will carry a geomagnetically induced current. In particular this is true of 3—phase power transmission lines that are grounded through the neutrals of wye-connected transformers (see figure 1.2). These very low resistance lines can connect surface points hundreds of kilometres apart, and so experience voltage drops of tens to hundreds of volts. Currents grounded through transformers in areas susceptible to storms can be in the hundreds of amperes. The GIC currents themselves cannot alone damage power transformers, but they are responsible for the currents that can. Figure 1.3 shows the non-linear transfer function that relates exciting current to applied voltage in a transformer. Under normal operating conditions, the exciting current and voltage remain in the central, almost linear, portion of the curve. GIC's in the windings bias the operating conditions in one direction (since the GIC frequency is nearly dc), and the system voltage variation that is still impressed upon the transformer draws an unusually large excitation current during one half of the ac cycle. The lower portion of figure 1.3 compares the excitation current drawn by the transformer during normal and GIC biased operations. Some of the resulting problems include a dangerous drop in transmission system voltage; the creation of harmonics of 60 Hz that may adversely affect relays; and the stressing of the transformer due to overheating, and due to the flow of magnetic flux through paths other than the steel core. Fig 1.2: Potential Differences at the earth's surface cause electric currents in three-phase power transmission lines that are grounded at two or more points by wye-connected transformers. (After Albertson and Van Baelen, 1970.) 5 Fig 1.3: GIC's bias the core of a transformer in one direction, so that during alternate half-cycles of the applied 60 Hz voltage, the exciting current reaches very high peaks as determined by the saturated portion of the curve. The dashed curve shows the excitation current during normal operation. (After Kappenman and ' Albertson, 1990.) 6 Long Term Goals Knowing that there are many different types of magnetic variations, one realizes that there is likely to be a similar variability in induced currents. The long term goals of the project are to understand the nature of the different forms of GIC's. This means characterizing spatial scales and occurrence frequencies for each type. The ultimate goal is to identify the magnetospheric sources for the currents, and to be able to predict the timing and size of future currents. Power engineers would then be able to use this information in system planning and in the design stages of transformer manufacturing. Overview of Previous Work Initial investigations of GIC's in power lines were initiated in the late 1960's by V.D. Albertson et al. (1973, 1974a, 1974b, 1979). They documented adverse effects due to geomagnetic activity, such as generation of harmonics, system voltage drops, and misoperation of relay units. Towards the project's long term goals, the UBC Aeronomy Group has been cooperating since 1979 with B.C. Hydro, studying GIC's in the 500 kV trans-mission system in central B.C. seen in figure 1.4. Boteler et al. (1982) studied the time spectral characteristics of currents measured at two 500 kV substations, Telkwa and Williston, and speculated on the spatial scale of the magnetosphere sources. They found that GIC amplitude varied directly with period. They com-pared the observed ratio of GIC amplitude vs surface electric field to the ratio that would be expected from a uniform electric field, and posed arguments to explain why there was not a good match. They pointed out that the spatial scale of the fields would explain the departures from uniformity seen in the observed 7 ratios: the ratio should be near zero for scales much smaller than the station sep-aration distance, as the sum of many opposing small scale fields would be near zero over long distances; should increase to a maximum corresponding to scale lengths equal to the separation distance and polarity changes occurring right at substation locations; and should decrease to a plateau for scales longer than the length of the system. Using the same data set, Lundby et al. (1985) developed two empirical formulae that allowed one to predict the rate of occurrence of GIC's for a given Kp — the planetary magnetic "storminess" index that indicates how disturbed the earth's magnetic field is during a three hour period. One formula describes the mean GIC amplitude (expressed as an hourly range value) as a simple exponential function of Kp. The other formula characterizes the scatter of individual points about the mean as a lognormal distribution. The two formulae are combined to give one describing the probability of occurrence of a GIC greater than a specified value, given a value of Kp. Boteler et al. (1989) used the historical time statistics of the Kp index, gathered since its definition by Battels (1949), to modify the Lundby et al. formula. The new formula gives the probability of occurrence over time of GIC's that exceed a specified value. Modifications to the formula were provided to account for the varying levels of geomagnetic activity during the 11 year sunspot cycle. Viljanen and Pirjola (1989) presented a method that estimates GIC's in a power grid given magnetometer readings. They calculated the electric field at the surface of a halfspace, assuming the source to be a spatially uniform time varying 8 Fig. 1.4: Northern B.C. Hydro 500 kV system. Series capacitors are installed in the lines from the Gordon M . Shram generating station to the Kelly Lake j substation. They are shown by the bars perpendicular to the power lines. (After Lundby et al, 1985.) 9 magnetic field. The earth's conductivity was considered uniform, but was allowed to vary with time as an increasing function of Kp. The non-physical conductivity function was justified based on Alberston and Van Baelen (1970), who showed that a plane wave current in the upper atmosphere produces the maximum possible surface electric field and a line current source produces the minimum possible surface electric field. Viljanen and Pirjola suggested the function of Kp was necessary to offset the overestimation of the electric field by using a plane wave model. The conductivity function was obtained empirically. Initial Goals of the Thesis At the outset, the goal of this thesis was to determine the spatial scale (the distance over which large changes occur) of the different types of GIC's seen west of Williston in the transmission network of figure 1.4. Knowing the scales would allow one to assess the probability of GIC's being greatly enhanced in the system due to the interaction of storm sizes with substation locations. 10 Chapter 2 Data and Initial Processing To quantify the spatial scale of the GIC disturbances, one must be able to track the storms throughout the grid, to detect and correlate onset times and compare amplitudes in each section. Monitoring modern transmission grids, however, would require a very large number of instruments, as a Hall effect transducer (HET) would be needed for each potential grounding point, and the B.C. grid currently has over two dozen such points in the 500 kV system. Most sites though are in areas not strongly affected by magnetic storms. The lines running from Skeena (SKA in figure 2.1) to Williston (WSN), are useful in that they are in an area that is prone to unusually high levels of magnetic activity, compared to the majority of North America: the stations lie between 58.5° and 59.5° geomagnetic latitude. As well, GIC's are blocked from entering the north-south lines by capacitor banks at McLeese (MLS) and Kennedy (KDY) substations. Therefore the 500 kV system is isolated at GIC frequencies from the majority of the province's transmission grid. It was initially believed that the 230 kV lines could be, to first order, neglected in the analysis; that GIC amplitudes would be significantly smaller than those on the 500 kV lines, as the 230 kV line resistances were believed to be much higher. (In fact, this was not the case, e.g., 0.064 ft/km versus 0.030 fi/km, but this problem will be discussed in chapter three.) The 138 kV lines could be disregarded since the substation transformers all had delta-connected 138 kV windings, which provided no grounding path (see figure 2.2). Therefore it was believed that there were only three lines of concern in the network, and that 11 Fig 2.1: Major B.C. Hydro transmission lines. GIC's on the 500 kV transmission lines in central B.C. are isolated from the rest of the province by capacitors at McLeese (MLS) and Kennedy (KDY) substations. The 230 kV lines were presumed to be of comparatively high resistance, and were therefore neglected in the analysis. 12 measuring GIC's at the four stations Skeena (SKA), Telkwa (TKW), Glenannan (GLN), and Williston (WSN) uniquely determines the electric fields. The power system can be treated, as shown in figure 2.3, as a dc network, since reactive effects at GIC frequencies are negligible. By measuring the currents grounded through one transformer in each substation, one can determine the total current in each station, the overhead line currents, and the corresponding electric fields. Such data were acquired in the summer of 1984, in the periods July 14th to 19th and July 31st to August 3rd. Currents grounded by one transformer in each substation were measured using split-core HET's wrapped around the neutral leads where they leave the transformers and run to the subsurface grounding grids. The data were recorded using F M tape units. As well, the east-west earth surface potential was recorded at Smithers, 40 km NW of Telkwa, as was the rate of change of magnetic field. Ten storms were selected from the records and were digitized at a sampling rate of 0.4 Hz. Plots of the current and magnetic data are shown in appendix E. The magnetic field plots were created by removing an average value from the output of the induction magnetometer, and then integrating that output using Simpson's rule. The easterly magnetic field was not measured. The electric fields were computed using the following scheme. The currents were first converted from currents through one transformer to currents through entire substations and were then resampled to common time points (still at 0.4 Hz), to account for timing offsets between the four clocks . The electric fields were then calculated direcdy using the dc network approximation. It was evident that during the recording period, there was always a GIC path 13 Fig. 2.2: During 1984, 138 k V transmission lines in the area ended in delta-wound transformers. The lower voltage lines are typically wye-connected, but no path exists through the transformer. 14 Initial Electric Field. Model: Fig. 2.3: Direct current approximation of power transmission system. Knowing the currents grounded at the four substations, one can immediately calculate the earth surface potentials. 15 north or south of Williston, since capacitors were shunted either at McLeese or Kennedy. In the cases where only one of a north or south path existed, the model became a four cell system. It was unclear how to treat the cases where both north and south paths existed; to begin with, an assumption was made that the north-south electric field was small compared to the east-west field, so that the two lines could be treated as resistors in parallel. This was known to be a source of error, but was not considered important, since the fields from Skeena to Williston were still explicitly determined. To convert from transformer currents to station currents, the percentage of total station current flowing through the measured transformer must be known. For each station, the electric network was redrawn by replacing each transformer by its dc resistance. From these networks, the current passing through the measured transformer, expressed as a percentage of the total current flowing to ground from the 500 kV node, was determined. Appendix A gives an example of how the percentages are determined. Determination of the percentages was hampered by a lack of information about the substations. The dc resistance of a transformer's high and low voltage windings depends on: the "tap" setting, the location on the high voltage winding at which the low voltage connection is made (these are autotransformers); the resistances corresponding to each tap; the temperatures at which the resistances are quoted; the temperature of the winding during the recording period; and the temperature coefficient of resistivity for the winding. Typically, of the five pieces of information needed, only two or three were available, since the data were either not recorded or not kept indefinitely. However, enough pieces were available so 16 that reasonable guesses could be made for most of the transformers, by using approaches such as using the temperature of one transformer and the temperature coefficient of a similar one found at another substation. Once the electric fields were determined, field signatures were to be identified and correlated across the grid. Initial plans were to produce static frequency spectra for the entire lifetime of each storm, to identify frequencies of interest, and to calculate dynamic frequency spectra so that the evolution of various components could be seen over time. The fields would then be filtered to remove all but the frequencies of interest, and the amplitudes and times of onset of the remaining traces compared to see if the extent of the source storm could be delineated. Results and Discussion The dc network approximation produced interesting results (see figure 2.4). The low frequency components east of Telkwa were markedly different from those to the west: eastward the components were an order of magnitude larger. This was not true of the higher frequencies: although not necessarily uniform, the values were comparable. The traces showed no obvious progressions of the storms in space. This led to two speculations: that the storm dimensions were larger than the size of the survey, and the subsurface conductivity structure was responsible for the variations in the strengths of the electric fields; or that the power line model in figure 2.3 is inadequate. The size of the low frequency amplitude discrepancies led to the validity of the model being questioned before the effects were ascribed entirely to deep earth structure. Thus a re-examination of the Hydro system was required. It was necessary to better address the effect of the shunted capacitors to the north and south, and 17 E 2 / A , 1 . i h3 1 I i '\ : . » - j / \ i , V t\ r A, \$ r. 1. , /J v l ty*™ E 1 f'> •.'.'v.. V pi1 few-, if-if »\. ,v V 8 8.5 Time in Hours UT Fig 2.4: Electric field results using model in figure 2.3. The solid, dotted, and dashed curves represent Ei , Ej, and E 3 , respectively. The long dashed curve represents both the fields from Williston to Gordon M. Shrum and from Williston to Kelly Lake (this curve was not considered correct, but had no effect on the calculation of the other three). Notice the very strong fields east of Telkwa that are not seen to the west. 18 to evaluate whether or not the 230 kV lines were of concern. V 19 Chapter 3 A Critical Evaluation of the Experiment To determine whether the 230 kV lines were of importance, it was necessary to verify which stations could ground dc currents. B.C. Hydro was asked for "one line operating diagrams" for all stations north of and including Kelly Lake (see figure 1.4). These diagrams display the winding characteristics of transformers, i.e., whether they are wye-connected and therefore capable of grounding dc currents. Figure 3.1 shows all stations containing such transformers. Obviously the task of calculating electric fields has become more complicated; of 21 potential grounding sites, only four were measured. Some of the 21 can be immediately neglected, because the electric fields between some of the stations in figure 3.1 do not affect currents at the measured stations. For example, the potential drop from Prince Rupert (RUP) to Skeena will affect currents measured at Skeena, whereas the drop from Prince Rupert to Falls River will not. Therefore there are 11 grounding points of concern (as shown in figure 3.2), and ten corresponding electric fields that control currents at the four measured sites. Five lines and six grounding points are associated with the 500 kV network, and the remaining five lines and five grounding points are on the 230 kV network. The 500 kV network was initially assumed to be the most important, carrying and grounding the majority of current. Of the 230 kV system, two or three of the lines — 2L99, 2L97, and 2L96 — were considered to be possibly insignificant, since they would not be much affected by east-west electric fields. The other two — 2L101 and 2L353 — would be affected. 20 Fort St.. John Soda Creek 500 kV lines 230 kV lines 60 kV lines Kelly Lake 100 Mile House Fig. 3.1: Stations North of Kelly Lake that were capable of grounding GIC's. Capacitors were shunted at Kennedy and McLeese. 21 Fig 3.2: Electric fields responsible for the GIC's measured at the four stations. Lines 5L2, 5L12, 5L61, 5L62 and 5L63 are 500 kV lines; 2L96, 2L97, 2L99, 2L101 and 2L353 are 230 kV lines. 22 The experiment monitored four of the six 500 kV grounding sites, and none of the others. If one of the monitored stations was at the very edge of the possible total network, then the current at that station would uniquely determine the electric field between it and the next station, since there would only be one possible route for the current to follow. But the four stations are in the interior of the network, so that current from one edge could theoretically travel right across the network to another edge, completely undetected. For example, current entering the system at Prince Rupert could travel all the way to Kelly Lake or Gordon M . Shrum without being seen by the four monitoring sites. If, however, the lower voltage lines are of a particularly high resistance, the collected data might be sufficient to determine the electric fields between Skeena and Williston. The usefulness of the data cannot be judged until the effects of the seven additional sites have been determined. Section 1 Resolution of the Experiment One could calculate the electric fields directly using the circuit model in figure 2.3, but those fields would be meaningless without associated error bars. The sizes of the errors depend most strongly on the assumptions implicit in the circuit model; altering the circuit would significandy alter the calculated fields. Since the true circuit was as in figure 3.2, quantifying the errors is not possible without knowledge of the currents at RUP (Prince Rupert), MIN, TAC, SVY, GMS, BLW, and K L Y . It is more instructive to assess how well one can distinguish the fields between Skeena, Telkwa, Glenannan, and Williston, using only the four data. To obtain a clear understanding of whether the unmonitored stations are really necessary to determine the solution, one must reformulate the problem. Consider 23 figure 3.2: it shows the ten power lines that affect the monitored stations, and the 11 grounding points. Each of the stations has been simplified from its component transformer resistances to one, two, or three equivalent resistances, depending on how the transformers are connected with the overhead power lines. A summary of these simplifications is in appendix A. In the cases of lines 2L96, 2L97, 2L99, 2L101, 2L353, 5L2, and 5L12, the line resistances have been incorporated into the grounding resistances of the stations. It is convenient to think of the system as a collection of ten cells: the electric fields between stations are referred to as E i through E 1 (), and the corresponding power lines as lines 1 through 10. The currents labelled Ii, I2, I3, and I4 refer to the currents flowing through the measured transformers at Skeena, Telkwa, Glenannan, and Williston, respectively. Each cell has a total loop resistance that is the sum of all resistances encountered by current flowing from ground at one station, along the power line, and back to ground at the next station. These loop resistances are referred to as Ri to Rio. As well, distances between stations are referred to as di to dir> Simple circuit theory states that the current being grounded through the measured transformer at Skeena is: TSKE is that percentage of total current grounded at Skeena that passes through the measured transformer. Similarly, the set of four currents can be represented 24 3. 1 or, in terms of loop resistances, 3. 2 in matrix notation as / = GE, 3. 3 where f = (I , , / 2 , I 3 , / 4 ) r -T*gF,d\ - T . s j f Rd2 TSKEda o o o o o o o "1 «2 "3 o o -TTF,r.d* ^rm.dt o o o o o o o o o - Tr^.ns<'< Taignd* TarhEf,d6 o o o o 0 0 0 0 0 ~Tw,J' Twu.2d7 Twu,*dn ~Twjr.T'fi -Twjr.ni\« . R & « Y H 8 H 9 RIQ The vector E contains the model parameters: 3.4 E = (E\,E2,Ez,E±,E$,E§,ET,E%,E$,Ei(i)r 3. 5 The row vectors of G can be used to obtain a measure of the resolution of the experiment (Backus and Gilbert, 1968). Resolution is a measure of how well one can determine and distinguish between model parameters: can one clearly and distincdy "see" the parameter, or is it smeared with noise from neighbouring model elements? If the image of a parameter is blurred with images of neighbouring parameters, is it centred on the desired parameter, or is it instead shifted to some other portion of the model? Therefore the question is whether a high quality resolution function can be developed to resolve each parameter of the model. A generalized notation will be used in the development, to avoid confusion between symbols for currents, electric fields, vectors, vector elements, and indices. The four data (the currents measured at the stations), and the model vector containing the ten electric fields, will be referred to as e = (ei,e2,e3,e4) 3. 6 —* T rh = (mi,m2,m3,m4,m5,m6,m7,m8,m9,mio) 25 respectively. The G matrix is treated as four row vectors, referred to as kernels: 91 (1) 91(2) 9 1 ( 3 ) 91(4) 91(5) 91(6) 9 l ( 7 ) 91(8) 91(9) 9 1 ( 1 0 ) 92(1) 9 3 ( 2 ) g 2 ( 3 ) 9 2 ( 4 ) 9 2 ( 5 ) 9 2 ( 6 ) 9 2 ( 7 ) 9 3 ( 8 ) g 2 ( 9 ) 9 2 ( 1 0 ) 9 3 ( 1 ) 9 3 ( 2 ) 93 (3) 9 3 ( 4 ) 9 3 ( 5 ) 9 3 ( 6 ) 9 3 ( 7 ) 9 3 ( 8 ) 9 3 ( 9 ) 93(10) 94(1) 94 (2) 94 (3) g 4 (4) 9 4 ( 5 ) 9 4 ( 6 ) 9 4 ( 7 ) g 4 ( 8 ) g 4 (9) 9 4 ( 1 0 ) 3.7 : ( 9 l , 92, 93, 9 4 ) T -So the data are inner products of the kernels and the model: e. = (g»m) 1,4 3. 8 The kernels contain the physics of the problem, relating the data to the model. To accurately determine ten model parameters without error, one needs to know the physics involved, and have ten independent accurate data. The data are independent if a matrix of ten corresponding kernel vectors cannot be reduced to a state where one or more rows or columns contains all zeros. The matrix could then be immediately inverted, providing a unique path from data space to model space, and the model determined. If fewer than ten independent data are available, the link is no longer unique, connecting a single point in data space instead to a region of model space, wherein the correct model lies. Figure 3.3 is a pictorial representation of the difference. The resolved model would be a result of the resolution function operating on m — the true, unknown model. Therefore the image, denoted by(m), is (m) = Am 3. 9 where A is the best possible resolution function. Each element of (m) will be a weighted average of the elements in the true model. Matrix A contains ten row 26 unique path determined from ten independent kernel functions reproduce the Fig 3.3: A completely determined problem uniquely links one point in data space with one point in model space (part a). An underdetermined problem (not enough data) maps a single point in data space to a region in model space, as the path is not fully known. Resolution is a measure of how well the linkage can be narrowed, and how small the range of solutions can be. One attempts to narrow the path linkage and minimize the range in model space wherein the true solution lies. 27 vectors A\ through Aio, each one designed to focus best upon the corresponding electric field. Each vector is a linear combination of the kernel vectors: / 4 4 4 4 \ Ak = £ a*j9i C1) • £ aki9j (2) £ c k j 9 j (9) , £ <*kj9j (10) , k = 1, 2 10 \ i = i i = i i = i i = i / 3.10 The alpha coefficients depend on the model parameter to be resolved, so that the full resolution function, is E <*ij9j (1) E <*\j9i (2) i=i i=i 4 4 E « 2 ^ i ( i ) E^-fl-iCi) A = j=i j=i E ^li^i (io) j=l 4 E « 2 ^ i (i) j=l M A~lo 4 4 E <*io.?#i(1) E^ioy^^l) ... E«ioyffi( l) . The best •'averaging function would simply be a 10x10 identity matrix - Mp -A" 3. 11 3. 12 A\Q since if used in equation 3.10, it would perfecdy resolve the model. To obtain the best possible resolution function, one minimizes the magnitude of (^Ak — Af^j. The magnitude is the sum of the squares of the differences between elements of the two vectors; that is, the I2 norm: $ f c=||(4-ijr)|2 *=i , 2 , . . . , io = ( A - A > ) - ( A - A > ) 3. 13 28 To solve for the alphas, the norm is minimized: = 2 E £ « W ( 0 - A * ( 0 U » ( 0 m = 1,2,3,4 1=1 \ \j=i 1 0 / 4 10 = 2 £ £ a * ^ i (o * » co - 2 £ Af (/) ^ m (o J=I y = i / /=i 4 10 = 2 E aki E # (*) 9 m ~ 2 g m j=i i=i 4 = 2 £ a * i ' _ 2 f l , m = °-These four equations can be expressed in vector form: 3. 14 Tak = Gk k = 1,2,...,10 3. 15 where r = 91 -91 91- 92 91 • gz 91- 54 92 - 91 92 - 92 92 • 9z 9~2- gk 93-g{ g~3 • g~2 g~z • gz g~z- gk gk • 91 gk- g~2 gk • 91 gk- gk 3. 16 <Xk = (oiki,ak2, ak3,ak4) Gk = (gi 0 ) , g2 (fc), gz (k), 54 (k))T. r is the matrix of inner products of the kernel functions. It is positive definite and symmetric, and is therefore invertable (the inverse of T is given in Appendix C), so the a's can be obtained directly: o?k = T 1Gk. 3. 17 These coefficients are used in 3.12 to obtain the k t h resolution vector. Note that the resolution matrix is independent of the data, depending only on the number 29 and form of the kernel functions. The number of kernels, however, is determined by the number of independent data. A common requirement for resolution vectors is that the sum of the elements of a vector equals unity (the unimodular condition). The rationale behind this is that a blurred model parameter image is a weighted average of all model parameters (although hopefully it is heavily weighted towards the desired one); to ensure that the averaging process does not magnify the image, the sum of the weights must be unity. Therefore another constraint can be added to the minimization: $k = (lk - i f ) • ( 4 - i>) + fa (1 - Ak (p) j k = 1,2,10. P 3. 18 Therefore, minimizing with respect to akm. r)<b 1 0 / 4 \ 1 0 = 2 E E "Wi (0 -AP(k,l)gm(l)\- fik J2 9m (p) a k ' m 1 = 1 V=1 J P = 1 3. 19 4 10 = 2 E A * J '9m)~ 2gm (k) - /?* ^2gm (p) = 0 j=l p=l Minimizing with respect to /?k> Therefore E ( E ^ ( P ) ] = 1 • 3.21 P = l \ g = l / Combining 3.22 with the four equations implied by 3.20, one obtains 2Tak - 2Gk - pkU = 0 . 3. 22 30 U is a vector of the sums of the elements of the kernel elements: T (10 10 10 10 \ £ * ( p ) , £ a ( p ) , £ * 3 ( p ) , X > ( p ) • 3.23 P = i P = i P=i P=i J In this case, then, the alpha vectors are: ak = r - 1 [Gk + f U j . 3. 24 Rearranging equation 3.21, one can show that a* • U = 1. Therefore, the fa's are: 2 ( l - U • r _ 1Gfc) fa = -^-=5 ^ 3. 25 These coefficients are used in equation 3.21 to obtain the averaging function. The program "RESOLUTION" (see appendix D) performs the above calcula-tions, producing resolution matrices with and without the unimodular constraint. Figure 3.4 is a representation of a perfect resolution matrix: the y-axis shows which electric field parameter is being resolved, and the x-axis represents the ten unknown electric fields. Each horizontal line represents a resolution vector, showing which model elements are included in the image of the desired param-eter; each element is shown as a vertical spike of magnitude proportional to the value of the element. The vertical scale is such that 90% of the separation be-tween horizontal lines represents a magnitude of 1.0. Perfect resolution would appear as a matrix with zeros everywhere, except for spikes of magnitude 1.0 on the diagonal. Figure 3.5 is the best resolution matrix possible if no unimodular constraint is used. The most obvious conclusion that can be drawn from it is that E3, E4 and Eg are the best resolved. This is not surprising since the experiment was designed to determine those fields only. However, also apparent is that the 31 three fields have been smeared by the neighbouring fields. The resolution vector for E 3 , for instance, has a strong spike corresponding to E 3 , but also has strong components from E i , E 2 , and E 4 , as well as a reasonable component from E^. The resolution vector for E 4 has the strongest self-component, but is smeared by E i , E 2 , E3 and E6. Ee's resolution is strongly affected by E 5 , and to a lesser but still respectable extent by E i to E4. Not surprisingly, E i , E 2 , E 5 , E7 , Eg, E 9 , and E10 are not well resolved. The resolution vectors do not have strong self-components, and the amplitudes in the vectors are not concentrated close to the diagonal. That is, the resolution vectors are not very narrow, and are not concentrating on the desired portion of the model. As well, vector 2 is simply a multiple of vector 1. Similarly, the vectors 7, 8, 9, and 10 are multiples of each other. Trying to force the resolution matrix to be close to an identity matrix (the delta criterion) produces strong elements along the diagonal, but it does not ensure that the energy in the other components is concentrated close to the diagonal. This causes more smearing than one would like, with the desired components containing elements from many of the other fields. An alternative to the delta criterion is the Backus-Gilbert spread criterion (Backus and Gilbert, 1968). This condition is designed to minimize sidelobe energy in the resolution vector by minimizing a function that is a sum of products of (i) the squares of the vector elements and (ii) a parabola centred at the desired 32 4 6 Model Parameter Number Fig 3.4: A perfect resolution matrix. Y-axis values show which model parameter is being resolved by each vector. Vertical lines represent the magnitude of each element in the vector (90% of the distance between horizontal lines corresponds to magnitude 1.0). This function would perfectly resolve the true model, as only the main diagonal has non-zero elements. Therefore (rh) would equal rh. 3 3 Electric Field Resolution ~i 1 <~ E E ra u rs 1 1 Z 3 4 5 LilocifiC Fiuld model parameter rvjn:i;er T 0 3 10 Fig. 3.5: Resolution of the ten electric fields from figure 3.2 using only data from Skeena, Telkwa, Glenannan, and Williston. The unimodular constraint was not used. E3, E4, and E6 are the best resolved but are smeared with noise from neighbouring fields. The large spike that appears in resolution vector 4 is the result of a positive amplitude of 0.8 for vector 4, and a negative amplitude of 0.3 for vector 3. 34 field component. The function to be minimized for the Ic* model parameter is 10 /10 \ 12 E (* - f c ) 2 (0)2+2/?* E A * (p) - 1 1=1 V>=1 I ( = Y § E c - *)2 (E j + (E ( E ( p ) j - 1 j 3. 26 Differentiating with respect to aymi 3 \ = I i E ^ " FE)2 ^ 2 E aW> O j (/) j +2/?* 53 </m (p). 3. 27 Differentiating with respect to fa: 0ft = 2 ( E Ea * i ^ C p ) - l ) • \p=i i=i / 3. 28 Setting both derivatives to zero, and solving for the coefficients: 3. 29 These coefficients are used in equation 3.12 to obtain the resolution vectors. Figures 3.6 and 3.7 show the resolution matrices produced using a unimodular delta criterion, and a spread criterion, respectively. They are displayed together because they suffer from the same problem: only having available four out of the total of ten kernel functions does not allow the sum of a resolution vector's elements to equal one without seriously disfiguring the function. Therefore the resolution matrices contain many very strong elements off the main diagonal. 35 . 2 4 6 Electric Field Model Parameter Number Fig 3.6: The best resolution possible if the sum of the elements of each vector equals unity. The extra constraint disfigures the function, adding strong off-diagonal components. 36 2 4 6 Electric Field Model Parameter Number Fig 3.7: The resolution matrix if the Backus-Gilbert spread condition is used. Again, strong off-diagonal components appear. The unimodular delta condition and the spread condition are therefore not considered to give appropriate resolution functions. 37 Discussion The resolution matrix in figure 3.5 makes it clear that the validity of explicitly determining the electric fields using the model in figure 2.3 is now in doubt. To assume that the four data were produced only as a result of E 3 , E 4 , and E6 would be suspect as the other fields, if they were of comparable size, would have strongly influenced the data amplitudes. It is important then to note here that the methods used to find the electric fields must change. The fields could still be determined explicitly if enough assumptions are made to make up for the lack of data. For example, one could assume that the north/south lines carried little or no current, and that the field from Glenannan tq Tachick equalled the field from Glenannan to Williston. However, this approach defeats the purpose of the experiment, which was to determine the spatial scale of the electric fields responsible for the GIC's. As well, estimating errors in the electric fields would become very difficult as the results would be strongly dependent on those assumptions. A logical way to proceed then is to search for a reasonable solution that will reproduce the data. It is therefore preferable to treat the experiment as an underdetermined inverse problem. This will allow an investigation of the range of electric field models that might have existed during the experiment. Section 2 Smallest Model Inversion Since the problem is underdetermined, there are an infinite number of solutions that fit the data; the fields produced using the model shown in figure 2.3 represent 38 one such solution. It is up to the investigator to choose a solution that seems reasonable, and reject those that do not seem likely. One approach is to quantify a certain aspect of the solutions, and choose the solution whose aspect quantity is attractive for one reason or another. For example, one could search for a model that has the smallest possible spatial or temporal derivatives, or one could find the solution that minimizes the surface area affected by a storm. Any optimization would be constrained so that the model obtained fits the data. A common approach involves the smallest model inversion. The objective function (the representation of the model aspect) is the sum of the squares of the model parameters, and if minimized, the model will be the "smallest" one that fits the data. It is in a sense the simplest model that reproduces the observations — usually preferable to a complicated solution — and is developed here. The objective function is initially the I2 norm of the model parameters. For a vector, that is equivalent to the inner product of the vector with itself: $ = | |m | | 2 = ( m , m ) . 3.30 To ensure that any model fits the data, the data equations are used as constraints, using a Lagrangian multiplier vector o7k = (ai, 0:2, «3,0:4) : 4 $ = (m, m) + 2 aj (ey — (g"j, m)) . 3.31 i=i The factor of 2 in front of the data constraints can be arbitrary, and is chosen here to avoid a factor of 1/2 later in the equation; it does not affect the final outcome. 39 The model, and therefore the objective function, are now perturbed: fh —¥ m + Srh $ —> $ + <5$ 4 .'. $ (m + £m) = (m + £m, m -f (5m) + 2 53 a i ( ej — (#}> ™ + <5m)) 4 = (rh, m) + 2 (rh, Srh) + <^$m, <5m) 4- 2 53**.? ( e i — 4 - 2 53 a i (g~j,6rr?) j=l = $ 4-3. 32 To first order, since the perturbation is taken to be small, 4 <S<& « 2(m, £m) — 2 53 a i (gj^rn^ • 3.33 i=i The model is minimized by setting the objective function perturbation to zero. Therefore, m — 53 » ^ m 1 — 0 3. 34 i=i / Since the model perturbation is arbitrary, the left hand element of the inner product must be zero. Therefore the smallest model is a linear combination of the kernel functions: 4 rh — 53 aj9j • 3. 35 The Lagrange multipliers can be found by substituting equation 3.35 into the data equations: e« = ( 9i>^2aj9i j i = i ' 2 ' 3 ' 4 3 - 3 6 40 Combining the four equations, e = Ta 3. 37 .-. a = T~1e . The T matrix here is again the matrix of inner products of the kernel functions. The alphas are substituted in equation 3.36 to obtain the smallest model. The program "SM_EFIELD" (see appendix D) produces the smallest electric field model that explains the data. The inputs to the program are the percentages, for each station, of the total station current being grounded through the measured transformer, the ten loop resistances; the ten distances; and the data files. The program calculates the inner product matrix and decomposes it into its eigenvalues and eigenvectors: r = RART , 3. 38 where R is a matrix whose columns contain the eigenvectors, and A is a diagonal matrix whose elements are the eigenvalues. At each time sample, the four data are read in and used to determine the alpha vector. Using equation 3.37, RARTa = e 3. 39 .-. ARTa - RTe . Now define e = RTe , and a = RTa. Therefore di = Y i = 1,2,3,4. 3. 40 Aj The di's are then unrotated to obtain the desired coefficients: a = Ra . 3. 41 The routine solves equation 3.35 separately at each time point so the model at t+At is independent of the model at t. 41 Results and Discussion The smallest model inversion is not based upon principles of physics. It simply searches for that set of electric fields that fits the data using minimal amplitudes. Figure 3.8 shows a typical problem with the results: half the fields are positive, and half are negative. In order to fit the large currents seen at Telkwa and Williston, the inversion prefers to split the polarities of the fields. For the lines on the east/west section of the grid (see figure 3.2), the fields west of Telkwa are made negative, and those to the east, positive. A similar splitting occurs for the northerly trending lines, making fields north of Williston opposite to those to the south. This arrangement of the fields is the smallest one that will reproduce the large currents at Telkwa and Williston, and the smaller currents at Skeena and Glenannan. The smallest model approach produced such polarity reversals for all nine storms. It is unlikely that the earth's conductivity structure could produce such reversals for storms whose scales are larger than the size of the survey. Therefore the results, if correct, would imply that the source storms always had polarity reversals centred over Telkwa. Since the storms occurred over a three week period, and were distributed in local time from 5:00 pm to 3:00 am, this implication was considered unlikely. The initial aim of the experiment was to obtain the spatial scale of the electric fields affecting the transmission grid. With that in mind, it was decided to find the solution that is closest to a uniform field; that is, closest to an electric field model whose spatial scale is longer than the entire size of the grid. Such a model would contain the minimum non-uniformity required by the data. Therefore a 42 1 8 8.5 Time in Hours UT Fig 3.8: Results from the smallest model inversion using data from event 3. Simply trying to minimize field amplitudes results in field polarities being split at Telkwa (for the east-west portion of the grid) and at Williston (for the north-south portion). Notice that E i , E2, and E3 oppose E4, E5, and Eg, and that E7 and Es oppose E 9 and E10. This type of result led to the conclusion that the smallest model was inappropriate. 43 minimum bound could be placed on the uniformity of the field, or equivalently, a maximum bound on the spatial scale. Section 3 Smallest Deviatoric Model The smallest deviatoric inversion looks for the data fitting model that is closest to an input model. The development is similar to that for the smallest model, involving only a change in what the model sees as the data. Assuming that an initial model is provided (that generally does not fit the data), the smallest deviation from that model is looked for such that the sum of the initial and deviatoric models reproduces the data. The final model is mp = mj + mn , 3. 42 where mj is the initial model, and mn is the deviatoric model. The final model must meet the condition that ej = (g*j,mi + mp) j = 1,2,3,4. 3.43 Therefore four new data are defined: fi = eJ ~ (dj, mi) - (<?j, m~i>) j = 1,2,3,4. 3. 44 The ej's are known, and the inner products of the kernels and the initial model can be calculated. The objective function to be minimized then is 4 $ = (nib, mb) + 2 £ OLJ (fj - (<jj, mD)) , 3. 45 j=i 44 with the second term on the right hand side ensuring that the final model fits the original data vector e . This function has the same form as that for the smallest model (equation 3.31), and so the development from here is identical to that shown from equations 3.32 to 3.37. Nevertheless, the inversion should be altered somewhat so that the elements of the model may be more closely compared. By doubling the number of model parameters, both northerly and easterly components can be determined for each of the ten cells. The data equations then become: m i 1 = 91(1) ... 3i (10) 5ri(ll) 54(1) 34(10) 34(H) 91(20) 34(20) m i o m n 3. 46 where the model vector now has 20 parameters, the first ten for northerly components, and the last ten for easterly fields. The kernel elements are similar to those from equation 3.4, except that, for example, /-i \ TsKEdi a 91 (1) = ^ cosdi 91(11) = Rl TsKEd Ri -sind\ 3. 47 f?i is the angle, taken clockwise, between true north and the trend of the i * power line. The initial model for the inversion is a uniform field that is produced by a least-squares fit to the data of a northerly and an easterly component. The least-squares fit is developed in appendix B, and only the final formulae are shown 45 here: 4 4 E ayiaya E °>3«yl 3.48 4 4 E ayi">.;2 E ayi ey. y=i y=i where E N and E E represent the northerly and easterly directed electric fields, respectively. The coefficients aji and aj2 refer to the sums of the first ten and last ten elements of the kernel vectors, respectively: o-ji Typical error handling in inversion schemes deals with errors on the data, and their effects on the final model. The kernel functions are assumed to be exact, so that solution errors are due only to data errors. That assumption does not hold for this experiment. The kernels contain information regarding the distances and the bearings between grounding stations, the simplified equivalent resistances for the power grid, and the percentages of station currents passed by the measured transformers. The resistances and percentages are all subject to errors, as mentioned in chapter two. The distances and bearings are considered exact, since the errors on them are comparatively small. There are 25 terms that go into the building of the kernel vectors and that have errors associated with them. Including the four data, there are 29 parameters that introduce errors in the final solution. The errors in the model can be approximated by using the linear terms of a Taylor series expansion centred about the location 46 EN E «?i E ah - E anai* L E °yi E E ah E oyi«j 10 E # ( P ) J = 1,2,3,4 20 J^fiiP) i = 1,2,3,4 P=II 3. 49 of the solution in model space: AmF » — — A p i + — — A P 2 + ... + -x Ap 28 + - 5 — A p 2 9 • 3. 50 dpi dp2 Op28 OP29 The full development of the error handling is given in Appendix C. Results Appendix F contains a full set of results from the smallest deviatoric inversion, presented in two different forms: as plots against time of either the northerly or easterly components of the fields (figures F . l to F.9); or as vector plots of the fields seen by power lines 1, 3, 4, 5, and 6 (figures F.10 to F17). Key plots of both types are reproduced in this section. In the first form, the northerly electric field component between Telkwa and Glenannan (E4N) is shown, along with the easterly fields between Skeena and Telkwa, Telkwa and Glenannan, and Glenannan and Williston (e3e, e4e, and E6E)- Only one northerly component is shown, as the variation in northerly fields across the network is small compared to the easterly variation. Of the ten easterly components, only e3e, e4e, and E 6 E are shown. The other 16 components are not shown as they would significantly enlarge the size of the thesis without transmitting much useful information; the four data do not allow them to be accurately determined. The second form is a display of the strength and direction of the total fields seen by the five predominantly easterly trending power lines in the network. Figures F.10 to F17 plot time along the x-axis, and distance east of Prince Rupert along the y-axis. Five traces appear: each is placed at the y-coordinate that corresponds to the centre of one of the power lines. The length of the vector is 47 proportional to the final electric field solution, and the direction is such that true north is to the left, parallel to the x-axis. The inversion results show that if the errors are included, one cannot formally consider the fields in events 1 to 5 to be distinct, as the sizes of the errors are such that the uncertainty envelopes around the solutions always overlap, as in figure 3.9. Therefore the experiment, using this formulation, cannot strictly prove any non-uniformities in those electric fields. Nevertheless, this does not preclude further observations about the solutions themselves. The results for events 7 to 10 are distinct, as in figure 3.10, where strong easterly fields as high as 1 V/km to the east of Telkwa are not seen to the west The results were more encouraging than those for the smallest model. The most significant improvement was that the directions of the fields east and west of Telkwa were no longer in opposition for most storms. Figure 3.11, a vector plot of the smallest model for event 5, shows the physically unrealistic opposing fields that are not evident in figure 3.12. The first plot also demonstrates a major difficulty with the smallest model: the directions assumed by the electric fields are constant and exactly parallel to the power lines. Therefore the smallest solution is too affected by the survey geometry to have any useful directional information. The majority of the results show E4E as the strongest easterly component, with E6E as the next strongest (see figures 3.9, 3.10). E3N, E4N, and E6N are usually almost equal. Also evident in all the storms is that E4E and E6E have almost exactly the same form, with EgE roughly 60% to 80% of E4E. The form of E3 is distinguished from E4E and E6E by the lower amplitudes of the long periods (figure 3.9b,c). The difference becomes extreme in the more 48 Fig 3.9: Event 1 • / i l i / / 1 1.5 2 Timo In Hours UT A , ^ — • N V 1 1.5 2 Time in Hours UT Figures 3.9, 3.10, 3.13, 3.14, 3.17, and 3.18: Smallest deviatoric inversion results. These are the closest to uniform results that fit the data. Part a is the northerly component of E 4 , and b, c, and d are the easterly components of E3, E 4 , and E6. The solid curves are the solutions, and the dashed curves represent the corresponding error bounds. Note that these do not represent the total possible range of electric fields that fit the data; rather, the errors show the range possible for results of this inversion — the smallest deviation from a uniform field. The improvement over the smallest model results is that the field directions are not always split at Telkwa and Williston. 49 Fig 3.10: Event 7 4 5 6 7 Time In Hours UT Time in Hours UT 51 Smallest Deviatoric E Field (V/km) on a s_ Smallest Deviatoric E Field (V/km) \ < r -1 U l > 3 5* Hours c - i \ • \ CV t » :-V. Fig 3.11: Event 5 smallest electric fields 500 g 400 cu Q< K 300 <D O C P-o 200 u CD o C 100 mtltii'ltiitl H i>»iiiii»i ///// ' cvent_05 smallest model, unrillered. amp - 40 : Sun Aug 5 14:53:02 1990 • IIP 7 / h n h > imiimi Win, 7//JMI» . 'IHHI'llUipil 10 10.5 Time in Hours U.T. 11 Figures 3.11, 3.12, 3.15, 3.16, and 3.19: Pictorial representations (without errors) of the strength and direction of the total fields E i , E3, E4, E5, and E6. The x-axis represents time, with the base of each arrow showing the time of occurrence for that particular, field orientation. The y-axis shows distance east of Prince Rupert; the y-coordinate of each trace distance east from Prince Rupert to the centre of the power line concerned. North is to the left, parallel to the x-axis. The length of each vector gives the field strength, using the provided scales. 53 Fig 3.12: Event 5 smallest deviatoric electric fields 54 powerful events 7 and 8, where east of Telkwa, powerful easterly electric fields that last for five to ten minutes are almost unnoticed to the west (figure 3.10). Events 9 and 10 are recordings of micropulsations that occurred at local standard times of 5:20 pm and 4:30 am, respectively. In the evening event, E 4 E and E 6 E (figures 3.13c,d) are very similar, with E6E of slightly smaller amplitude. E3E, on the other hand, has elements that also similar, but the 70 s ringing in E4E and E6E from 1.32 UT to 1.4 UT is largely unseen in E3E due to stronger higher frequencies. This is not true of a pulsation seen at the beginning of event 3, where E3 is comparable to, if not stronger than, E4 and E6 (see figure 3.13.2). The morning event (figures 3.14c,d) again shows E4E and E 6 E to be very similar. In this case, however, E 3 E is very similar as well, although advanced in time by ten seconds, or one third of a period. Because of this advance, the fields are indeed distinct, including the error bars. Discussion Although E 3 , E 4 , and E6 cannot formally be considered distinct during much of the recording period, it is appropriate to make suggestions based purely on the solutions themselves, neglecting the error bars, as long as one remembers that the suggestions are speculative. The justification for this is that although the error bars show the range of solutions to be large enough to obscure any differences, the smallest deviatoric inversion results are nevertheless the best guesses at the true solutions, and so therefore have some value. The similarity of fields E4 and E$ is common to all events: from the large scale structures to the fine details, the only difference is amplitude, not the overall signature of the traces. This suggests that, between Telkwa and Williston, no major 55 Fig 3.13: Event 9 57 7.85 7.9 7.95 Time in Hours UT Fig 3.13.2: Micropulsation seen at beginning of event 3. This short period fluctuation differs from the rest of the storm in that E 3 is stronger than E 4 and E$. The data have been subjected to a trapezoidal bandpass frequency filter (corners at 0.01, 0.012, 0.054, 0.056 Hz). 58 59 60 differences in source characteristics existed during the storms; and that the earth response to uniform long period (one minute or longer) magnetic perturbations, when averaged over 150 km, is roughly homogeneous. E 3 , however, often does not mimic the other two fields: there are differences between the field signatures to the east and west of Telkwa in every event. These differences are due either to source storm, earth response, or inversion process effects. Inversion process effects refer to any differences that truly did not exist during the storms, and that arise only because the type of solution for which one searches is inappropriate. Nothing ensures that if the inversion produces the true solution at one time, it will do so at another. That is, a uniform field approximation may be reasonable for parts of the storms, and unreasonable for others, but the inversion will always find the solution that is closest to uniform. Considering the error bars, events 1 to 5 do not show any formal difference between E 3 , E 4 , and E6 , whereas events 7 to 10 do. In both cases one must choose whether the results likely represent the fields that really existed during the recording periods, and if they do, whether any observed non-uniformities are storm or earth induced The gross structures of E 3 , E 4 , and E6 in events 2 and 4, best seen in figures 3.15 and 3.16, resemble one another quite well. The similarity of the traces and the reasonableness of the uniform field assumption suggests that the results may well be close to the true, unknown solutions. The major differences occur at 3.65 UT and 3.93 U T in event 2 (see figure 3.17), where a strong peak in E4E and E6E is not seen in E 3 E ; and in event 4(figure 3.18) where phase shifts exist between low frequency components of E3 and E4. The validity of these results — whether 62 or not they are inversion induced — is uncertain. The initial opposition of E3 to E 4 and E6 in events 1, 3, and 5 (see, for example, figure 3.12) falls into a similar category: the effect may well be real, and either storm or earth induced; or an artifact of the inversion. It is suggested that, of the three choices, the effects were either earth or inversion produced, since they occur in all three storms. E 3 , E4 and E 6 are similar during the remainders of the three storms, suggesting again that the fields may have been close to uniform. Results for these five events are not likely to be grossly in error. They suggest the storms were larger than the survey area, as the majority of each event shows the form of E3 to be similar to E 4 and E6. One possible explanation for the initial opposition of the fields next to Telkwa in events 1, 3, and 5 is the following: if E 3 and E4 point in the same direction, but E 4 is much larger, the inversion may produce a smallest model that splits field directions at Telkwa rather than the true solution, because the true solution may well be farther from a uniform field than a split field model is. Essentially, then, the initial opposition of the fields may be inversion induced. Results from events 7 and 8 are less believable, and likely do not represent the true solution. The low amplitudes of E3 compared to E 4 and E6 are probably due to the inversion producing a solution similar to a smallest model as that solution which is closest to a uniform field. This is evident in figure 3.19, where one can see E i always opposing E4 and E6 , while E3 is very quiet in comparison to the rest of the plot. The production of an unbelievable solution occurs when input data consist of very large transformer currents. Fields for events 7 and 8 are in a way similar to what one might expect from 63 evenl_02 unfillered. imp = 40 : Sun Aug 5 17:21:57 1990 500 i 1 1 , , . , . . i . . , , , . . 4^ Time in Hours U.T. Fig 3.15: Event 2 Fig 3.15: Event 2 66 Fig 3.18: Event 4 500 400 300 200 100 event_08 unlillercd. imp = 20 : Mtm Aug 6 13:19:49 1990 9.5 Time in Hours U.T. 10 Fig 3.19: Event 8 converging auroral electrojets, but is likely that they do not mimic the true fields. Two main auroral electrojet vortices form during magnetic storms (Iijima and Nagata, 1972), one centred in the early morning, and one in the late afternoon sector, (see figure 3.20). The convergence of the two vortices occurs from 9:00 pm to 10:00 pm local time, with eastward ionospheric currents from the afternoon sector meeting westward currents from the morning side near 60° latitude. The resultant electric field polarities expected on the ground would be similar to those seen in figure 3.19 — eastward to the east of the convergence point, and westward to the west. It is implausible though that results from events 7 and 8 are the true solutions, as that would require that the convergence point remained above Telkwa from 8:00 pm to 3:00 am local time, an unlikely occurrence. These presumably inappropriate solutions throw doubt on the solutions for the other events, but do not discount them. Several approaches could be taken to narrow the range of data fitting solutions. The most obvious choice is to force E3E to equal the electric fields measured at Smithers. Another is to incorporate the magnetic data. Unfortunately, problems are involved with both routes. The electric field data are currently stored only on F M tapes and corresponding paper playbacks. Problems during the experiment lead to data only being collected for the second recording period, missing events 1 to 5. Time constraints did not permit the digitization and incorporation of the data into the analysis. As well, there is a question of scale involved. The resolution of a perfectly measured experiment would be approximately 130 km. Requiring E3E, a solution averaged over this scale, to equal the Smithers data that were collected on a scale of 72 SUN Fig 3.20: Auroral electrojets that form during the growth and expansion phases of a magnetic storm. The vortices have opposing senses of rotation, forming a zone of confusion where they meet between 9:00 and 10:00 pm local time. (After Iijima and Nagata, 1972.) 73 50-100 m, might force E3E to more closely reflect the local conductivity structure near Smithers, rather than the average structure between Skeena and Telkwa. Comparison of the results to the Smithers data is inconclusive: portions of the paper records resemble E 3 E , other portions resemble E 4 E , while other large sections resemble neither. Incorporation of the magnetic data into the inversion would require a model of the earth's conductivity structure. With such a model, one could estimate the surface electric fields that would result from a given magnetic field signature. Gough (1986) discusses a conductive structure beneath the Intermontane and Omenica belts of B.C. whose northern edge terminates at Williston (see figure 3.21). The structure's upper surface is estimated to be in the 15-30 km depth range, and its depth extent to be from 30 to 70 km. The conductivity is estimated as on the order of 0.1 S/m. Such a structure would attenuate the magnetic fields observed in this study. If the difference in amplitude between E 4 and E$ is real, one can speculate that the Canadian Cordilleran Regional Conductor may be the cause. As shown here, however, it would not explain the lower amplitudes of E 3 . Neither of these routes can be considered inappropriate: both would help narrow the range of solutions that fit the data. It was felt, however, that the time required to implement them would likely be more than the time remaining to complete the study. Therefore it was decided to propose a new experiment that would be capable of explicitly determining the electric fields responsible for the GIC's in B.C. Hydro's system. Notes on such a proposal appear in the next chapter. 74 Fig. 3.21: Schematic representation of conductive structure (named the Canadian Cordilleran Regional conductor) in the Intermontane and Omenica tectonic provinces, with locations of power lines superimposed. Depth to the conduc- ' tor's upper surface is estimated at 15 to 30 km, and its depth extent is believed to be 30 to 70 km. The conductor may be the cause of the reduced amplitude of the electric field between Glenannan and Williston (E 4), compared to that between Telkwa and Glenannan (E6). (After Gough, 1986.) 75 Chapter 4 A Redesigned Experiment This chapter proposes an experiment to define the factors that control GIC's in the B. C. hydro system. Those factors include characteristics of the source magnetic storms, and the conductivity structure of the earth in the area. Ideally, one would like to measure currents at all sites capable of grounding GIC's, so that overhead line currents at magnetic storm frequencies can be properly calculated. The potential differences between stations could then be explicitly determined, if provided the physical status of the power system. Knowing the potential drops, however, is not sufficient. Each power line allows the calculation of one component of the electric field, averaged over the distance between two stations; but the power lines are not parallel. Therefore only the inner products of the true averaged electric fields with the orientations of the power lines could be compared. One could not, with only loop current information, compare across the network the easterly or northerly components of the induced fields. Therefore it would be necessary to deploy magnetometers to determine how the two perpendicular components relate to one another. That would allow the calculated inner products of the fields and the power line bearings to be properly converted to northerly and easterly components. As well, the use of a large number of magnetometers would provide much more accurate knowledge of the spatial variations of the surface magnetic field. The time varying portions of the magnetic fields would be the result of processes both in the magnetosphere and in the earth. The two effects can be separated by deploying magnetometers over a large two dimensional grid. 76 Monitoring the magnetic field over an area approximately 250 km by 600 km would, by the method oudined by Weaver (1964), allow one to determine the six components X j , Yi , Z j , X e , Y e and , Z e , where i refers to internal sources and e to external. Obviously a very large number of magnetometers might be required (60 instruments if only a 50 km spacing is used), so the spatial scale of magnetic disturbances should be investigated first. A test experiment run before the GIC project (with as many magnetometers as possible), would determine how dense the coverage should be, avoiding later unnecessary duplication of effort. If the scale is so small that an inordinate number of instruments is required for a two dimensional grid, one must settle on ad hoc assumptions regarding the relative contributions of internal and external sources, and monitor fields only along the power lines. Large pools of magnetometers do exist though, and have been used in similar magnetovariational studies (e.g., Camfield et al., 1971; Gough etal., 1982). Those studies used 46 and 33 magnetometers, respectively. Placement of HET's Since the recording of the data in 1984, B.C. Hydro has added one more line that is capable of carrying GIC's. Wye wound 138 kV transformers at Stewart, 200 km north of Prince Rupert, provide a GIC path north from Skeena (see figure 4.1). (Aiyansh's transformers are delta wound, so GIC's cannot be grounded there). Therefore 22 sites now exist north of and including Kelly Lake where GIC's can be grounded. Not all of these need to be monitored. Regardless of the status of the capacitors at Kennedy (KDY) and McLeese (MLS), by conservation of currents the number reduces to 12: Glenannan, Tachick, and all the stations to 77 the west. Notice that the field between Glenannan and Williston can be explicitly determined without measuring currents at Williston. Using fewer than 12 instruments risks not being able to properly distinguish the electric fields between stations. Twelve HET's, however, may not be available. With that in mind, the effect of each station's status (monitored or not) on the resolution was assessed, with the intention of defining how best to arrange a limited number of instruments. The formulation used in section 2 of chapter 3 can be used to monitor the resolution as the number and placement of HET's is varied around the network. Figure 4.2 shows how the electric field subscripts were changed in this approach to include the extra stations. Fields 3, 4 and 6 from chapter 3 are referred to here as fields 9, 10 and 12, respectively. Figure 4.3 shows the resolution matrix of an adequately monitored network. Twelve resolution vectors contain elements for each of the 12 electric fields affecting measurements at the stations; each vector in this case contains only one non-zero element: the main diagonal amplitudes are 1.0. With only 11 HET's, one station must be neglected. Figure 4.4 shows a poor choice: not measuring currents at Kitimat (datum 6) smears the resolution vectors for Eg, E 7 , E 9 , E10, and E12. One cannot avoid a certain amount of smearing, but the best arrangement would neglect one of Falls Paver, Green River, or Diana Lake. Figure 4.5 shows the resolution obtainable with Falls River neglected. In fact, one could use only nine instruments, neglecting all three stations, with the resultant resolution shown in figure 4.6. E 9 , E10, and E12 would be slightly smeared by E 3 , but each by the same amount. Therefore the error introduced into 78 Fort St.. John 500 k V lines 230 k V lines 138 k V lines 60 k V lines K e l l y Lake Fig.4.1: Possible GIC paths. A 138 k V system, installed from Skeena to Stewart, has experienced GIC related problems since 1984. Soda Creek <p-100 M i l e House 79 Fig.4.2: Updated schematic of power grid: electric field subscripts were changed to account for effects of all stations. Field El2 can be determined without measuring currents at Williston, 80 1 0 Model Parameter Number Fig 4.3: The resolution matrix of the experiment if all 12 data are collected: all fields are perfectly determined. 81 the fields would be constant for E 9 , E10, and E12. These considerations should be kept in mind when deciding upon the moni-toring of transformer neutral currents. Financial constraints may dictate that one of these alternate deployments be used, but a preferable approach would be to measure every station. Summary Regardless of whether a one or two dimensional magnetometer grid is used, a comprehensive experiment designed to monitor all aspects of the GIC phenomenon would be considerably larger than that performed in 1984. Acquiring eight more HET's and the related recording equipment would cost approximately $10,000 in total. The number of magnetometers available would determine the feasibility of a two dimensional survey. For an existing power system, it is likely sufficient, as far as the power companies are concerned, to know that large currents exist at certain locations; they are not as concerned about why they occur. The value of such an experiment, however, would be in confirming the GIC controlling factors for a known system, thereby allowing one to predict where currents would cause problems for future power transmission systems currently in the planning stages. 82 ~1 T" _ j 1 lo IZ 1 r Model Parameter Number 10 Fig 4.4: Resolution for a poor choice of placement of 11 HET's. Neglecting to monitor Kitimat (datum 6) smears images of E 7 , Eg, E9, E10, and En-83 IZ , I i , i , L 5 10 Model Parameter Number Fig 4.5: A good choice for placement of 11 HET's. Neglecting Falls River does little damage to resolution of electric fields. 84 4 6 Model Parameter Number Fig 4.6: Resolution of 12 fields using nine instruments. The best placement would neglect Falls River, Green River, and Diana Lake. 85 Conc lus ions The initial goal was to determine the spatial scales of the electric fields responsible for the GIC's in B. C. Hydro's 500 kV system. The first attempts to do so suggested that the long period electric fields east of Telkwa were much stronger than those to the west. The difference in amplitudes was so large that the model used to calculate the fields was suspected to be inadequate. This proved to be true, as capacitors north and south of Williston were shunted during the recording period, and as the roles of the 230 kV lines in carrying GIC's were more important than originally thought. Although it was not possible to uniquely determine the fields, reasonable solutions were found for some storms by finding a set of electric fields that were close to uniform and that reproduced the data. The results suggested that the size of the source storms was larger than the size of the grid. Reasonable solutions were not found for particularly strong storms. The solutions that were closest to a uniform field required the field directions to be in oppostion to the east and west of Telkwa, and this was not considered likely. It was felt more probable that the fields were grossly similar in direction, but significantly different in amplitude, implying that the source scale was larger than the transmission grid, and that the earth was principally responsible for the field variations. Fields E4 and E6 were always similar in form, whereas E3 was not. It was not determined whether this was an earth induction or an inversion process effect, but it was felt that it was likely not a source effect. 86 -A comprehensive approach to determining what controls the GIC's in the power lines would require significant effort. It would be necessary to collect grounding current data at 12 substations, and magnetic field data at perhaps 50 to 100 positions. Confirming these controlling factors would allow one to predict electric field amplitudes and spatial scales in areas where new power transmission systems are to be constructed. 87 List of References Albertson, V. D. and Van Baelen, J. A., Electric and magnetic fields at the earth's surface due to auroral currents, IEEE Trans, on Power Apparatus and Systems, PAS-89, 4, 578-584, 1970. Albertson, V. D., Clayton, R. E. , Thorson, J. M . , and Tripathy, S. C , Solar-induced-currents in power systems: cause and effects, IEEE Trans, on Power Apparatus and Systems, PAS-92, 2, A1X-M1, 1973 Albertson, V. D., and Thorson, J. M . , Power system disturbance during a K-8 geomagnetic storm: August 4, 1972, IEEE Trans, on Power Apparatus and Systems, PAS-93, 4, 1025-1030, 1974a. Albertson, V. D., Miske, S. A., and Thorson, J. M . , The effects of geomagnetic storms on electrical power systems, IEEE Trans, on Power Apparatus and Systems, PAS-93, 4, 1031-1044, 1974b. Albertson, V. D., and Kappenman, J. G., Magnetic storm effects in electrical power systems and prediction needs, in Solar Terresttrial Predictions Proceed-ings, edited by R. F. Donnelly, 2, 137-148, Boulder, Colo., 1979. Backus, G., and Gilbert, F., The resolving power of gross earth data, Geophys. J. Roy. Astr. Soc, 16, 169-205, 1968. Bartels, J., The standardized index, Ks, and the planetary index, Kp, IATME Bull. 12b, 97, IUGG Pub. Office, Paris, 1949. Boteler, D. H., Watanabe, T., Shier, R. M . , and Horita, R. E. , Characteristics of Geomagnetically Induced Currents in the B.C. Hydro 500 kV System, IEEE Trans, on Power Apparatus and Systems, PAS-101, 6, 1447-1456, 1982. Boteler, D. H. , Watanabe, T., Horita, R. E. , Chen, G. M . , and Butler, D., Prediction of Geomagnetically Induced Current Levels in the B. C. Hydro 500 kV System, Proceedings of Solar-Terrestrial Predictions Workshop, London, 88 U. K., Oct. 1989, (in press). Butler, D. B., unpublished Fortran code, Dept. of Geophysics and Astronomy, University of British Columbia, 1990. Camfield, P. A., Gough, D. I., and Porath, H., Magnetometer array studies in the northwestern United States and southwestern Canada, Geophys. J. R. Astr. Soc., 22, p. 201-221, 1971. Gough, D. I., Mantie Upfiow Tectonics in the Canadian Cordillera, J. Geophys. Res., 85, 6113-6156, 1980. Gough, D. I., Bingham, D. K., Ingham, M . R., and Alabi, A. O., Conductive structure in southwestern Canada: A regional magnetometer array study, Can. J. Earth Sci., 19, 1680-1690, 1982. Iijima, T., and Nagata, T., Planet Space Sci., 20, 1095-1112, 1972. Kappenman, J. G. and Albertson, V. D., Bracing for geomagnetic storms, IEEE Spectrum, March, 27-33, 1990. Lundby, S., Chapel, B. E . , Boteler, D. H. , Watanabe. T., and Horita, R. E . , Occurrence Frequency of Geomagnetically Induced Currents: A Case Study on a B.C. Hydro 500 kV Power Line, J. Geomag. Geoelectr., 37, 1097-1114, 1985. Merrill, R. T., and McElhinny, M . W., The Earth's Magnetic Field, Academic Press Inc., 1983. Viljanen, A. and Pirjola, R., Statistics on Geomagnetically-Induced Currents in the Finnish 400 kV Power System Based on Recordings of Geomagnetic Variations, J. Geomag. Geoelectr., 41, 411^420, 1989. Weaver, J. T., On the Separation of Local Geomagnetic Fields into External and Internal Parts, Zeitschrift fur Geophysik, 30, 29-36, 1964. 89 Appendix A Monitored Transformer Percentages At each substation, only one transformer neutral was monitored by a Hall effect transducer. Any GIC flowing in that neutral could be related to a total GIC flowing through the entire station, by comparing the resistances of all transformers and reducing the substation wiring diagrams to simple equivalent circuits. Figure A . l is the operating one line diagram for Glenannan substation. At frequencies near zero, reactive effects are negligible, so transformers and shunt reactors can be replaced by their dc resistances, and the other equipment (current and potential transformers, circuit breakers, and surge arrestors) neglected. Figure A.2 shows how the station is then represented, and how the percentage of total station seen by the measured transformer is determined. 90 -!H*' — D -Hi' | — —a—]i f v. u f-r -•Hii-I -mH1' z o H C O i—i J F i g A . l : Operating one line diagram of Glenannan. The station is connected to Telkwa and Williston by lines 5L62 and 5L6T, respectively. Transformers T I and 17 step the voltage down from 500 k V to 230 kV. Transformers T5 and T I 1 then reduce it to 138 kV. From there, the voltage is reduced to 60 k V by T3 and T4. A l l six transformers arc "wye connected, and can therefore ground GIC's, as can shunt reactors S R X 5 and SRX5N. 91 Glenannan Transformer Resistances S R X 5 R P S 500 kV level SRX5NR 230 kV level 138 kV level S R 5 X R = 3.28 O. +/-4% S R 5 X N R = 4.78 Cl +/-4% T 1 H = 0.337 Q +/-4% T 1 L = 0.161 Q +/-4% T 2 H = 0.337 Cl +/-4% T 2 L = 0.161 Cl +/-4% G R N D R = 0.45 Cl +1-1% T 1 1 H = 1.25 Cl +/-2% T 1 1 L = 1.56 +1-2% T 5 H = 0.217 n +/- 1 1 % T 5 L = 0.276 Cl +1-9% T3 = 1.57 Cl +/- 1 5 % T 4 = 1.16 Cl +1-4% O O O percentage of total current flowing from 500 k V bus to ground seen by T l = 40 +/- 3 percentage of total current flowing from 230 k V bus to ground seen by T I L = 41 +1-4 Figure A.2 : The transformers and shunt reactors from the one line diagram are shown here by their d.c. resistances. The HET monitored T I L , which saw 40 % of the total current grounded by the station. 92 Appendix B The Least-Squares Uniform Electric Field  The smallest deviatoric inversion requires a starting model, which was chosen as a uniform electric field. Finding the uniform field that comes closest to reproducing the data is a simple minimization of the sum of the squares of the differences between the observed data, and those data that would result from a uniform field (the predicted data). For a field with a northerly arid an easterly component (EN and EE), whose dimensions are larger than the size of the transmission grid (as shown in figure B.l), the four predicted data would be: epl an 0-12 021 «22 ePz 031 032 a 4i «42. EN EE B. 1 B. 2 where the ay's are the sums of the first and last halves of the kernel vectors from equation 3.47: 10 i=i 20 l=n Therefore the objective function to be minimized is: 4 B. 3 i=l 93 Minimizing, dEN 0EE = -2 53 ( e « ~ ailEN ~ ^EE) an = 0 i=l 4 = -2 53 ( e i - CLUEM - CHIEE) ai2 = 0 1=1 Solving the two equations for for EN and EE, one obtains the following: EN EE 1 . x 4 4 / 4 E « i l E ai2 ~ ( E a«l««2 «=l4 1=14 \ t = l 4 / 4 E a\l E ° i l e i _ E a«'l ai2 E a«2e, i=l t=l i=l t=l 4 4 4 4 E a i l E ai2e, - E a t l O i 2 E a i l e « ' t=l 1=1 t'=l t=l B. 4 B. 5 94 Fig B. l : A uniform electric field would induce currents at Skeena, Telkwa, Glenannan and Williston, whose relative amplitudes are controlled only by the loop resistances in each cell. Appendix C Handling of Errors in Inversion This appendix details how errors on the input parameters affect the final model produced by the smallest deviatoric inversion. Equation 3.51 stated that the estimated error in the smallest deviatoric solution is the sum of the partial errors due to each input parameter. That equation is restated here: A - dmp . dmp A dmp . dmp Amp fa ——Api + ——Ap2 + ... + -5 Ap28 + - 5 Ap29 • C. 1 dpi Op2 Op28 Op29 The 25 parameters containing the resistances and transformer percentages, together with the four data, comprise the input parameter vector: P — (TsKE, TTEL,TGLE5,TGLE2,TWIL5,TWIL2,RRUP, RMIN , RSKESI RSKEO, RST, RTEL,RTG, RGLES, RGLE2,RGLE0, RTAC, RGW, RwiL5, R\VJL2I RWILO, C. 2 RsVY, RGMS, RBLW, RKLY, ei , e 2, e 3, e 4) = (Pl>P2,P3,—,P27,P28,P29) Parameters pi to p6 are the percentages of total station currents flowing through the measured transformers. There are two each for Glenannan and Wiliston, as the percentages vary depending on whether the source of the current is in 500 kV or 230 kV bus. Parameters P7 to P25 are the various resistances used to build the loop resistances. The last four parameters are the four data. 96 One therefore needs the partial derivative of each model element with respect to each input parameter. The derivatives of the final model are the sums of the derivatives of the initial and the deviatoric model: draft _ dm/, dmj)i dpk dpk dpk C. 3 Dealing with the initial uniform field model first, one must differentiate the northerly and easterly fields. For simplicity, the numerators and denominator given by equation B.5 are referred to as A, C, and B (The limits on the summation signs have been dropped for brevity; the limits are j=l to 4): dEN d {J2a2j21±2ajiej-J2ajiaj2Ylaj2ej\_ d A dpk *J2 =  $P*V E « j i E « ? 2 - ( E « i i « i 2 ) 2 J 9P*B daj2\ v 2 (daji + O.J2 E {2a* f daj VT" dpk B-A dpk E W J ' + E ^ E daj2\ daj i dpk ej + aji dej dpk ( daj2 ^ \dpZai2 + a j 2 ap7 j ^ aj2Cj" Ea*a" E( -Q^'i E 2a>i daj i dpk aj2 + ajl da j2 dpk •IB'' dp dpk C. 4 97 dEE _ d ( E a )\ E aj2ei ~ E a i i a j 2 E Q j i e i \ a c dpk dpk B : J E 2 a i ^ E w + E 4 E « w E I (9a,i dey e,- + a , i apt apfc C. 5 d p * /j These equations require the derivatives ^ , -§^-, and . These are simply k < 25 a ? =(0,0,0, o f dp* = (1,0,0,0)J = (0,1,0,0)^  = (0,0,1,0^ = (o,o,o,iy daji dpk daj2 dpk 10 z=i 20 = E d9j (I) dpk dgj (Q dpk k = 26 k = 27 k = 28 k = 29 j = 1,2,3,4 3= 1,2,3,4 Therefore the derivatives of the initial model are dm/,. dEx dpk dpk dEE dpk i = 1,2, ...,10 i = 11,12, ...,20 The derivative of the smallest deviatoric model is a little more involved: 9 m n . d f < -dpk dpk C. 6 C. 7 C. 8 C. 9 98 where / is the vector of the difference between the observed data and those data that would be predicted by a uniform field, and hi is the i * column vector of the kernel matrix: tii = (gi(i),g2(i),g3(i),gi(i)f i = 1,2,..., 19,20. C I O The predicted data are: ev = Gu C. 11 where the elements of u are U i = EN i = 1,2, ..,9,10 = EE i = 11,12, ...,19,20 Continuing with equation C.9: The second term above is C. 12 dpk dpk V ) dp, C. 13 /V - i fV ®±L- (v~lf\T (dgi (*) dg2 (Q dg3 (i) dg4(i)\ V ' dPk ~V JJ ' V dPk ' dpk ' dPk ' dPk J • ^ Developing the first term of equation C.13: V 7 -h^i-iV-^e-e^.h, dpk dpk a r - 1 s i . de\T c far-*^ , ,de ^T dpk dpkj \ dpk p dpki C. 15 Differentiating the predicted data: dev dG _ du •u + G -dpk dpk dpk where | ^ = ^ » = 1 , 2 , 9 , 1 0 ' C. 16 dpk opk dEv = — - i = 11,12,...,19,20 . dpk 99 The derivative of the kernel matrix is simply dG dpk dpk 39i(l) dpk dpk dpk dpk dgt(19) dpk dgi(20) dpk dgt(20) dpk C. 17 It is straightforward to show that the inverse of T is: C.18 C.19 (911922 — 9?2) (933944 — S31) — 9119^944 922(933 944 — 934) —912(933944 — 914) 912923944 —912923934 — 9 1 2 ( 9 3 3 9 4 4 - 9 3 4 ) 9ll(933944 — 934) —911923944 911923934 912923944 —911923944 9 4 4 ( 9 1 1 9 2 2 — 9 1 2 ) —934(911922 — 912) —912S23934 911923934 — 9 3 4 ( 9 1 1 9 2 2 — 9 1 2 ) 933 (gil 922 — 912) _ 911923 where gij = (gi,g~j) - The elements of the inverse must be determined separately. Therefore it is convenient to define a second matrix, removing r — 1 ' s common factor: 7c C. 20 where jc = (911922 - 9 i 2 ) (933944 - 534) _ 511923544 The derivatives of -yc are die -1 x d p k [{911922 - g\2) (933944 - 934) - 911923944]' (933944 - 934) + dgn dg22 _ dgi2 922 + 911-5 2#i2 dpk (911922 - 912) dpk dff33 944 + 933 dpk . #944 C. 21 dpk - 2534 dg 34 dpk , 0 0Q23 , 2 9gu 911 ( 2^23-^—944 + 923 dpk dpk dgn 2 ~dp~;9^u 100 which depend on the derivatives of I\ Those derivatives are straightforward: T OPk opk dpk) = 2 9G G T dpk C. 22 Then the derivatives of the individual elements of the inverse are as follows: U L 11 _ dpk d922 / 2 \ , (^933 , ~X (933944 - 934 j + 922 I ~X 944 + dpk \dpk 933 dg. 44 dpk 2#34 dg 34 dpk 0 ^923 2 ^ 944 Z923~^— 944 — 923 7c + i ^ 7 c dpk C. 23 0 , 1 12 ^912 / 2 \ , /^933 . dgu -T. (933944 - 934j + 912 I -T. 944 + 933"^ Opk y \dpk Opk " 2fir34 • #934 \ dT^1 _ dT$ dpk dpk 9Tji dpk K l c + Vx2dpk r i 3 \dgn (dg23 dg44: ~X = -X 923944 + 912 I -X 944 + 923^ OPk L °Pk \ OPk Opk , + T n dpk 9pk 7c dpk dT^l [ dgu (dg23 , %>4\1 [-Wkg2m*-gi<Wk9M + g2ZWJ\ + T u dPk dpk dpk d?22l [ 5 9 i i ( 2 \ . /^933 , dgu C. 24 C. 25 C. 26 C. 27 101 dpk dpk dpk dgn fdg23 , dg± a 023044 - 9U "5 044 + 023 . Opk \Opk opk d?2Z dpk 'dgii dgn , / dg23 , dgu^ -T. 023034 + #11 -T, 534 + 023"^ . opk \ opk opk , + 2 4 dpk U L 24 dpk dpk dpk dg. 44 dpk (911922 - gj2) + gu ( 7 c d 2 3 1 2 ^ 5 1 2 ) 17c+ r 3 3 dgn , dg22 -z—g22 + 0 1 1 - 5 — OPk OPk idle dpk C. 28 C. 29 C. 30 C. 31 dpk U L 43 #034 / 2 \ , fdgu , dm -dp~k~{9n922 ~ 9 l 2 ) + m ydp~k~922 + 9n~dp~k~ -29l2dVk912)} 7 c + udPk dpk dpk 0T4l \dg33 ( 2 \ . (d9n ~dp-k~ = [dp~k~ { 9 n 9 2 2 " 9 u ) + 9 3 3 ydp~k~922 + 9 1 1 dg: '22 dpk „ dgn\ dgn 2 d# 23 7c + r 4 4 C. 32 C. 33 102 Appendix D Computer Code The following is a list of the programs (all written in Fortran) directly responsible for the results shown in this thesis. Working copies and code for all can be obtained from the author (Butler, 1990). 1. EFIELD: produces the electric fields that would result from using the model shown in figure 2.3. 2. SM_EFIELD: produces the smallest model that would fit the four data at Skeena, Telkwa, Glenannan, and Williston. 3. SDEV_EFLD_FFT: produces a uniform field that most closely fits the data, and the smallest deviation from that model that fits the data. 4. RESOLUTION: analyses the resolution of the four data, given the resistances of the power lines and transformers, by applying delta and unimodular delta criteria. 5. RESOL_SPREAD: similar to RESOLUTION, except that a spread criterion is used. 103 Appendix E Data This appendix contains the input GIC and magnetic field variation data. Fig-ures E. 1 to E.9 display current through the four measured transformers. Current at Skeena is shown by the solid curve, and the dotted, dashed, and long dashed curves represent current at Telkwa, Glenannan, and Williston, respectively. Figures E.10 to E.18 are the results of integrating the output of an induction magnetometer that was located at Smithers, 40 km NW of Telkwa. 104 1 1 1 1 1 1 11 r r, / V —u£ r 1 l i V. > ^-.^ \ ; 1 V. 1 \,1 A w f 'I' V --y| V- r\. u V \, / \ vt -v < — — (I .1: \/ V f; tr; "j 3 3.5 4 4.5 5 Time in hours UT Fig. E.2: Transformer current for event 2. Date of storm: 1984-07-14. 106 10 Fig. E.3: Transformer current for event 3. Date of storm: 1984-07-15. 107 10 f/J a, B < a <D U l - i d >-. <u 6 o C E-i -5 -10 f. \ \* \ V?' \ y — ' — - A * w i l l , h \ • \ , \—r 'Z \ \ 1 \ i i m ft* \ A 1 / 1 I I — / — /I • I / v 4.5 5.5 Time in hours UT Fig. EA: Transformer current for event 4. Date of storm: 1984-07-17. 108 - 1 0 1 0 1 0 . 5 Time in hours UT 1 1 Fig. E.5: Transformer current for event 5. Date of storm: 1984-07-17. 109 20 -20 Time in hours UT Fig. E.6: Transformer current for event 7. Date of storm: 1984-08-01. 110 * */ i t : 1 ' "< i k. 1 N i •J <^ /+t • < \ , y A u iL M m & if, •VV:1 V I f * 9 9.5 10 10.5 11 Time in hours UT Fig. E.7: Transformer current for event 8. Date of storm: 1984-08-01. i l l 1.5 ->-> C cu u 3 0.5 u o c/> C CO E-1 -0.5 1.5 Time in hours UT Fig. E.8: Transformer current for event 9. Date of storm: 1984-08-02. 112 1 r- " / im 1 1 IL. f K \ i . „ ) ' * V A 1 / M V x \ W A , A V I - M i V I 1 1 . • \ * / " V •" :f!t • ! : • r ..„..,»/'• 11 11.5 Time in hours UT 12 g. E.9: Transformer current for event 10. Date of storm: 1984-08-02. 113 Fig. E . l l : Magnetic field variation at Smithers for event 2. 40 \ i A f \ A L J / \ / J \ i / / J I / I, V-V / u Time in Hours UT 114 Fig. E.12: Magnetic field variation at Smithers for event 3. Fig. E.14: Magnetic field variation at Smithers for event 5. Fig. E.16: Magnetic field variation at Smithers for event 8. Magnetic F i e l d ( gammas) Appendix F Smallest Deviatoric Inversion Results  The results of the smallest deviatoric inversion are contained here, in two different forms. Figures F . l to F.9 show the northerly component of E 4 , and the easterly components of E 3 , E 4 , and E6. Figures F.10 to F.18 are vector plots of the magnitude and directions of the fields seen by the five east-west trending power lines (lines 1, 3, 4, 5, and 6). 119 Fig. F . l : Smallest deviatoric fields for event 1. Fig. F.2: Smallest deviatoric fields for event 2. Fig. F.3: Smallest deviatoric fields for event 3. ""'i:k • \ \ . . . . y ' \ \ 8.5 Time in Hours UT 8.5 Time in Hours UT 124 125 126 Fig. F.5: Smallest deviatoric fields for event 5. > -. X " . . X /- ' n 1 # *f~' ":'v \\ / i '•• 10 10.5 Time in Hours UT •/••-:V'i L. .11. f/-«i i*. • J n r •<,.< •:•< V y 10.5 Time in Hours UT 128 Smallest Deviatoric E Field (V/km) t o ( X. ) r ... 1 1 H <> i ' 1 I 1 Fig. F.6: Smallest deviatoric fields for event 7. U —. —- A, ) V Y f Y v J 1 A 5 6 7 Timo In Hours UT t — if' \ , i > • • •S t 4 • 5 6 . 7 Timo in Hours UT 130 Smallest Deviatoric E Field (V/km) o • •< -/ % • < \ < y—--•-Smallest Deviatoric E Field (V/km) o j J •4 < ? > \ r \ Fig. F.7: Smallest deviatoric fields for event 8. r r A W A •I 9.5 10 Time in Hours UT 10.5 \ I. . > i N — L— •ran* $ . .?./< i / 9.5 10 Time in Hours UT 132 Smallest Deviatoric E Field (V/km) Smallest Deviatoric E Field (V/km) Fig. F.8: Smallest deviatoric fields for event 9. Fig. F.9: Smallest deviatoric fields for event 10. t u 1 h rr h / s 'N - A/ i f A A \A A V V \ A V p /V i I vX V s • ^ V V V \ ? i "fl j f r. V V 11.3 11.4 11.5 Tima In Hours UT 11.6 11.7 > ; | 1 f f JU \ i j l ft) VI r \ ft ¥ vv M l ill V ». f V HI II V t ! ' 1 ' 11.5 Time in Hours UT 11.7 136 0.01 -u.ui 11.3 11.5 Timo In Hours UT 11.6 •u.u i 11.5 Time in Hours UT 137 evenLOl, scale - 100 mm/(V/km) : Wed Sep 12 16:55:32 1990 00 500 400 Cu a, 0 K 300 cu o a U g O 200 u CO o C (0 -K> CO 100 - — - W W I I ««e>S2gg 1.5 Time in Hours U.T. Fig. F.10: Vector plot of smallest deviatoric fields from event 1. « Fig. F.12: Vector plot of smallest deviatoric fields from event 3. * * evenL07, scale = 40 mm/(V/km) : Wed Sep 12 17:16:29 1990 4 5 6 7 Time in Hours U.T. Fig. F.15: Vector plot of smallest deviatoric fields from event 7. evcnt-08, scale - 40 mm/(V/km) : Wed Sep 12 17:19:51 1990 500 | 1 1 , , 1 1 , 1 , , 1 , 1 , , , , 1 , , , , , , 0 * 1 1 1 ' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9 9.5 10 10.5 11 Time in Hours U.T. Fig. F.16: Vector plot of smallest deviatoric fields from event 8. 

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