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Numerical weather prediction for electrical transmission lines Campbell, Margaret 2018

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Numerical weather prediction for electrical transmission linesbyMargaret CampbellB.Sc. with specialization in Atmospheric Sciences, University of Alberta, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Atmospheric Science)The University of British Columbia(Vancouver)February 2018c©Margaret Campbell, 2018AbstractJoule heating from electrical currents causes the conductor temperature of a transmission lineto increase. Weather can further heat or cool the line. Wind speed and direction have the largesteffect (CIGRE, 2006), followed by air temperature. Utility companies need to know the maximumcurrent they can transmit without exceeding critical temperature thresholds for transmission linesafety (e.g., excess sag or metallurgical damage).The maximum transmittable electrical current for safe transmission (ampacity) must be esti-mated from wind speed, direction, temperature, insolation, and maximum conductor temperature.Power utilities apply this thermal rating to all powerlines. Traditional thermal rating methods donot monitor the weather surrounding powerlines, but assume relatively constant weather, leadingto either overly conservative or unsafe thermal ratings. Dynamic thermal ratings (DTRs) take intoaccount varying weather conditions in an effort to more realistically represent ampacity variations.To demonstrate the potential of DTR forecasts based on numerical weather prediction (NWP)forecasts to improve powerline safety, increase transmission capacity, and provide power utilities ameans of advanced planning, this thesis 1) evaluates and compares seven bias-corrected, calibratedDTR forecast configurations to two traditional thermal rating methods to determine the most skillfulDTR forecast method as well as to show the usefulness of probabilistic forecasts. 2) Determines rawDTR forecasts along a powerline to assess the degree and cause of spatial DTR forecast variability.The most skillful DTR forecasts start with bias-corrected NWP forecasts from which DTRsare calculated and combined into an ensemble average, which is then bias-corrected again and cal-ibrated. The 1st, 5th, and 10th DTR forecast percentiles are safer than traditional thermal ratingmethods, while the 20th - 50th DTR forecast percentiles allow higher transmission capacity. Ex-tensive temporal and spatial DTR forecast variability along a powerline results from wind speedforecast variability. Based on this research, it is recommended that utility companies use hourlyDTR forecasts at their transmission line to maximize both current and safety.iiLay SummaryThe maximum electrical current safely transmittable through a powerline depends on its tem-perature. If the powerline overheats, it can be damaged or sag too low to the ground. The weatherconditions surrounding a powerline, namely wind and air temperature, have a large impact on itstemperature. If wind speeds are high and air temperatures low, powerline temperatures will belower, and more power can be safely transmitted. By knowing the weather around a powerline, andthe maximum allowed temperature, its maximum electrical current limit can be found. Presently,electrical current limits of powerlines are constant and conservative.This thesis uses wind speed and temperature forecasts to forecast electrical current limits by thehour, in an effort to more realistically represent a powerline’s transmission capacity limit. The aimis to increase the safety and electrical transmission capacity of the power grid, ultimately reducingcosts for the consumer.iiiPrefaceDr. Roland Stull developed the original idea for this project. The methodology of calculatingquasi-static thermal ratings was adopted from Lu (2014). All NWP forecasts used in this thesisare archives of past NWP forecasts from WFRT’s operational SREF. To obtain these forecasts, theauthor adopted NWP forecast retrieval codes originally written by Drs. Henryk Modzelewski andDavid Siuta. Dr. Roland Stull derived the catenary approximation (Appendix B). Dr. Greg Westproduced wind and temperature forecast maps with gridded NWP forecasts used to calculate DTRforecasts along the casestudy powerline (Figs. 5.2-5.6, 5.13-5.17; Chapter 5). The author performedall data retrieval and analysis, and wrote all codes to produce DTRs. Dr. Roland Stull and Dr. GregWest offered guidance, editing, and refinement of procedures.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Ampacity and the evolving electrical grid . . . . . . . . . . . . . . . . . . . . . . 11.2 Weather effects on ampacity and conductor temperature . . . . . . . . . . . . . . . 21.3 Modern practices of power companies . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Dynamic thermal ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Numerical weather prediction forecasts for DTRs . . . . . . . . . . . . . . . . . . 71.6 Thesis objective and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Thermal ratings and case-study site selection . . . . . . . . . . . . . . . . . . . . . . 92.1 Thermal rating equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Maximum conductor temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Catenary approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Wind direction errors in mountainous terrain . . . . . . . . . . . . . . . . . . . . . 142.5 Case-study site and powerline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15v2.6 Thermal rating versus ampacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1 Calculating DTRs with point forecasts . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Calculate DTRs with gridded forecasts at every point along line . . . . . . . . . . 253.3 Calculate STRs and QSTRs with weather observation statistics . . . . . . . . . . . 274 DTR forecasts at weather-observation stations . . . . . . . . . . . . . . . . . . . . . 314.1 Characteristics of DTR forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Deterministic forecast verification . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2.1 Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2.2 MAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.3 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3 Probabilistic forecast verification . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.1 PIT histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.3.2 CDR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.3 CRPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4 Assessing DTR forecast safety and efficiency . . . . . . . . . . . . . . . . . . . . 585 DTR forecast evolution mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.1 Winter example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.2 Summer example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.4 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.1 DTR forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2 How to implement DTR forecasts operationally . . . . . . . . . . . . . . . . . . . 976.3 Limitations and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101A Radial Temperature Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108viB Catenary Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110C Conductor Thermal Expansion Sensitivity Study . . . . . . . . . . . . . . . . . . . . 112D Wind Direction Forecast Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114E Verification metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116viiList of TablesTable 3.1 Metadata for weather stations with observations used to calculate DTR observa-tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Table 3.2 NWP ensemble details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Table 3.3 Members of gridded forecast ensemble. . . . . . . . . . . . . . . . . . . . . . . 26Table 3.4 Characteristics of weather stations whose maximum daily temperatures wereused to calculate monthly QSTRs for the case-study powerline region. . . . . . 28Table 3.5 List of the maximum monthly 85th percentile of maximum daily temperaturesfor each month, and which weather station they were observed at. . . . . . . . . 28Table C.1 The difference between each variable when calculated at 35◦C and -35◦C. . . . 113viiiList of FiguresFigure 2.1 Schematic of the case-study BC Hydro 360 kV transmission lines, 3L2 and 3L5(pink, surrounded by black box). Map courtesy BC Hydro. . . . . . . . . . . . 15Figure 2.2 Schematic of the Bridge River hydroelectric system. Map courtesy BC Hydro. 16Figure 3.1 Weather station locations relative to the case-study powerline (black line). . . . 19Figure 3.2 The six bias-correction techniques applied to DTR forecasts. 1) Applies bias-correction to each ensemble member’s weather forecast, 2) applies bias-correctionto each ensemble member’s DTR forecast, calculated from raw weather fore-casts. Yellow box = EDBW, green box = BEDBW, red box = DEBW, bluebox = BDEBW, purple box = EBDW, and orange box = BEBDW. Acronymsexplained in text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 3.3 Schematic of the different ordering scenarios of the thermal rating (TR) meth-ods when determining the extended hit rate. . . . . . . . . . . . . . . . . . . . 25Figure 3.4 Map showing the weather stations surrounding the case-study powerline withmaximum daily temperature observations used to calculate monthly QSTRs. . 29Figure 4.1 Scatterplots of DTR as a function of bias-corrected wind speed (x-axis) and airtemperature (colouring) over the study period for the following locations [DTR,temperature, and wind observation or forecast method]: (a) Agassiz [observa-tions], (b) Pemberton Base [observations], (c) Agassiz [EDW], (d) PembertonBase [EDW], (e) Agassiz [BEDBW], and (f) Pemberton Base [BEDBW]. Figs.(c - f) are based on NWP forecasts. . . . . . . . . . . . . . . . . . . . . . . . 33Figure 4.2 Timeseries of BEDBW DTR forecasts, DTR observations, QSTRs, and STRsover a 20-month period at (a) Agassiz and (b) Pemberton Base weather stations. 34ixFigure 4.3 Boxplots of the deterministic DTR forecasts and observations at (a) Agassiz,and (b) Pemberton Base (outliers not included). Ideally, the DTR forecast box-plots will have the same width and median as the ampacity boxplot. . . . . . . 36Figure 4.4 Timeseries of all deterministic thermal rating forecast methods and observedampacities at Agassiz weather station for 5-day periods in (a) January, and (b)July. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 4.5 Timeseries of all deterministic thermal rating forecast methods and observedampacities at Pemberton Base weather station for 5-day periods in (a) January,and (b) July. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 4.6 Boxplots of the difference between the 99th and 1st percentiles of all DTRforecasts at (a) Agassiz, and (b) Pemberton Base. . . . . . . . . . . . . . . . . 40Figure 4.7 47-hour forecast of EDW initialized on 30 January 2016 at 0100 PST, at Pem-berton Base weather station. Namely, this plot shows uncalibrated probabilities.In this and the next two figures, the 50th percentile corresponds to the determin-istic forecast. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 4.8 47-hour forecast of BEDBW initialized on 30 January 2016 at 0100 PST, atPemberton Base weather station. Namely, this illustrates a calibrated forecastfor winter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 4.9 47-hour forecast of BEDBW initialized on 30 July 2016 at 0100 PST, at BigSilver 2 weather station. Namely, this illustrates a calibrated forecast for summer. 43Figure 4.10 Boxplots of the biases of each forecast method at each weather station. Boxplotsare color-coded by location. Biases closer to zero are better. . . . . . . . . . . 44Figure 4.11 Forecast bias of each thermal rating method at each forecast hour for Agassiz.Forecast hours start at 0100 PST. Biases closer to zero are better. . . . . . . . . 45Figure 4.12 Timeseries of the bias-corrected DTR forecast methods at each forecast hourfor D’arcy. Biases closer to zero are better. . . . . . . . . . . . . . . . . . . . 46Figure 4.13 Boxplots of the MAEs of all forecast methods. Smaller MAE is better. . . . . 47Figure 4.14 Mean absolute error of all the DTR forecast methods at each forecast hour at(a) Big Silver 2, and (b) D’arcy. Smaller MAEs are better. . . . . . . . . . . . 48Figure 4.15 Boxplots of correlation coefficients of each forecast method at each location.Correlation values closer to +1 are better, and 0 indicates no skill. . . . . . . . 50Figure 4.16 Timeseries of correlation coefficients of all thermal rating methods at each fore-cast hour at Agassiz. Correlation values closer to +1 are better, and 0 indicatesno skill. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51xFigure 4.17 Correlations of 24-hour (a) air temperature and (b) wind speed forecasts withobservations at Agassiz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 4.18 Raw EDW forecast PIT histograms for (a) Agassiz and (b) Big Silver 2. Aflatter distribution is better. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 4.19 Calibrated BEDBW forecast PIT histograms for (a) Agassiz and (b) Big Silver2. A flatter distribution is better. . . . . . . . . . . . . . . . . . . . . . . . . . 54Figure 4.20 Boxplots of the CDRs (dimensionless) of all DTR forecast methods at eachlocation. Smaller values (equal or less than 1) are better. . . . . . . . . . . . . 55Figure 4.21 Boxplots showing the CRPSs for each DTR forecast method at each location.A smaller CRPS is better. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 4.22 Timeseries of CRPS of all DTR forecast methods at each forecast hour at (a)Agassiz and (b) Pemberton Base. Smaller CRPS is better. . . . . . . . . . . . 57Figure 4.23 The miss rate for each DTR forecast configuration for all probabilistic forecastdistribution percentiles, the QSTR and the STR at Big Silver 2. Values closerto the straight line segment are better. . . . . . . . . . . . . . . . . . . . . . . 59Figure 4.24 The difference between the miss rate and its corresponding DTR forecast per-centile, for each calibrated, bias-corrected DTR forecast configuration, at BigSilver 2. Values closer to zero are better. . . . . . . . . . . . . . . . . . . . . . 61Figure 4.25 The difference between the miss rate and the DTR forecast percentile it wascalculated with, for each bias-corrected DTR forecast configuration, for the 1st,5th, and 10th percentiles, at Big Silver 2. Values closer to zero are better. . . . 62Figure 4.26 The extended hit rates of BEDBW forecasts, the QSTR, and the STR for eachforecast percentile, at Big Silver 2. Larger values are better. . . . . . . . . . . 63Figure 4.27 The extended hit rates for each DTR forecast configuration percentile relativeto the QSTR and the STR, at Big Silver 2. Larger values are better. . . . . . . . 65Figure 5.1 NCEP reanalysis of mean sea level pressure, atmospheric thickness, and 500hPageopotential heights valid on 1000 PST on January 18, 2016. . . . . . . . . . . 67Figure 5.2 Wind and air temperature forecasts from the MM5 model, with 4 km horizon-tal resolution, initialized with GFS initial conditions, valid for 0100 PST onJanuary 18, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 5.3 Wind and air temperature forecasts from the MM5 model, with 4 km horizon-tal resolution, initialized with GFS initial conditions, valid for 0600 PST onJanuary 18, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69xiFigure 5.4 Wind and air temperature forecasts from the MM5 model, with 4 km horizon-tal resolution, initialized with GFS initial conditions, valid for 1200 PST onJanuary 18, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 5.5 Wind and air temperature forecasts from the MM5 model, with 4 km horizon-tal resolution, initialized with GFS initial conditions, valid for 1800 PST onJanuary 18, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Figure 5.6 Wind and air temperature forecasts from the MM5 model, with 4 km horizon-tal resolution, initialized with GFS initial conditions, valid for 2300 PST onJanuary 18, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 5.7 DTR forecasts (amperes) along the case-study powerline for 0100 PST initial-ized on January 18, 2016. The magenta diamond with the black outline showsthe location of of the lowest DTR forecast along the transmission line. ThisDTR forecast limits the amount of current allowed for the whole line. . . . . . 73Figure 5.8 DTR forecasts (amperes) along the case-study powerline for 0600 PST initial-ized on January 18, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 5.9 DTR forecasts (amperes) along the case-study powerline for 1200 PST initial-ized on January 18, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 5.10 DTR forecasts (amperes) along the case-study powerline for 1800 PST initial-ized on January 18, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 5.11 DTR forecasts (amperes) along the case-study powerline for 2300 PST initial-ized on January 18, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 5.12 NCEP reanalysis of mean sea level pressure, atmospheric thickness, and 500hPageopotential heights valid on 1000 PST on August 19, 2016. . . . . . . . . . . 78Figure 5.13 Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 0100 PST on August19, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 5.14 Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 0600 PST on August19, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 5.15 Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 1200 PST on August19, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 5.16 Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 1800 PST on August19, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82xiiFigure 5.17 Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 2300 PST on August19, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 5.18 DTR forecasts (amperes) along the case-study powerline for 0100PST, initial-ized on August 19, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 5.19 DTR forecasts (amperes) along the case-study powerline for 0600PST, initial-ized on August 19, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 5.20 DTR forecasts (amperes) along the case-study powerline for 1200PST, initial-ized on August 19, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 5.21 DTR forecasts (amperes) along the case-study powerline for 1800PST, initial-ized on August 19, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Figure 5.22 DTR forecasts (amperes) along the case-study powerline for 2300PST, initial-ized on August 19, 2016. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 5.23 Boxplot of the lowest DTR forecast along the powerline at each forecast hour,on each forecast initialization day in August 2016 and January 2016. . . . . . . 90Figure 5.24 Hourly minimum DTR forecasts along the powerline, at each forecast hour oneach forecast initialization day in (a) January and (b) August 2016. The winterSTR and QSTR for January is plotted for January 2016 (a), and the summerSTR and QSTR for August is plotted for August 2016 (b). . . . . . . . . . . . 92Figure A.1 Timeseries of conductor core and surface temperature (◦C) over the 20-monthstudy period at Pemberton Base. . . . . . . . . . . . . . . . . . . . . . . . . . 109Figure B.1 Schematic and notation of 3-segment catenary. . . . . . . . . . . . . . . . . . 111Figure D.1 Scatter plot of the ensemble average of hourly raw wind direction forecastsversus hourly wind direction observations at D’arcy for the 20-month studyperiod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure E.1 Cumulative distribution functions of three forecasts (labelled 1 - 3) and the ob-served step-function (thick black line) of variable y (Wilks, 2006) . . . . . . . 117xiiiAcknowledgmentsI must first express my endless thanks to my research supervisor Prof. Roland Stull who pro-vided the means and funding for my thesis project, and whose creativity, flexibility, and supportallowed me to grow and prosper as a graduate student. I also wish to thank my committee members,Drs. Doug McCollor and Stephen Mitchell for their inputs and feedback.I am deeply grateful to all members of the Weather Forecast Research Team whose insight,knowledge, and friendship allowed the success of my work. I would especially like to thank Dr.Greg West who went above and beyond to provide any and all resources, support, teachings, andadvice I may have needed, on top of pushing me to strive for academic excellence. I thank PedroOdon who always made time to discuss mathematical derivations and statistical analysis techniqueswith me. Many thanks go to former team member Dr. David Siuta for his patience, expertise, andlimitless time spent explaining NWP concepts and forecast verification procedures, helping withany small problem I was having, and, most importantly, his camaraderie as a cubicle neighbor. Iam very grateful to Dr. Rosie Howard for her unwavering friendship, support, and assistance withany aspect of my project whenever asked. I would like to thank Tim Chui, Roland Schigas, and Dr.Henryk Modzelewski for their technical support regarding computing and NWP resources.I wish to acknowledge the funding for this research provided by Mitacs, BC Hydro, and theNational Sciences and Engineering Research Council of Canada, as well as the computer supportfrom the Weather Forecast Research Team.I am also grateful for my extended family and many close friends for whom I am so lucky tohave encouraging and supporting me.Finally, I would like to thank my mother, Kathy, who taught me to pursue anything I waspassionate about, who made me believe I was capable of anything I put my mind to, and whose loveand support I credit for everything I have achieved.xivChapter 1Introduction1.1 Ampacity and the evolving electrical gridThe ampacity of a powerline is the maximum current that the conductor can carry under theprevailing weather conditions before that powerline exceeds its maximum allowed conductor tem-perature (Billinton and Koval, 1970; Deb, 2000; Phillips et al., 2014; Arroyo et al., 2015; Hoseket al., 2011). The maximum conductor temperature of a powerline is the maximum temperature theconductor can reach before sustaining annealing damage, or before thermal expansion (temporaryand/or permanent) causes it to sag below vertical ground clearance limits (Douglass and Edris, 1996;Reding, 1994). Maximum allowed conductor temperatures of 75◦C - 150◦C are typical (Lawry andFitzgerald, 2007; IEEE, 2013a,b). Annealing is a metallurgical process where the internal crystalstructure of the conductor reduces due to high temperature, causing loss of strength and increasedelectrical resistance of the conductor (IEEE, 2013a). The susceptibility of the conductor to anneal-ing and resulting degradation depends on the metals used − aluminum alloy and steel-reinforcedaluminum are most common (IEEE, 2013a; Thrash Jr, 2018).Some transmission lines frequently and continuously carry currents close to their ampacitybecause of (1) new private energy-generation sources that are often in remote locations, whereonly smaller transmission lines are available to transmit power (Lawry and Fitzgerald, 2007; Schellet al., 2011); (2) increased power demand (Cradden and Harrison, 2013; BCHydro, 2017); and(3) increased renewable energy and distributed generation resource integration (Natural ResourcesCanada, 2017b; The Canadian Trade Commissioner Service, 2017). Wind power is a challengefor power operators: there must be enough capacity in the connecting powerlines to transmit allthe possible power generated, as wind power typically cannot be stored (Siuta et al., 2017). Solarpower generation is also a challenge as it is coincident with high incoming solar radiation that can1heat powerlines and limit power transmission (Denholm et al., 2016), as discussed in section 1.2.Transmission lines that have lost strength due to annealing can have a significantly decreasedlifetime, necessitating extremely costly replacements (Heckenbergerova et al., 2013). Taking fulleradvantage of existing transmission capacity will reduce the need to build new lines, and potentiallyreduce extremely costly repairs and replacements.Smart grid strategies and technologies are being put in place to increase the transmission ca-pacity of existing transmission lines, as well as to generate and move power more efficiently, ef-fectively increasing the efficiency of the electrical grid as a whole (Natural Resources Canada,2016; Office of Electricity Delivery & Energy Reliability, 2018b). Smart grid technologies includedemand response (Office of Electricity Delivery & Energy Reliability, 2018a), autonomous mi-crogrids disconnected from the large scale electrical grid (Natural Resources Canada, 2012), anddistributed generation. Demand response (or demand side management) means power consumptionis adjusted, either by the consumer, the utility, or third-party management software in an effort toreduce demand peaks, and fill in demand lows that strain the limits of the grid (Strbac, 2008). Dis-tributed generation refers to having multiple electrical generation sources that are geographicallydispersed connected directly to the distribution network (low voltage power lines as opposed to thelarge-scale, high-voltage transmission network; Pepermans et al. (2005)).Smart grid technologies are not the only available methods to increase transmission capacity. Inrecent years, dynamic thermal rating analyses of existing transmission lines are also being conductedto better anticipate and forecast ampacity (Ampacimon, 2015; Nexans, 2017; USi Overhead T&DSolutions, 2016; Deb, 2000; Douglass et al., 2000; Ciniglio and Deb, 2004; Cloet et al., 2010;Nguyen et al., 2013; Schell et al., 2012). Ampacity prediction is the focus of this thesis.1.2 Weather effects on ampacity and conductor temperatureDue to the ever changing environment (weather, load, generation), the associated changingampacities, and the power utilities’ desire avoid annealing and sag, power utilities must estimate theampacity of their lines. This thermal rating, the maximum current allowed in a powerline so theline does not exceed its maximum allowed conductor temperature, is applied to each powerline bythe power utility to give a safe upper bound on transmittable electrical current.Factors affecting conductor temperature are: 1) conductor material properties (primarily elec-trical conductivity), 2) conductor diameter, 3) conductor surface condition (primarily IR emissivityand solar-radiation absorptivity), 4) weather conditions, and 5) electrical current that causes Jouleheating (IEEE, 2013b). All factors except weather conditions are constant or controlled by opera-tors, making the weather the most variable factor in determining a safe amount of current to transmit2through the line. The weather conditions surrounding a powerline at any given time have either anet heating or cooling effect on the conductor surface temperature, changing the radial temperaturegradient inside the conductor, and thus changing the conductor temperature as a whole (Clairmontet al., 2012). Conductors have typically been designed to operate below their maximum allowedtemperature, at, or within 5 to 15 degrees of ambient air temperature (IEEE, 2013a; Loudon et al.,2017).Weather variables that influence conductor temperature include: air temperature, wind speed,wind direction, insolation, and precipitation. Wind speed and direction have the largest effect(CIGRE, 2006), followed by temperature. If high wind speeds are blowing perpendicular to theconductor, temperatures are low, and precipitation is present, then conductor temperature is low.Conversely, if low wind speeds are blowing parallel to the conductor, temperatures are high, andinsolation is strong, then conductor temperature is high. For example, on days that are sunny, hot,and calm, a conductor will heat up much faster with a given current than on a cloudy, rainy, cool,and windy day. This means that when prevailing weather conditions are conducive to cooling,more power can be transmitted through a powerline safely. When the weather conditions are lessconducive to cooling, a powerline cannot transmit as much power without risking exceeding themaximum allowed conductor temperature.The North American Institute of Electrical and Electronics Engineers (IEEE) and the Interna-tional Council for Large Electric Systems (CIGRE) have standards that present a set of equationsrelating the conductor characteristics and weather conditions to the conductor temperature and elec-trical current (IEEE, 2013b; CIGRE, 2002). Using either one of these standards, when the conductorcharacteristics, maximum conductor temperature and weather conditions are known, a thermal rat-ing can be calculated. The standards are very similar, with a few exceptions, and have been foundto give results with negligible differences (Arroyo et al., 2015; Castro et al., 2017). Neither stan-dard includes the cooling effects of precipitation because of the difficulty in determining its localintensity and the number of factors needed to model it (Pytlak et al., 2009; Michiorri et al., 2015).Repairing or replacing conductors is difficult and extremely expensive (on the order of $1.3M/kmCAD; Alberta Energy (2009)), therefore power utilities have typically been very conservative, in thesense of restricting currents so as to never come close to damaging conductors. Similarly, gettingapproval to build new powerlines is increasingly difficult due to environmental concerns (Hecken-bergerova et al., 2013). These are strong motivations for a more cost-effective method for maximiz-ing usage of existing powerlines while still minimizing the chance of exceeding ampacity limits.31.3 Modern practices of power companiesHistorically, the common practice for power utilities has been to apply a static thermal rating(STR) to their powerlines (Lopez and Coullon, 2013). An STR is calculated with either the IEEE orCIGRE standard, assuming constant, conservative weather conditions that are the least conduciveto line cooling (i.e., “worst-case” conditions). Usually one STR is calculated for summer, one forwinter (Deb, 2000). Typical “worst-case” weather conditions assumed are high temperatures for thatseason (usually 10◦C for winter, 40◦C for summer), low wind (0.6 m/s) speed blowing perpendicularto the line, and full solar radiation (IEEE, 2013b; Lu, 2014; Douglass, 1988; Reding, 1994). Thegoal of using these “worst-case” weather condition inputs is to ensure that the resulting STR willnot allow the line to overheat and exceed its true ampacity, no matter what the weather conditions.STRs are used because they are easy to implement and they ensure a high level of safety.Weather conditions and conductor temperatures do not need to be monitored, therefore sensors donot need to be installed on the powerlines (Deb, 2000). The benefits of an STR are compromisedby the typical assumption of perpendicular wind direction relative to a conductor, as perpendicularwind flow, regardless of wind speed, always results in a maximum cooling of a conductor (Douglass,1988). Therefore, assuming perpendicular wind is assuming unrealistically high cooling of the line,meaning STRs are not as conservative and low risk as intended.STRs are very common and many power companies are content with the assumption that am-pacity is constant in time and along the transmission line, when in reality weather conditions (andtherefore ampacity) are neither constant in time nor space (Deb, 2000). Due to the fixed valuesof low wind speeds and high air temperatures, STRs greatly underestimate the true ampacity oftransmission lines virtually all the time, causing the underutilization of transmission line capacity(Lawry and Fitzgerald, 2007; Lopez and Coullon, 2013; Cloet et al., 2010). A conservative STRfurther reduces transmission capacity as operational electrical loads are typically much less than theoperational thermal rating (Douglass and Edris, 1996). Along with excessive under-rating of lines,STRs can result in higher energy prices, excessive curtailment of renewable energy generation, andcan discourage development of renewable energy and distributed generation (Lopez and Coullon,2013).In an effort to increase transmission system efficiency, several studies have developed thermalratings that take into account local cooling effects by determining statistical probabilities of local,historical weather. In Billinton and Koval (1970), the joint probabilities of all possible weather con-dition combinations and weather and electrical-current combinations were calculated from weatherdata over one year. Reding (1994) derived a probabilistic model by accounting for probabilitiesof exceedances of vertical clearance limit, maximum conductor temperature, and thermal rating,4given a weather probability model determined from air temperature and wind speed data collectedin the Pacific Northwest of the United States. Heckenbergerova et al. (2013) determined a “typi-cal meteorological year” from weather data spanning a time period much longer than a year fromwhich the cumulative distribution functions of ampacity were determined. In British Columbia, Lu(2014) calculated thermal ratings with monthly averages of air temperature over 10 years, whilekeeping wind speed and direction constant. The timescale of probabilistic thermal ratings based onhistorical weather data vary; they can be applied for certain times of day, or on a weekly or monthlybasis. These thermal ratings have been called probabilistic STRs (Billinton and Koval, 1970; Red-ing, 1994; Heckenbergerova et al., 2013), as well as quasi-dynamic thermal ratings (Lu, 2014). Thisthesis will refer to thermal ratings using statistical averages of historical weather conditions overvarying time periods as quasi-static thermal ratings (QSTRs).Heckenbergerova et al. (2013) found that QSTRs cause the maximum allowed conductor tem-perature to be exceeded more often than typical STRs. This is because the QSTRs considered inthat study were slightly less conservative, resulting in higher ratings making for more efficient op-erations, but more exceedances. Even slightly less conservative conditions assumed in an STR canresult in alarmingly large increases in risk (Douglass, 1988).While there are advantages to using STRs and QSTRs, (namely, they are simple and safe) theresulting costly transmission system inefficiencies provide a large incentive to investigate alternativethermal rating methods. STRs and QSTRs do not reflect day-to-day and hour-to-hour variations ofweather surrounding powerlines, thus, leading to underutilization of lines during some hours, andputingt conductors at risk during other hours. Therefore, to take full advantage of the transmissioncapacity of the powerline, as well as to ensure the powerline does not exceed its true ampacity, thereis a need for a thermal rating that accounts for the higher frequency variations of weather.1.4 Dynamic thermal ratingsDynamic thermal ratings (DTRs) are calculated using real-time observed or forecasted conduc-tor temperature dependant variables to capture the true variability of ampacity at high frequency.They allow power companies to take better advantage of the transmission capacity of the lines,while also better representing when the lines are at risk of exceeding thermal limits.Methods for determining real-time DTRs vary widely. Monitoring equipment, often placed onthe transmission line, estimates the true line ampacity in real time via weather-based, temperature-based, or sag and tension-based means (Douglass et al., 2000; Deb, 2000). Weather-based real-timeDTRs input weather observations to the IEEE (IEEE, 2013b) or CIGRE (CIGRE, 2002) standardequations, while the other methods measure conductor temperature, current, voltage, sag or tension5(USi Overhead T&D Solutions, 2016; Ampacimon, 2015; Nexans, 2017). These measurements,in combination with the known current in the line, are used to determine the real-time DTR. Forweather-based real-time DTRs to be successful, it is best to have weather stations installed wherethey will best measure the weather affecting the line (i.e. on the transmission line; Douglass et al.(2000); Ciniglio and Deb (2004)). Weather-based real-time DTRs are usually less expensive thanother methods (Gentle et al., 2012).Multiple studies have shown that weather-based DTRs can increase line capacity by 5% to200% over traditional STRs (Douglass and Edris, 1996; Lawry and Fitzgerald, 2007; Schell et al.,2011, 2012; Nguyen et al., 2013). Seppa (2007) found that DTRs exceed STRs by 10% to 15% for95% of the time, and by 20% to 25% for 85% of the time. Cloet et al. (2010) found an increase intransmission capacity over STRs 99% of the time with an “ampacimon” sensor.Taking advantage of the increase in ampacity on short time scales, while respecting the ther-mal limit, reduces the need for frequent repairs or replacement. Widespread implementation ofDTR systems has been recommended to maximize cost savings as much as possible (Douglass andEdris, 1996). Deb (2000) found that DTRs can prolong the life of conductors by 10 years, deferringinvestments in the construction of new transmission lines by at least the same amount of time. Insummary, Douglass et al. (2000) identifies the two primary benefits of DTRs: (1) increased trans-mission capacity, and (2) increased reliability. Both of these decrease costs through: (1) increasingsystem efficiency, (2) reducing maintenance costs, and (3) reducing the need for physically increas-ing system capacity (Michiorri et al., 2015). Secondary benefits include increased renewable energypenetration, reduced greenhouse gas emissions from carbon-based generation, and increased adapt-ability of the grid to climate change (Pytlak et al., 2011; Cradden and Harrison, 2013; Michiorriet al., 2015).By nature, DTRs regularly bring a conductor closer to its true ampacity and are more com-plicated to implement and maintain. There is a “common knowledge” that they are riskier, moreexpensive, and more work (Gentle et al., 2012), although recent studies, including this one, disputethe claim of increased risk. Current practices of power operators would have to be greatly modifiedin order to operationally implement DTRs (Cloet et al., 2010). Heckenbergerova et al. (2013) statesDTRs have not been widely adopted because they require increased equipment and communicationcosts, although Seppa (2007) says that the implementation costs of DTRs is a fraction of the cost ofthe alternative of electrical equipment upgrades. Gentle et al. (2012) states the following problemswith sensor-based DTRs: (1) sensors installed on a powerline take point measurements and do notrepresent the conditions at every span, (2) installing more sensors could require outages, which areexpensive and difficult to organize, and (3) tension-based sensors give only average line tempera-ture and sag, which may not be representative of local at-risk spots. The Empire State Electrical6Energy Research Corporation found that DTRs are easy to implement if measurement of equipmenttemperature is not required (Douglass and Edris, 1996).Despite the drawbacks, the benefits provide more than enough incentive to adapt and improveDTR methods. Real-time DTR systems are currently in place at a few locations worldwide, withmany more studies and evaluations underway. The next logical step, is to extend these real-timeDTRs into the future using weather forecast data.1.5 Numerical weather prediction forecasts for DTRsNumerical weather prediction (NWP) has become increasingly skillful over the past decades(Bauer et al., 2015; Stern and Davidson, 2015). Products derived from NWP output have becomemore accurate and sophisticated. The economic value of weather forecast information is well-established (e.g., Murphy (1976, 1977); Lee and Lee (2007); Matte et al. (2017)). Deterministicand probabilistic forecast information has been proven useful for energy applications, and proba-bilistic information has generally been found to provide greater utility to a broader range of users(Milligan et al., 1995; Mylne, 2002; McCollor and Stull, 2008). Likewise, NWP output can pro-vide economic (and safety) benefits when used as input to DTR forecasts to optimize transmissionefficiency (Michiorri et al., 2015).Energy market trading is becoming increasingly competitive. Utilities and power producersnow arrange the purchase and sale of electricity in advance, which requires knowing ampacityin advance as well. Knowing future ampacities allows power system planners to accommodatepower demands hours and days ahead, plan for maintenance, accommodate outages, and plan marketoperations (Deb, 2000; Lopez and Coullon, 2013; Ciniglio and Deb, 2004; Douglass, 1988; Hoseket al., 2011; Ringelband et al., 2013; Cloet et al., 2010; Michiorri et al., 2015). It is important thatthe powerline operator can plan ahead so as to avoid the need of load shedding or quick startups oflocal generation (Douglass, 1988), both of which incur large costs or fees. DTR forecasts can enablethe purchase of more economical electricity from remote locations, as well as to help schedulepowerline needs from multiple power producers (Deb, 2000). Cloet et al. (2010) state that DTRforecasts are more important than real-time DTRs, and Lopez and Coullon (2013) state that DTRforecasts are needed to meet the needs of the grid and fully realize the potential of DTRs. Ciniglioand Deb (2004) recommend using gridded NWP output to fill in thermal ratings at locations alonga transmission line where weather observations are not available.Using NWP output to calculate DTR forecasts is becoming more prevalent (Ampacimon, 2015;Schell et al., 2011; Cloet et al., 2010; Nguyen et al., 2013; Lopez and Coullon, 2013; Ringelbandet al., 2013). Lopez and Coullon (2013) discuss a detailed approach of how to integrate NWP-based7DTR forecasts into power system operations. If ampacity and/or on-transmission-line weather ob-servations are available, statistical post-processing can be applied to raw NWP output to improveforecasts, and to create probabilistic forecasts for risk assessment. There is no industrial standardof risk tolerance for transmission lines − it is up to the discrepancy of the power utility. For maxi-mum safety of transmission lines, risk tolerances are commonly between 1 - 5% (Heckenbergerovaet al., 2013; Zhang et al., 2008; CIGRE, 2006; Hosek et al., 2011; Reding, 1994). Ampacimon(Ampacimon, 2015), a company that started from a European DTR research program, has recentlystarted selling DTR forecasts created from NWP forecasts with machine-learning based statisticalpost-processing applied. Nguyen et al. (2013) describe a probabilistic DTR forecasting system im-plemented in Belgium that uses NWP forecasts, on-line Ampacimon sensors, and post-processingto give DTR forecasts with only 2% risk. Ringelband et al. (2013) uses an ensemble of weatherforecasts to produce skillful probabilistic ampacity forecasts, and finds the 1st percentile forecastscan increase transmission capacity.The field of DTR forecasting, however, is still in its nascent stages, with more research neededinto best methodologies and viability in geographically-disparate regions.1.6 Thesis objective and outlineThis thesis will use ensemble NWP forecasts to create probabilistic DTR forecasts for a high-voltage transmission line in British Columbia (BC). The DTR forecasts will be compared to STRsand QSTRs. The goal is to show that DTR forecasts calculated with ensemble NWP forecasts canallow for increased electrical transmission capacity over conventional thermal rating methods inBC, while maintaining a high level of safety.Chapter 2 explains the equations, assumptions, and input variables used to calculate thermalratings in this thesis. Chapter 3 discusses the methods and datasets used.The objective of this thesis will be demonstrated through two paths. First, in Chapter 4, theskill of deterministic and probabilistic DTR forecasts calculated with NWP output is determinedat key weather station locations in order to show the skill of ensemble NWP forecasts to calculateDTRs.Next, in Chapter 5, gridded NWP forecasts are horizontally and vertically interpolated to thepowerline to show that DTR forecasts can be calculated at high spatial resolution along the power-line, with the lowest DTR line segment forecast applied as a limit for the whole line. DTR forecastsare compared with STRs and QSTRs in chapters 4 and 5.Discussion and conclusions are presented in chapter 6.8Chapter 2Thermal ratings and case-study siteselection2.1 Thermal rating equationsDTRs, QSTRs, and STRs will be calculated using the IEEE Standard 738 (IEEE, 2013b),hereby referred to as the IEEE 738. The IEEE 738 presents a set of equations that relates air tem-perature, wind speed, wind direction, solar radiation, electrical current, and conductor temperature.Equations for steady-state and transient thermal ratings are given. Steady-state thermal ratings rep-resent the state of the conductor when the electrical current in the conductor and meteorologicalforcings are constant or slowly varying and, therefore, when the conductor temperature is relativelyconstant. Although neither meteorological conditions nor current is constant in the real world, it isstandard practice for power utilities to apply steady-state thermal ratings, as opposed to transientthermal ratings, to powerlines. Therefore, this project will calculate steady-state thermal ratings.A steady-state thermal rating is described by an energy-balance equation (per unit length alongthe transmission line) relating the environmental conditions surrounding the powerline with thejoule heating of the line itself:qc+qr = qs+ I2R(Tavg) (2.1)where qc is the convective cooling (W/m), qr is radiative cooling (W/m), qs is solar heating(W/m), and I2R(Tavg) is the joule heating (W/m) made up of current, I (amperes), and resistanceper unit length at the average conductor temperature, R(Tavg) (Ω/m). Hereafter, steady-state thermalratings will be referred to simply as thermal ratings.9The resistance of the line at the average conductor temperature is determined by linear inter-polation:R(Tavg) ={R(Thigh)−R(Tlow)Thigh−Tlow}× (Tavg−Tlow)+R(Tlow) (2.2)where R(Tlow) and R(Thigh) are the resistances per unit length (Ω/m) of a specific conductor ata low temperature, Tlow (◦C), and a high temperature, Thigh (◦C), respectively. For a Drake ACSR(case-study powerline conductor; CME Wire and Cable (2016)), R(Tlow) is 7.283×10−5 (Ω/m) andR(Thigh) is 8.688×10−5 (Ω/m) for a Tlow of 25◦C, and a Thigh of 75◦C, respectively. These valuesare used for all thermal ratings calculated in this thesis.The convective cooling term, qc, takes into account the cooling effects that wind has on the con-ductor. For any wind speed, the IEEE 738 recommends choosing the higher value out of equations(2.3) and (2.4) as qc:qc1 = Kangle×[1.01+1.35×N0.52Re]× k f × (Ts−Ta) (2.3)qc2 = Kangle×0.754×N0.6Re × k f × (Ts−Ta) (2.4)where Kangle is a wind direction factor determined from the angle between the wind directionand the conductor, NRe is the Reynold’s number, k f is the thermal conductivity of air (W/m ◦C), Tsis the surface temperature of the conductor, and Ta is the ambient air temperature (◦C).Kangle is given by:Kangle = 1.194− cos(φ)+0.194cos(2φ)+0.368sin(2φ) (2.5)where φ is the angle between the wind and the conductor (◦). Kangle is only valid for values ofφ between 0◦ and 90◦. Therefore, each φ is converted to its equivalent between 0◦ and 90◦.The Reynold’s number is calculated as:NRe =D0×ρ f ×Vwµ f(2.6)where D0 is the outer diameter of the conductor (m), ρ f is the air density (kg/m3), Vw is thewind speed (m/s), and µ f is the dynamic viscosity of air (kg/m s).For wind speeds of 0.2 m/s and lower, calm conditions are assumed. In this case, qc is set asnatural convection, qcn, caused by rising air that was warmed by the hot wire:10qcn = 3.645×ρ0.5f ×D0.750 × (Ts−Ta)1.25 (2.7)The radiative cooling term, qr, derived from the Stephens-Boltzmann law, considers the netcooling effect due to the long-wave radiation emitted and absorbed by the conductor:qr = 17.8×D0× ε×[(Ts+273100)4−(Ta+273100)4](2.8)where ε is the emissivity of the conductor, set to 0.8 (IEEE, 2013b).The solar heating term, qs, is determined from the latitude and elevation above sea level of theconductor, time of day and time of year:qs = α×Qse× sinθ ×A′ (2.9)where α is the absorptivity of the conductor, set to 0.8 (IEEE, 2013b), Qse is the total heat fluxdensity at any elevation above sea level (W/m2), θ is the angle of the sun relative to the conductor(◦), and A′ is the projected area of the conductor (m2/linear m). qs assumes a clear sky, and does nottake into account any effects due to clouds.θ is given by:θ = arccos[cos(Hc)× cos(Zc−Zl)](2.10)where Hc is the altitude of the sun (degrees), Zc is the azimuth of the sun (degrees), and Zl is theorientation of the powerline (degrees, bearing from North). Hc and Zc depend on the latitude, hourof the day, and day of the year. Qse depends on Hc, one of two sets of constants depending if the airsurrounding the line is deemed clear or industrial (i.e., polluted), and the elevation above sea level ofthe conductor. The case study powerline is in rural British Columbia, far from a large metropolitancenter, therefore the air surrounding the line is deemed clear. To account for increased heat flux atelevations above sea level, a factor, m, is applied when calculating Qse. The IEEE 738 gives valuesof m for four elevations above sea level: m = 1.00, 1.10, 1.19, and 1.28 when the elevation abovesea level of the conductor is 0, 1000, 2000, and 4000 meters, respectively. To obtain the values ofm at elevations in between those provided a polynomial line was fitted to the m values provided.There are two problems associated with equation (2.10) as given in the IEEE 738: 1) it as-sumes flat terrain and equal-height transmission towers such that a straight line drawn betweentwo adjacent tower tops is perfectly horizontal, and 2) it assumes no sag in the powerline, so thatthe powerline is also perfectly horizontal. In section 2.3, a corrected version of equation (2.10) isproposed that is relevant to mountainous terrain, such as that found in the case-study area in BC.11During the night, when solar heating is zero, the equation for Hc as well as equation (2.10) canbe negative, which results in negative values of qs. This is unphysical, therefore qs is set equal tozero when this occurs.To calculate a thermal rating, equation (2.1) is rearranged to solve for current:I =√qc+qr−qsR(Tavg)(2.11)The thermal rating, I, is the maximum current that can be in the line given the specified maxi-mum safe conductor temperature.2.2 Maximum conductor temperatureThere is no agreed upon maximum safe conductor temperature for each type of conductor,therefore it is up to the discretion of the power utilities to set a maximum conductor temperature fortheir powerlines depending on their judgement, needs, and practice (Heckenbergerova et al., 2013;IEEE, 2013b). Over the past several decades, the range of maximum conductor temperatures usedhas increased from between 50◦C to 75◦C, to between 95◦C to 150◦C (IEEE, 2013b; Clairmontet al., 2012) as new alloys have been used in the transmission-line wires.Maximum conductor temperatures prevalent in the industry today are high enough that thetemperature difference between the conductor core and surface cannot be ignored. The IEEE 738takes into consideration the thermal gradient between the conductor core and the conductor surface,whereas previous IEEE 738 editions did not. The IEEE 738 specifies that if the temperature dif-ference is greater than 10◦C, then closer attention should be paid to the high temperature effectsof the conductor core on the whole conductor. Therefore, while in previous editions of the IEEE738 only one conductor temperature was specified to calculate qc, qr, and R, the updated IEEE 738specifies that conductor surface temperature, Ts, must be used to calculate qc and qr, and the averagetemperature of the conductor surface and core, Tavg, is used to calculate the resistance, R(Tavg).The equation relating the conductor surface temperature and conductor core temperature pre-sented by the IEEE 738 is:Tcore−Ts = I2R(Tavg)2pikth×[12− D2coreD20−D2core×(lnD0Dcore)](2.12)where Dcore is the diameter of the conductor core (mm or m), D0 is the diameter of the entireconductor (mm or m), kth is the effective radial thermal conductivity (W/m ◦C), Ts is the conductorsurface temperature (◦C), and Tcore is the conductor core temperature (◦C). Both Dcore and D0 are12known for a Drake ACSR (case-study powerline conductor; CME Wire and Cable (2016)) and areused for all thermal ratings calculated in this thesis.Experimental studies have been conducted to determine appropriate values of kth depending onthe conductor type and diameter (Clairmont et al., 2012). For this study, kth will be set to a value of1.0 (W/m ◦C), as deemed appropriate for ACSR (aluminum conductor-steel reinforced) conductorswhen there is little to no tensile strength in the aluminum strands of the conductor, i.e. the conductoris operating at high temperatures (IEEE, 2013b).To account for the radial thermal gradient, this study will specify a maximum safe average con-ductor temperature the conductor can reach before sustaining damage or sagging too low. Equation(2.12) can be reduced to:Ts = Tavg− (qc+qr−qs)× 0.5−X4pikth (2.13)where qc and qr depend on Ts, and X is defined as:X =D2coreD20−D2core×(lnD0Dcore)(2.14)The Newton-Raphson numerical method (Press et al., 1992) is used to determine Ts by choosinga constant maximum Tavg, a first guess of Ts, and a precision tolerance of 10−6. The first guess of Tsis chosen as 95◦C. The maximum safe Tavg for this study is set to 100◦C − a temperature within therange of commonly used maximum conductor temperatures (Douglass, 1988; IEEE, 2013b), andone that is used in sample calculations (IEEE, 2013b). These values are used for all thermal ratingscalculated in this thesis. Appendix A presents an example of the resulting Ts and Tcore using thismethod.2.3 Catenary approximationPresently, the IEEE 738 assumes a conductor is a horizontal straight line; it does not take intoaccount the true catenary shape of a conductor hanging between two towers on sloping terrain.The angle between incoming solar radiation and the conductor, and between wind direction and theconductor, depends on the elevation angle of the conductor. Therefore, the straight line conductorassumption made by the IEEE 738 leads to inaccurate solar heating and convective cooling. Thisis especially true in mountainous terrain, such as BC, where sloped transmission line segments arecommon.This thesis improves the IEEE 738 equations by including an approximation to the catenaryshape of the conductor when calculating thermal ratings along the full extent of the case study pow-13erline (Chapter 5). For each intra-tower span the conductor is split into three straight segments ofequal length, from which the elevation angles of each segment are determined. The approximationto the catenary shape is described in full in Appendix B. For simplicity, zero thermal expansion ofthe conductor is assumed. A sensitivity study showing negligible thermal expansion of the conduc-tor over a range of operating temperatures is detailed in Appendix C.Thermal ratings are calculated at the midpoint of each segment of the catenary between everypair of towers. Latitudes and longitudes of all towers of the case study powerline are known; fromthese coordinates the bearings from North of each span between towers were determined, and thecoordinates of the midpoint of each segment were found. The latitude, longitude coordinates ofeach transmission tower and each segment midpoint were overlayed on the Canadian Digital Eleva-tion Model (CDEM, Natural Resources Canada (2017a)) in QGIS (QGIS, 2015) to extract terrainelevations (m).Equation (2.10) is augmented to include the effects of a non-horizontal segment of a powerline:θ = arccos[cos(Hc)× cos(Zc−Zl)× cos(Hp)+ sin(Hc)× sin(Hp)](2.15)where Hp is the elevation angle of the conductor (degrees), corresponding to any one of α , β ,or γ found by methods described in Appendix B.To include the catenary effect on convective cooling, the angle between the wind direction andthe conductor is calculated with:φ = arccos(cos(Hw)× cos(Zw−Zl)× cos(Hp)+ sin(Hw)× sin(Hp)) (2.16)where φ is the angle between the wind and the conductor (degrees), Hw is the elevation angleof the wind relative to the terrain (degrees), and Zw is the azimuth angle of the wind (degrees).Wind direction has both vertical and horizontal components. For simplicity, we assume the verticalcomponent is terrain following. Therefore, Hw is always zero and equation (2.16) reduces to:φ = arccos(cos(Zw−Zl)× cos(Hp)) (2.17)2.4 Wind direction errors in mountainous terrainTypically, wind direction is used as an input for thermal ratings. However, when forecasted-versus-observed wind direction was analyzed for the study weather stations in mountainous BC,very poor correlation between wind direction forecasts and observations was found (details in Ap-pendix D).14Due to these results, only hourly air temperature and wind speed forecasts and observationsare used as time-varying inputs to calculate DTR forecasts and ampacity estimates (section 3.1), re-spectively. Wind direction forecasts and observations are neglected, and a constant wind direction of30◦ relative to the powerline for QSTRs and DTRs is used as input to calculate DTR forecasts. Thevalue of 30◦ was chosen so as to include some cooling due to wind direction, which is a conservativemiddle ground between parallel (minimum cooling) and perpendicular (maximum cooling). Thisis line with Heckenbergerova et al. (2013) and Michiorri et al. (2015), both of which recommendusing a small wind direction angle relative to the powerline in order to be conservative.2.5 Case-study site and powerlineThis case study will focus on two BC Hydro 360 kV transmission lines, 3L2 and 3L5 (Fig.2.1). Line 3L5 begins at the Rosedale substation near Agassiz, BC, running northward to UpperHarrison Terminal (UHT) substation. Line 3L2 continues northwards then eastwards from UHTto the Bridge River Terminal (BRT) substation in Shalalth, BC. Both line 3L2 and 3L5 are DrakeACSR conductors (CME Wire and Cable, 2016). These sections of transmission line were chosenfor this DTR study because they are sometimes at risk of violating their thermal limit.Figure 2.1: Schematic of the case-study BC Hydro 360 kV transmission lines, 3L2 and 3L5(pink, surrounded by black box). Map courtesy BC Hydro.The Bridge River hydroelectric system is comprised of four generating stations (Fig. 2.2;15BCHydro (2018)). Operational challenges affecting the Bridge system include: (1) an increasingnumber of new independent power projects (IPPs) that tie into the UHT substation, producing powerthat must be transmitted using existing transmission infrastructure. (2) One of the transmission linesused to transmit power out of the Bridge system, 2L91, is out of service due to damage from aforest fire. Only 2L90 is currently energized. (3) The storage capacity of the Downton reservoirwas decreased by 50% due to an updated seismic assessment on the LaJoie Dam. (4) The system’sgenerating units were derated from 480 MW down to 360 MW. Combined with (3), this meansthat the operating units must be run almost continuously to move water out of the system andavoid spilling at the Terzhagi Dam. These factors have made the Bridge system difficult to manage(McCann, 2016) − the type of system where increased available transmission capacity (via DTR)could make for easier operations in situations where transmission is thermally limited using an STR.Figure 2.2: Schematic of the Bridge River hydroelectric system. Map courtesy BC Hydro.2.6 Thermal rating versus ampacityThroughout this thesis, a thermal rating refers to the calculated maximum current allowed inthe conductor, which is a function of the maximum allowed temperature of the conductor and theenvironmental conditions. For example, a DTR forecast refers to a DTR that was calculated usingwind speed and air temperature NWP forecasts, as well as a specified maximum conductor temper-ature, as inputs into the thermal rating equations. The DTR forecast is the electrical current thatwould bring the conductor to the specified maximum conductor temperature under those environ-mental conditions. QSTRs and STRs are also types of thermal ratings. When discussing thermal16rating methods, this refers to all or any combination of DTR forecasts, QSTRs, or STRs. The am-pacity, true ampacity, or observed ampacity, of the line refers to the actual instantaneous amountof current that the line can hold without reaching a temperature that would cause the line to bedamaged or sag too low. Thermal ratings are an estimate of ampacity.17Chapter 3Methods3.1 Calculating DTRs with point forecastsTo determine if skillful probabilistic DTR forecasts can be produced for a powerline in BC,hourly DTR forecasts are calculated using post-processed point forecasts of air temperature andwind speed from a NWP ensemble. These are evaluated against an estimation of the true ampacityat each forecast location.Ampacity observations are not available along the case-study powerline. Instead, hourly DTRscalculated with 2-m temperature and 10-m wind speed observations from four weather stationsnearby the powerline (Fig.3.1, Table 3.1) are used as a proxy for the true ampacity at those fourlocations near the line. NWP model output will be post-processed using, and evaluated against, theseobservations. It is assumed that DTR forecast skill at the transmission line would be nearly identicalto that of DTR forecasts at the four stations near the line (determined herein), thus providing anestimate of DTR forecast skill if weather observations had been available on the line. Withoutobservations, post-processing of NWP output, for improved forecast skill, is not possible.18Figure 3.1: Weather station locations relative to the case-study powerline (black line).19Table 3.1: Metadata for weather stations with observations used to calculate DTR observations.WeatherstationOperatedbyAvailableobservation datesLatitude,Longitude(◦)Elevationabove sealevel (m)Bearing ofnearestpowerlinespan fromNorth (◦)Distancetopowerline(km)Agassiz EnvironmentCanadaJanuary 1, 2015 -August 31, 201649.2500,-121.766716 322.87 0.387Big Silver 2 BC Ministryof ForestsJanuary 1 -April 19, 2015;July 27, 2015 -August 31, 201649.6908,-121.8594561 339.46 10.431PembertonBaseBC Ministryof ForestsJanuary 1 -April 19, 2015;July 27, 2015 -August 31, 201650.3058,-122.7286204 30.66 4.576D’arcy BCMinistryof ForestsJanuary 1 -April 19, 2015;July 27, 2015 -August 31, 201650.5217,-122.4981346 306.60 0.86420Table 3.2: NWP ensemble details.Model InitialconditionsHorizontalgrid length(km)Forecastinitializationhour (UTC)Forecastlength (h)MM5GFS 0.5◦36, 12, 4 000084NAM 32km 60WRF-ARWVersion 3.6.1GFS 0.5◦ 108, 81, 36,27, 12, 9 0000180NAM 32km 84WRF-ARWVersion 3.7.1GFS 0.5◦36, 12, 40000 168, 168, 84NAM 32km 0000 84, 84, 60GEM 0.24◦ 0000, 0300, 0600 168, 168, 84NAVGEM 0.5◦ 0000 168, 168, 84WRF-NMMVersion 3.6.1GFS 0.5◦ 36, 120000180NAM 32km 36, 12, 4 84WRF-ARWVersion 3.7.1 (runon Google Cloud)GFS 0.5◦ 27, 9 0000, 0300 168Hourly post-processed ensemble point forecasts of 2-m air temperature and 10-m wind speedfor each weather station location were obtained from an ensemble of NWP model forecasts (Table3.2). The models used are either the Advanced Research core or the Nonhydrostatic MesoscaleModel core of the Weather Research and Forecast model (WRF-ARW, Skamarock et al. (2008);WRF-NMM, Janjic et al. (2014), respectively), or the 5th Generation of the Pennsylvania StateUniversity-National Center for Atmospheric Research Mesoscale Model (MM5; Grell et al. (1994)).Forecasts are initialized at times specified in Table 3.2 for each day in the study period (January 1,2015 - August 31, 2016). For each member, forecast hours prior to 0900 UTC are discarded toavoid numerical model spin-up inaccuracies. 47-hour forecasts are used for evaluation (0900UTCto 0700UTC two days later, 0100 PST to 2300 PST the following day).Each weather station is assumed to be representative of weather directly on the case-studypowerline, and thermal rating equation inputs for the powerline are approximated based on eachweather station location. The line orientation, Zl , is taken as the bearing from North between thetower pair with the closest straight line distance to each weather station. The elevation above sealevel (ASL) of the conductor, He, is taken as the tower height added to the terrain elevation ASL ofeach station. The latitude of each weather station is used as the powerline conductor latitude input.Because there are only four weather stations, spaced widely apart along the case-study powerline,that are not directly on or nearby the powerline (Fig.3.1), the exact catenary shape of the powerline21conductor and its elevation angle are not known. Thus, for simplicity, the conductor elevation angleis set to zero. The julian day and hour of the day of each hourly forecast and observation pair areused as julian day and hour of the day inputs.Figure 3.2 shows the six bias-correction configurations applied to DTR forecasts to test waysto improve forecast skill. The naming convention of the bias-correction techniques is based onthe concept of operators: bias correction (B), ensemble averaging (E) and dynamic thermal rating(D). For example, a bias correction of a weather (W) forecast is: B(W). An ensemble average ofbias-corrected weather forecast is E[B(W)]. A DTR calculated from the result is DE[B(W)]. And asecond bias correction of DEBW is B(DE[B(W)] ).Figure 3.2: The six bias-correction techniques applied to DTR forecasts. 1) Applies bias-correction to each ensemble member’s weather forecast, 2) applies bias-correction toeach ensemble member’s DTR forecast, calculated from raw weather forecasts. Yellowbox = EDBW, green box = BEDBW, red box = DEBW, blue box = BDEBW, purple box= EBDW, and orange box = BEBDW. Acronyms explained in text.The colored boxes in Figure 3.2 labeled with the name of each bias-correction configurationindicate the resulting deterministic DTR forecasts that are compared to the weather observations.EDBW is the bias-correction technique where the ensemble average is taken of DTR forecasts cal-culated from individual ensemble member’s bias-corrected weather forecasts (yellow box). DEBW22is obtained by calculating the DTR forecast from the ensemble average of bias-corrected weatherforecasts (red box). EBDW is the technique where the ensemble average is taken of bias-correctedDTR forecasts, calculated from raw weather forecasts (purple box). EDBW, DEBW, and EBDWare bias-corrected a second time in COMPS (defined below) to give the techniques BEDBW (greenbox), BDEBW (blue box), and BEBDW (orange box), respectively. All post-processed DTR fore-cast configurations are compared with raw DTR forecasts, which are termed EDW.Bias-correction was applied to weather and DTR forecasts using the Component-based Post-Processing System (COMPS; Nipen (2012)). An additive bias-correction technique was used forair temperature forecasts and a multiplicative bias-correction technique (degree-of-mass-balance)was used for wind speed and DTR forecasts. Bias-correction in COMPS is applied using a movingwindow that weights the most recent forecast/observation pairs most, with weighting falling off withan e-folding time of 30 days. This is applied to forecasts from each ensemble member individually.Probabilistic DTR forecasts are created by dressing each deterministic DTR forecast configu-ration with a Gaussian distribution. The spread of the Gaussian distribution is specified accordingto the historical relationship between the ensemble mean and its error. That is, the variance of thedistribution is specified as:σ2 = mx+bWhere x is the ensemble mean, and m and b are constants determined by regression againstthe square of past ensemble mean error. Thus, for all DTR forecast configurations except EDW, thespecified spread is a function of both the historical error of the ensemble mean, and the spread of theensemble (if a spread-error relationship exists). For EDW probabilistic forecasts, σ2 is estimateddirectly from the square of past forecast errors of the ensemble mean (Siuta et al., 2017).Having a reliable probabilistic forecast means that the forecasted probability of an event matchesits frequency of occurrence. While the method described in the previous paragraph ensures that thevariance of the Gaussian distribution is correct, the shape of the Gaussian distribution may not beappropriate for the distribution of possible DTRs. Reliability of the DTR Gaussian distributionswas examined, and indeed some were not reliable, so a calibration scheme (Nipen and Stull, 2011)was applied in COMPS. The scheme applies corrections to the raw probability distribution so that,over the adaptive training period, the new, calibrated (reliable) probabilities match the percentageof calculated ampacities that fall below each forecast probability.To compare QSTR and STR skill with DTR forecast skill over the 20-month data period, sea-sonal STRs and monthly QSTRs are evaluated against calculated hourly ampacities on an hourlybasis.23Bias, mean absolute error (MAE), and Pearson correlation are determined for each DTR fore-cast configuration (Fig. 3.2), as well as QSTRs and STRs. Then, each DTR configuration is dressedwith a calibrated probability distribution as described above, and probability integral transform(PIT) histograms, calibration deviation ratios, and the continuous ranked probability score (CRPS)are used to evaluate reliability and probabilistic forecast skill. All metrics are described in AppendixE. Tukey honest significant difference (HSD) tests determined the statistical difference between themetric means for each thermal rating method (except PIT histograms). A Tukey HSD test looksfor the difference between two means relative to an expected difference over a studentized rangedistribution (Miller Jr., 1981).A miss is typically defined as an event being observed, but not forecasted, while a hit is typ-ically defined as an event that was observed and forecasted. An event is defined as an observedquantity being either greater than or less than a set threshold (e.g., precipitation greater than 25 mm,temperature less than 0◦C). A contingency table is typically comprised of four categories, for ex-ample: hit (fcst ≤ threshold, obs ≤ threshold), miss (fcst > threshold, obs ≤ threshold), false alarm(fcst≤ threshold, obs > threshold), and correct rejection (fcst > threshold, obs > threshold). In thisthesis, however, the threshold is equal to whatever the observed, true ampacity is at a given time,and we can substitute in “observation” for “threshold” in the above inequality tests. An observationcannot be greater or less than itself, that is, in this case an event can not be observed or not observed.Therefore, for an observation, the contingency table collapses to just two options: hit (fcst < obs),or miss (fcst > obs). Miss rate, in this thesis, is then defined as (misses)/(all forecast-observationpairs), and hit rate is defined as (hits)/(all forecast-observation pairs).To assess the safety of the DTR forecasts, hit and miss rates are analyzed for multiple DTRforecast probabilities, for all DTR forecast configurations, at all weather stations. If a given DTRforecast value is less than its corresponding observation, the forecast is not exceeding the transmis-sion line capacity and is safe, and therefore is deemed a hit. However, if the forecast value exceedsthe observation, the DTR forecast is indicating the line can transmit more power than its true am-pacity, putting the line at risk of thermal overload, and is deemed a miss. In this case the miss rateis used as another metric for evaluating reliability. If the forecasts are reliable, the percentiles of theprobabilistic forecasts should equal the percentage of times the observations fall below that forecast.In other words, observations should fall below the DTR forecast value corresponding to the 95thpercentile of the forecasted probability distribution 95% of the time. Therefore, any percentile ofa calibrated probabilistic DTR forecast distribution, p, is the same as its hit rate. Conversely, itscomplementary percentile, 1-p, is the same as its miss rate. The miss rate can be viewed as therisk associated with using that thermal rating, and the hit rate can be viewed as the success of thatforecast to be safe.24In an effort to assess the capability of DTR forecasts to be safe as well as to increase transmis-sion capacity relative to STRs and QSTRs, a combined version of hit and miss rates is defined inthis thesis. Each percentile of each probabilistic DTR forecast configuration is compared with theQSTR and the STR. For each comparison, four scenarios are taken into account where each of thethermal ratings is tested to see if it is the highest (most efficient) thermal rating method out of allthree, while still being safe (Fig. 3.3). Scenario 1 takes into account when all thermal ratings arehits (all safe, below true ampacity), but TR1 is higher than TR2 and TR3 (highest rated ampacity,most efficient). Scenarios 2 and 3 illustrate the case where one of TR2 and TR3 is a miss (unsafe,above true ampacity), while the other, as well as TR1, are hits (safe), but TR1 is best due to itshigher rating. Finally, scenario 4 takes provides an instance where both TR2 and TR3 are misses,but TR1 is a hit, so TR1 is automatically best. All scenarios make up the extended hit rate, definedas the percentage of occurrences where a given thermal rating method (DTR forecast, QSTR, STR)is best, i.e. the highest rating while still below the true ampacity.Figure 3.3: Schematic of the different ordering scenarios of the thermal rating (TR) methodswhen determining the extended hit rate.3.2 Calculate DTRs with gridded forecasts at every point along lineDTR forecasts are computed using raw hourly gridded NWP forecasts from a smaller ensembleof gridded weather forecasts that are available for January and August 2016 (Table 3.3). Forecastswere again initialised at 0000 UTC, with the first 9 forecast hours discarded to account for spin-up.25Table 3.3: Members of gridded forecast ensemble.Model Initial conditions Horizontalgrid lengths (km)Initializationtime (UTC)MM5GFS 0.5◦36,12,4 0000NAM 32kmWRF - ARWVersion 3.7.1GFS 0.5◦36,12,4 0000NAM 32kmForecasts from 0900 UTC (0100 PST) to 0800 UTC (0000 PST) the next day were used to makea 24-h forecast horizon for each initialization date. This allows for a full diurnal cycle of DTRforecasts to be analyzed.The catenary shape approximation of the powerline is added into the IEEE 738 equations (asdiscussed in section 2.3) and DTR forecasts are calculated for the midpoint of each conductor seg-ment between each pair of transmission towers. Gridded temperature and wind speed forecasts arehorizontally and vertically interpolated to the latitude, longitude, and height-above-ground of themidpoint of each segment between each tower pair (Appendix B, Fig. B.1) for the entire case studypowerline. Linear interpolation was used for vertical interpolation of air temperature forecasts,while the power law was used for vertical interpolation of wind speed forecasts. Air temperatureand wind speed were interpolated to 1-m above and below the segment midpoint height. These val-ues were then averaged to get the value at the segment midpoint height. A first order, bi-cubic splinepiecewise interpolation was used for horizontal interpolation of both weather variable forecasts.Further, along with air temperature and wind speed forecasts, powerline segment characteristicswere used as input to the thermal rating equations described in section 2.1. The hour and and julianday of year was used for each forecast hour. The line orientation (bearing from North), Zl , latitude,and elevation angle, Hp, for each segment is calculated based on the catenary approximation (section2.3, Appendix B). Transmission tower height, as well as the elevation angle, terrain elevation below,and latitude, longitude coordinates, of each segment midpoint determines the height above sea level(ASL; used to determine the sun angle), and the height above ground level (AGL; needed for verticalinterpolation of NWP forecasts) of the midpoint of each segment. To accommodate typical verticalclearance regulations of powerline conductors (8 m; DMD & Associates Ltd. (2005)) as well asvertical interpolation of wind speed forecasts where the lowest forecast is at 10 m, the minimumheight AGL of any segment midpoint is 11 m. All transmission-tower heights are assumed to beequal. From field measurements, tower height is set to 40 m.Gridded DTR forecasts were calculated for each individual ensemble member, using the inter-polated air temperature and wind speed forecasts of that member. In order to be conservative and26assume conservative environmental conditions least conducive to cooling the line, the lowest DTRforecast of all ensemble members was chosen as the final DTR forecast for each point along theline, at each forecast hour.The percentiles of all DTR forecasts, of all dates in January and August, at all points along thepowerline, are used to inform the range of the scale when plotting DTR forecasts in space (Chapter5). DTR forecast values are sorted into bins between the 1st, 10th, 20th, 30th, 40th, 50th, 60th,70th, 80th, 90th, and 99th percentiles for the scale on spatial DTR plots.Weather observations are not available anywhere on the case study powerline, therefore, nobias-correction of weather or DTR forecasts is possible, probabilistic DTR forecasts cannot be cre-ated (or calibrated), and the skill of each segment’s DTR forecast cannot be evaluated. All DTRforecasts in Chapter 5 are raw, and are referred to as EDW DTR forecasts, as per the naming con-vention outlined in section 3.1.It is assumed that, if weather observations had been available on the case study powerline, thenthe most skillful bias-corrected DTR forecast method (out of all methods assessed in Chapter 4)could be applied to the interpolated gridded forecasts along the line, and they would have the sameforecast skill as determined in Chapter 4.3.3 Calculate STRs and QSTRs with weather observation statisticsSeasonal STRs and monthly QSTRs are calculated for the case-study powerline. STRs areintended to be more conservative than the QSTRs. Two STRs are computed, one for summer (Aprilthrough September) and one for winter (October through March), as is common practice (Billintonand Koval, 1970; Deb, 2000; Lu, 2014). Air temperatures of 40◦C and 10◦C are used as the airtemperature inputs for the summer STR and winter STR, respectively, as is standard practice (Lu,2014).A QSTR is calculated for each month of the year using daily temperature maxima obtainedfrom five weather stations nearby the case-study powerline (Fig.3.4, Table 3.4). For each weatherstation, for each month, the 85th percentile of the distribution of daily temperature maxima wascalculated. The highest monthly 85th percentile of daily temperature maxima out of all weatherstations was used as the temperature input for calculating each monthly QSTR (Table 3.5). This ismeant to be conservative − the higher the temperature input, the less environmental cooling felt bythe line, and the lower the thermal rating. This method is meant to replicate (Lu, 2014).For the STRs, the julian day of the year is set to the summer and winter solstices of 2016 (June21 and December 21) since the solar angle is greatest and least for Canadian latitudes, maximumand minimum insolation intensity, respectively. For QSTRs, the julian day of the 15th day of each27Table 3.4: Characteristics of weather stations whose maximum daily temperatures were usedto calculate monthly QSTRs for the case-study powerline region.WeatherStationOperatedbyAvailableobservation datesLatitude, Longitude(◦)Elevationabove sealevel (m)Agassiz EnvironmentCanadaJanuary 1, 1986 -March 2, 201749.24, -121.76 15.0Cheakamus BC Hydro January 2, 1960 -January 31, 201750.08, -123.03 640.0Pemberton EnvironmentCanadaApril 1, 1986 -March 11, 201750.31, -122.73 204.3Shalath BC Hydro October 2, 1984 -September 30, 201650.73, -122.24 290.0Stave Lake BC Hydro January 2, 1960 -January 31, 201749.56, -122.32 330.0Table 3.5: List of the maximum monthly 85th percentile of maximum daily temperatures foreach month, and which weather station they were observed at.MonthMaximum 85thpercentile ofdaily maximumtemperatures (◦C)Weather StationJanuary 10.0 AgassizFebruary 12.5 AgassizMarch 15.3 AgassizApril 20.8 PembertonMay 26.5 PembertonJune 30.3 ShalalthJuly 34.5 ShalalthAugust 33.5 Pemberton, ShalalthSeptember 28.0 PembertonOctober 19.8 ShalalthNovember 12.5 AgassizDecember 9.4 Agassiz28Figure 3.4: Map showing the weather stations surrounding the case-study powerline with max-imum daily temperature observations used to calculate monthly QSTRs.month was used as the julian day input. For STRs, the latitude input is that of the transmissiontower furthest south of the case-study powerline, and the elevation ASL of the conductor, He, isset to zero. For QSTRs, the mean latitude and elevation ASL of all weather stations is used as thelatitude and conductor elevation inputs.Other inputs to the thermal rating equations (Chapter 2) are the same for both the STRs andQSTRs. Wind speed is set to 0.6 m/s, a typical conservative value of wind speed for STRs (IEEE,2013b; Lu, 2014). The orientation of the line is set to 0◦, and the wind direction angle relative tothe line is set to 90◦, ensuring wind direction is perpendicular to the powerline, as is also the normin STR calculations (Lu, 2014; Reding, 1994; IEEE, 2013b). Hour of the day is set to 1200 LST,when the sun is at its highest point in the sky and Earth is receiving the strongest solar radiation.The elevation angle of the conductor is set to zero, as neither the STRs nor the QSTRs take intoaccount the catenary shape of the line nor the terrain slope.29Monthly QSTRs and seasonal STRs are compared with DTR forecasts and ampacities, in Chap-ter 4, and are compared with the minimum EDW DTR forecast along the powerline at each forecasthour in Chapter 5. In each case, an hourly timeseries of the QSTRs and the STRs is made applyingthe appropriate seasonal STR and monthly QSTR to each DTR forecast and observation hour.30Chapter 4DTR forecasts at weather-observationstationsFigures from only one of Big Silver 2, Pemberton Base, and D’arcy weather stations will bepresented as forecast and observation traits between them are very similar. These figures will becompared to Agassiz, which has different behavior. If there is no important differences between anyof the locations, then only one figure from one station will be presented.Forecast hour conveniently corresponds with local time. For example, for a 47-hour forecast,forecast hours 1 - 23 correspond to 0100 - 2300 PST of the first forecast day, and forecast hours 24- 47 correspond to 0000 - 2300 PST of the second forecast day.4.1 Characteristics of DTR forecastsIn this section, deterministic and probabilistic DTR forecasts, based on NWP models, are com-pared with observed ampacity, QSTRs, and STRs, which are based on weather-station observations.Figure 4.1 shows that wind speeds have a positive linear relationship and air temperatures havea negative linear relationship with DTRs, respectively. DTRs increase as wind speed increasesand as air temperature decreases, showing that higher wind speeds and colder air temperatures areconducive to powerline cooling, thus allowing for a higher DTR. Instances when the air temperatureis high can still give higher DTRs, provided the wind speed is also high. Conversely, DTRs can alsobe high when wind speed is low, if the air temperature is low. These relationships are consistentwith forecasts and observations at each weather station.Agassiz has much faster wind speed forecasts and observations than Big Silver 2, PembertonBase, and D’arcy, resulting in higher DTRs by up to 800 A (Fig. 4.1). At all stations, DTRs (fore-31casts and observations) are sometimes different at times of identical wind speed and air temperatureconditions, illustrating that solar radiation also influences the DTR.Bias-correction widens the range of raw temperature forecasts, that is too narrow at all stations(cf. Figs. 4.1c and 4.1d), making it more similar to the observed temperature range (Figs. 4.1aand 4.1b). At Agassiz, raw wind speeds are underforecast, while at all other stations they aresignificantly overforecast. Bias-correction increases DTR forecast skill by increasing (decreasing)forecasted wind speeds at Agassiz (all other locations).Hourly ampacities vary greatly throughout the 20-month study period at all stations (Fig. 4.2).Although there is a seasonal cycle of ampacity, with higher ampacities in the winter and lower am-pacities in the summer, daily and hourly variations are of a similar magnitude to seasonal variations.The seasonal cycle is expected, as cooler air temperatures and regular stronger winds in the winterallow for higher ampacities, and vice versa in the summer. However, the lowest DTRs in the winterare only 100 - 200 A higher than the lowest DTRs in the summer, and the highest DTRs in thesummer are of similar magnitude lower than the highest DTRs in the winter. This means that it ispossible to have environmental conditions that give similar extremes of high and low DTRs at alltimes of year.Agassiz shows the largest hourly variation and the largest range of both wind speeds and am-pacities out of all weather stations. DTR magnitudes at Agassiz are approximately 1800 A, while atthe other locations they are approximately 1300 A.As expected, the QSTRs and the STRs do not represent any of the sub-monthly or sub-seasonalvariation of the line’s ampacity, respectively (e.g., Fig. 4.2). Perhaps more importantly, they alsodo a fairly poor job of providing a safe thermal rating (lower limit of ampacity). From April toSeptember, at each weather station, there are only a few instances when ampacities are lower thanthe summer STR. From October to March, however, there are many instances when ampacitiesare much lower than the winter STR. All QSTRs from April to September are greater than thesummer STR, and therefore observed ampacities fall below the QSTRs more often and by a greatermagnitude than the summer STR. From October to March, QSTRs are closer to the winter STR, andboth overestimate the amount of environmental cooling, thereby consistently exceeding observedampacities by large amounts.The summer STR is a better estimate of a lower limit of ampacity than the winter STR, mean-ing, if seasonal STRs are used operationally, there would be a higher chance of ampacity violationsin the winter than in the summer. The monthly QSTRs vary more smoothly over the data periodthan the seasonal STRs, better representing the annual cycle of environmental conditions, but areless conservative (except in October and March), possibly incurring more thermal limit violationsyear-round.32llllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllll lllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllll llllllllllllllllllllll lllllll ll lll lllllllllllllllllllllllllllll l lllll llllllllllllllllllllll llllllllllllllllllllllllllllll llllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllll llllll ll llllllllllllllllllllll lll llllllllllllllllllllllllllllllllllllllllllll lll lllll lllllllllllllllllllllllllllllllllllllllll lll lllllll lllllllll llllllllllll lllllllllllllllllllllll lllllllllllllllll llllllllllllllllllll llllNumber of points =  14031500100015002000250030000 10 20 30 40 50 60Wind Speed (km/hr)Dynamic Thermal Rating (amperes)Temperature ( °C )lll −3.70 −   7.30  7.30 −  16.80 16.80 −  35.20Agassiz, observations(a)llllllllllllllll llllllllllllllllllll llllllllllllllllllllllllllllllllllll llll lll lllllllllll lllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllll llll lllll lllllllllllllllllllllllllllllllllllllllllllllllll llll llllllllll llllllllllllllllllllll l lllllllllllllllllllllllllll lllllll llllllllll lll llllllll llllllllllllll lllllllllll lllllllllll lllllll lll llllllllllllllllllllllllllllllllllllll lllllll llllllllll llllllllll llllllNumber of points =  9770500100015002000250030000 10 20 30 40 50 60Wind Speed (km/hr)Dynamic Thermal Rating (amperes)Temperature ( °C )lll−14.70 −   1.30  1.30 −  14.00 14.00 −  36.60Pemberton Base, observations(b)lllllllllllllll lllllllllllllllllll llllllllll lllllllllllllllll lllllllllllll llllllllllll llllllllll llllllllllllll llllllllll llllllll ll llllllllllllllllllllllllllllll ll llll llllllllllllllllllll l llllll llllllll llllllll lllll ll llllllllllllll ll lll ll lllllllllllllllllllllllllllll ll lllllllllll lll llll l llllllllllllllllllllll lllllllllllllllll llllllllll ll llllllllllllllllllllllllllllllll llllllllllllllllllllll lllllllllllllllllllll lll llll llll lll lllllllllllllll lllllllllll lllllllllllllllllllllll lllll ll lllll llllllll llllllllllllllllllll ll lllll l llllllllllllllllllll lllll l l llllllllllll ll ll llllllll l ll llllllllllllllllllllllllllll llllllllllllllllll lllllllllllll llllllllllllllll llllllll lllllllll llllllllllllllNumber of points =  14007500100015002000250030000 10 20 30 40 50 60Wind Speed (km/hr)Dynamic Thermal Rating (amperes)Temperature ( °C )lll −6.78 −   5.22  5.22 −  15.06 15.06 −  30.73Agassiz, EDW(c)llllllllllllll lllllllllllllllllllllll ll ll llll llll lllllllllllllllllllllllllllllllllllllllllllllllll llllll llllllllllllllllllllll lllllllllllllllllllllllllllllllllll llll llllllllll llllllllllllllll lllllllll llllllllllllllllllllllllllll llllllllllllllllllllll lllllllllllllllllllllll llllll lllllllllllll lll llll llll llllllll llllllllllllllllllll lllllllll llll lll lllllll llllllllll lllllllllll lllll ll lll lllll lllll llllllll lllllll lllll l lllllll llllllll llllllllllllllllllllllllllll lll lllllllllllll lllllllll lll lllllll ll llllll lllllllllllllllll lllllll llllllllllllllll llllllllllllll lll llll llllllllllllll lllllllllllllllllllllllllllllllllllllllllllll llll llllllllllllll ll lll llll lllllll lllllllllllllllllNumber of points =  14007500100015002000250030000 10 20 30 40 50 60Wind Speed (km/hr)Dynamic Thermal Rating (amperes)Temperature ( °C )lll−15.21 −   0.74  0.74 −  12.38 12.38 −  29.05Pemberton Base, EDW(d)lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllll lllllllllllllllllllllllllllllllllllllllllllll lllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllll llllllllllllllll l lll lllllllllllllllllllllllllllllllll llll lllllllllllllllllllllllllllllllllll llllllllllllllll lllllllllllll lllllll llllllll llllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllll lllll llllllllllllllllllllllllllllllllllllllllllllllllllllllll lllll llllllllllll llllllllllllllllllllllll ll ll lllllllllllllllllllllllllllllllllll llllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllll lllllllllllllllllllll llllllll llllllllllllllllll l llllllllllll lllllllllll ll lllllllllllllllllllllll llllllll lllllll llllll ll llllll lllllllllll lllll lllll llllllllllllllllllllllll lllllllllllllll lllllllll lllllllllllllllllllllllllllllllllllll lllll lll llllllllllll l lllllllllllll lllllllllllllllll llllll lllllllllllllllllllllllllllllll llllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllNumber of points =  14007500100015002000250030000 10 20 30 40 50 60Wind Speed (km/hr)Dynamic Thermal Rating (amperes)Temperature ( °C )lll −6.30 −   5.25  5.25 −  17.14 17.14 −  35.38Agassiz, BEDBW(e)llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llll lllllllll lllllll lllllllllll lllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllll lll llll lllllllllllllllllllllll ll llllll lllllll llllllll l llllllllllll llllllllllllllllllllllllllllllllllll l lllllllllllllllllllllllllllllllllll llll llllllllllllll ll ll llll lllllllllllll l llllll llll llllllllllllllll llllll lllllllllllllll lllllllllllll lllllllll ll llllll lllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllll llllllllllllll llllllllllllllllllll llllllllllNumber of points =  14007500100015002000250030000 10 20 30 40 50 60Wind Speed (km/hr)Dynamic Thermal Rating (amperes)Temperature ( °C )lll −9.88 −   0.44  0.44 −  13.92 13.92 −  48.56Pemberton Base, BEDBW(f)Figure 4.1: Scatterplots of DTR as a function of bias-corrected wind speed (x-axis) and airtemperature (colouring) over the study period for the following locations [DTR, temper-ature, and wind observation or forecast method]: (a) Agassiz [observations], (b) Pem-berton Base [observations], (c) Agassiz [EDW], (d) Pemberton Base [EDW], (e) Agassiz[BEDBW], and (f) Pemberton Base [BEDBW]. Figs. (c - f) are based on NWP forecasts.3302004006008001000120014001600180020002200240026002800300032002015−01−01−002015−03−01−002015−05−01−002015−07−01−002015−09−01−002015−11−01−002016−01−01−002016−03−01−002016−05−01−002016−07−01−002016−09−01−00Thermal Ratings (amperes)99th50th01stOBSQSTRSTRStation: 679, Config: BEDBW(a)02004006008001000120014001600180020002200240026002800300032002015−01−01−002015−03−01−002015−05−01−002015−07−01−002015−09−01−002015−11−01−002016−01−01−002016−03−01−002016−05−01−002016−07−01−002016−09−01−00Thermal Ratings (amperes)99th50th01stOBSQSTRSTRStation: 2827, Config: BEDBW(b)Figure 4.2: Timeseries of BEDBW DTR forecasts, DTR observations, QSTRs, and STRs overa 20-month period at (a) Agassiz and (b) Pemberton Base weather stations.34While the QSTRs and STRs are not particularly safe, the vast majority of the time they aresafer than necessary. In other words, ampacities greatly exceed the STRs and the QSTRs much ofthe time. In terms of optimizing the transmission system, this means there are a lot of times whenthe thermal rating could be much higher, giving opportunity to transmit more electricity.All percentiles of BEDBW forecasts represent the hourly and daily variability of ampacitymuch better than the QSTRs and the STRs (Fig. 4.2). Observed ampacity and the 50th percentileBEDBW forecasts have (most likely) similar means, as would be expected of a bias-corrected fore-cast. However, the forecasted variability is too low − there are many instances when ampacitiesare higher or lower than the BEDBW DTR forecast. The other DTR bias-correction methods havethe same variability (not shown). This could be a function of the ensemble mean averaging outdifferences of individual members that may have higher variability.1st and 99th percentile probabilistic DTR forecasts encompass the vast majority of ampacitiesthroughout the study period. At Agassiz, the spread of the probabilistic DTR forecasts, shown bythe 1st and 99th percentiles, is much larger in the winter than in the summer (Fig. 4.2a). Thislarge annual variation in spread is not seen at the other weather stations (e.g., Fig. 4.2b). The1st percentile of probabilistic BEDBW forecasts is a much safer lower limit of ampacity than theQSTRs and STRs throughout the study period at all stations.The width of each boxplot in Figure 4.3 represents the spread of the 50th percentile DTR val-ues, for each DTR forecast configuration and observed ampacity, over the 20-month study period.Overall, the bias-corrected DTR forecast spreads match that of the observed ampacities well, anddo a much better job than the raw EDW forecasts. This indicates the bias-corrected forecasts cap-ture the variance fairly well. The mean and spread of the EDW forecasts are higher than observed,illustrating their large positive bias at each location. Ampacities have a larger spread at Agassizthan at any other weather station, as was seen in Figures 4.1 - 4.2. The spreads of all bias-correctedDTR forecast methods are very similar at Agassiz, but all have a more narrow spread than observedampacities. At Big Silver 2, the spreads of BDEBW, BEDBW, DEBW, and EDBW best representampacities, although underforecast slightly, while the distributions of BEBDW and EBDW bestrepresent the spread of ampacities at Pemberton Base, but overforecast slightly. At D’arcy, the ob-served spread is smaller than that of BEBDW and EBDW, and larger than that of BDEBW, BEDBW,DEBW, and EDBW. No single DTR forecast configuration best represents ampacity spread at alllocations, but there is virtually no forecast bias from any configuration.Figures 4.4 and 4.5 zoom in on sample 5-day periods in January and July at Agassiz and Pem-berton Base to compare all DTR forecast methods with observed ampacities in different seasons.The deterministic forecasts shown in these figures correspond to the 50th percentile of the proba-bilistic forecasts. In each month, at each weather station, hourly ampacities show considerably more35BDEBWBEBDWBEDBWDEBWEBDWEDBW EDWOBS70010001300160019002200AgassizThermal Rating (amperes)(a)BDEBWBEBDWBEDBWDEBWEBDWEDBW EDWOBS70010001300160019002200Pemberton BaseThermal Rating (amperes)(b)Figure 4.3: Boxplots of the deterministic DTR forecasts and observations at (a) Agassiz, and(b) Pemberton Base (outliers not included). Ideally, the DTR forecast boxplots will havethe same width and median as the ampacity boxplot.variability than the forecasts. There is large hourly variation of ampacities at all stations, howeverampacity magnitudes are the largest at Agassiz (Fig. 4.4), as was already seen in Figures 4.1 - 4.3.At Agassiz, larger hour-to-hour variability in January could be more difficult for the DTR forecaststo represent.Overall, there is very little difference between bias-corrected DTR forecast methods acrossall seasons and stations. At Agassiz, there are forecast hours in January when the DTR forecasts ofDEBW, EBDW, and EDBW are different from those of BDEBW, BEBDW, and BEDBW, but withineach group, the forecasts are very similar. This is also seen in 5-day snapshots in October and April(not shown). At Pemberton Base and D’arcy, BEBDW and EBDW forecast higher and lower DTRsthan the rest of the bias-corrected DTR forecast configurations at multiple forecast hours.36lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllll lllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll ll llll lllllllllllllllllllll lllllllllllllllllllllllllllllllllllllll ll llllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllll lllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllll llllll lllllll lllllllllllllllllllllllllll llll7009001100130015001700190021002300250027002016−01−152016−01−162016−01−172016−01−182016−01−192016−01−20Thermal Ratings (amperes)llllllllEDWOBSBDEBWBEBDWBEDBWDEBWEBDWEDBWQSTRSTRStation: 679(a)lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllll lllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllll ll llll llllllllllllllllllll llllllllllllll lllllllll llllllllll lllll lllll llllll l l lllllllllllllll7009001100130015001700190021002300250027002016−07−202016−07−212016−07−222016−07−232016−07−242016−07−25Thermal Ratings (amperes)llllllllEDWOBSBDEBWBEBDWBEDBWDEBWEBDWEDBWQSTRSTRStation: 679(b)Figure 4.4: Timeseries of all deterministic thermal rating forecast methods and observed ampacities at Agassiz weather stationfor 5-day periods in (a) January, and (b) July.37llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllll llll lllllllllllllllll llllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllll lllll lllllllllllll lllllllllll lllll ll llllll lll lllll lllllll l lllllllll llll llllllll llllllllll lllll lllllllllllll llllllllll llllllll lllll l l llllllll llll llllllllllllll llllllllll lllllllll llllllllllll llllllllll llll ll llllll llll llllll llllllllllllllll lllll llllllllll lllllllll lllllllll l l llllllllllllllll ll7009001100130015001700190021002300250027002016−01−152016−01−162016−01−172016−01−182016−01−192016−01−20Thermal Ratings (amperes)llllllllEDWOBSBDEBWBEBDWBEDBWDEBWEBDWEDBWQSTRSTRStation: 2827(a)llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllll lllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllll llllllllll lllllllllllllllllllllll llllllll llll lllllllllllllllll lllllllllllll lllllllllllllllll ll llllll llll l lll lllllllll lll lll lll l llllll l llll llllllll lllllllllllll llll l lllllllllllllllll lllll lll llll llll llllllll l7009001100130015001700190021002300250027002016−07−202016−07−212016−07−222016−07−232016−07−242016−07−25Thermal Ratings (amperes)llllllllEDWOBSBDEBWBEBDWBEDBWDEBWEBDWEDBWQSTRSTRStation: 2827(b)Figure 4.5: Timeseries of all deterministic thermal rating forecast methods and observed ampacities at Pemberton Base weatherstation for 5-day periods in (a) January, and (b) July.38At each weather station, in each 5-day period, there are multiple forecast hours when all bias-corrected deterministic DTR forecasts are well above the observed ampacity, and hence are not safe.This is motivation for making probabilistic DTR forecasts. EDW forecasts are consistently higherthan the bias-corrected DTR forecast configurations throughout the 5-day snapshots at each station(except in January at Agassiz; Fig. 4.4).Figures 4.4 and 4.5 show that observed ampacities are regularly much larger than the STR andthe QSTR, as was seen in Figures 4.4 and 4.5. This is also seen during 5-day snapshots in Octoberand April, and also at Big Silver 2 and D’arcy (not shown), indicating unrealized line capacity. Atother times, bias-corrected DTR forecasts are sometimes less than the QSTR and the STR. Thisshows the potential of DTRs to more accurately represent “worst-case” environmental conditionsat times when those worst-case conditions might be occurring, whereas QSTRs and STRs give thesame rating no matter what the day-to-day variations are in conditions.While 5-day timeseries are not necessarily representative of the entire 20-month data period,these snapshots are used to illustrate that the DTR forecasts follow ampacities with mixed success.In most cases the DTR forecasts roughly follow ampacity (Fig. 4.5b), whereas in some cases theDTR forecasts perform poorly (Fig. 4.4a).On annual timescales, wind speed and temperature have similar contributions to DTR variabil-ity (Figs. 4.1 - 4.2). However, day-to-day, temperature ranges are much smaller, and wind speed isthe dominant contributor to ampacity variability (Figs. 4.4 and 4.5).Figures 4.4 and 4.5 show that although in many cases the overall magnitude and trend of thethermal rating is well-captured, the deterministic bias-corrected DTR forecasts do not fully capture1) the substantial hour-to-hour variability of ampacity, and 2) the magnitude of that variability.There are many instances when the DTR forecasts are different from the ampacities by severalhundred amperes. Due to these imperfect, deterministic forecasts, and the risks involved, calibratedprobabilistic DTR forecasts are a logical alternative.Calibration adjusts the probabilistic forecast distribution to the appropriate amount of spreadgiven the past forecast error. The DTR forecast distribution spread is decreased if the past squarederrors of the ensemble mean are less than the empirical raw ensemble spread, and vice versa. Thedifference (i.e., spread) between the 99th and 1st percentiles of the forecast distribution decreasessignificantly after bias correction and calibration are applied to EDW DTR forecasts at most loca-tions (section 3.1; Fig. 4.6). There was a smaller average decrease in spread at Agassiz, wherecalibration mainly increased the variability of spread, meaning the uncertainty of ampacity variessignificantly throughout the 47-hour forecast period and throughout the year. All bias-corrected,calibrated DTR forecast configurations show very similar spread at Agassiz. At the other locations,BEBDW and EBDW have larger spread than BDEBW, BEDBW, DEBW, and EDBW, all of which39BDEBWBEBDWBEDBWDEBWEBDWEDBW EDW040080012001600200024002800Agassiz99th − 1st percentile DTR forecasts (amperes)(a)BDEBWBEBDWBEDBWDEBWEBDWEDBW EDW040080012001600200024002800Pemberton Base99th − 1st percentile DTR forecasts (amperes)(b)Figure 4.6: Boxplots of the difference between the 99th and 1st percentiles of all DTR fore-casts at (a) Agassiz, and (b) Pemberton Base.are very similar. As the spread of DTR forecasts is a function of both the ensemble mean and thesquared error of the ensemble mean (section 3.1), if all bias-corrected DTR forecast configurationsare calibrated, then those with wider spreads have ensemble means (i.e., deterministic forecasts)with higher error.Figures 4.7 - 4.9 show 47-hour probabilistic DTR forecasts. EDW probabilistic forecasts havewide, smooth spread (Fig. 4.7), while the spread of the bias-corrected probabilistic forecasts, suchas BEDBW (Fig. 4.8), is much more variable hour to hour, and is often much sharper than that ofEDW forecasts. This is a result of the different distributions applied to the raw and bias-correcteddeterministic DTR forecast to generate probabilistic forecasts. Probabilistic forecasts provide valu-able information to planning engineers, showing them when there is less confidence (wider spread)and more confidence (smaller spread) in DTR forecasts, which can help them determine the level ofrisk they want to take.These timeseries (Figs. 4.8 - 4.9) illustrate that BEDBW forecasts are more calibrated (ampac-ities verify at various percentiles throughout the probability envelop) than EDW forecasts (almostall of the ampacities verify below the 50th percentile).The QSTRs and STRs are sometimes higher than ampacity, putting the line at risk of thermal40l l l l l l ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l lll l ll l lllllllll llllll ll l l ll llllllllllll lllll ll l ll lllll02004006008001000120014001600180020002200240026002800300032002016−01−30−002016−01−30−062016−01−30−122016−01−30−182016−01−31−002016−01−31−062016−01−31−122016−01−31−182016−02−01−00Thermal Ratings (Amperes)llllllllllllllll01st05th10th20th30th40th50th60th70th80th90th95th99thOBSQSTRSTRStation: 2827, Config: EDWFigure 4.7: 47-hour forecast of EDW initialized on 30 January 2016 at 0100 PST, at Pember-ton Base weather station. Namely, this plot shows uncalibrated probabilities. In this andthe next two figures, the 50th percentile corresponds to the deterministic forecast.overload (Figs. 4.4, 4.5, 4.8, and 4.9). By contrast, the probabilistic DTR forecasts, and the 1st per-centile of the DTR forecast probability distribution, are safe throughout the forecast period (belowobserved ampacities). This is a good example of how the QSTR and STR are not as safe as the 1stpercentile DTR forecast. The 5th and 10th percentile forecasts are also safer than the QSTR and theSTR on average throughout the forecast period, and are only higher than the ampacity at two hoursout of 47 (which is to be expected with 5% and 10% risks).In Figure 4.9, BEDBW forecasts, and ampacity, increase during the day, and the 1st percentileforecast is less than the ampacity, but greater than the QSTR and the STR. This provides a rareexample of when 1st percentile DTR forecasts (1% risk tolerance) are safe and allow for highertransmission capacity than the traditional thermal rating methods. Throughout the forecast period,ampacities, and the majority of the forecasted DTR probabilities, are greater than the QSTR andSTR, indicative of unrealized available capacity.In the morning and overnight periods the 1st percentile of the BEDBW forecast distribution issometimes lower than than the QSTR and STR. The dip in observed ampacities during these timesindicates these lower DTR forecasts are appropriate. In other words, the DTR forecasts are safer41l l lll ll l l ll l lllll ll ll llll l lll l l l l l l l l l ll l l l ll l llllllll llllll ll l l lllllllllllll llllll ll l ll lllll02004006008001000120014001600180020002200240026002800300032002016−01−30−002016−01−30−062016−01−30−122016−01−30−182016−01−31−002016−01−31−062016−01−31−122016−01−31−182016−02−01−00Thermal Ratings (Amperes)llllllllllllllll01st05th10th20th30th40th50th60th70th80th90th95th99thOBSQSTRSTRStation: 2827, Config: BEDBWFigure 4.8: 47-hour forecast of BEDBW initialized on 30 January 2016 at 0100 PST, at Pem-berton Base weather station. Namely, this illustrates a calibrated forecast for winter.during these riskier environmental conditions. Note that in this case the lower ampacities occurovernight when temperatures are typically cooler. The NWP input correctly forecasts that lowwinds will cause less of a cooling effect that more than cancels out the effect of cooler nighttimetemperatures. This is something that can be forecasted by NWP, whereas a planner may assumethermal ratings can always be higher at night.Other bias-corrected probabilistic DTR forecasts are very similar to BEDBW, varying slightlyin forecast values and spread (not shown). As was seen in Figure 4.4, the spread of the DTR forecastprobabilities is slightly wider for BEBDW and EBDW forecasts at Big Silver 2, Pemberton Base,and D’arcy. Therefore, the spread of their distributions in products similar to Figures 4.7 - 4.9 wouldbe slightly wider, which would lower DTR forecasts of the lower percentiles. This could lead toslightly safer forecasts, but forecasts that would allow less current to be put through powerlines.Figures 4.8 - 4.9 are examples of the bias-corrected, calibrated, probabilistic DTR forecastproducts that could be given to planning engineers for short-term planning (excepting the ampac-ities, which would be unavailable at forecast issuance). These probabilistic DTR forecasts wouldallow planners to get an idea of the possible range of the thermal ratings over the next 47 hours,including reliable lower limits to transmission capacity.42l l ll lllll l ll l ll lllll ll l l l l l l lllll l ll l ll llll l l l ll l llll llllll lllllll lll l l llllllllll lllllll02004006008001000120014001600180020002200240026002800300032002016−07−30−002016−07−30−062016−07−30−122016−07−30−182016−07−31−002016−07−31−062016−07−31−122016−07−31−182016−08−01−00Thermal Ratings (Amperes)llllllllllllllll01st05th10th20th30th40th50th60th70th80th90th95th99thOBSQSTRSTRStation: 185, Config: BEDBWFigure 4.9: 47-hour forecast of BEDBW initialized on 30 July 2016 at 0100 PST, at Big Silver2 weather station. Namely, this illustrates a calibrated forecast for summer.The deterministic, ensemble-average DTR forecasts are clearly not as useful as probabilisticDTR forecasts, as they do not account for the extreme uncertainty of ampacity. This section providesevidence that probabilistic DTR forecasts yield more useful thermal ratings that properly account forthe inherent uncertainty. This means safer thermal ratings, while allowing for higher transmissioncapacity when weather permits.4.2 Deterministic forecast verificationIn this section, deterministic raw and bias-corrected DTR forecasts, as well as QSTRs andSTRs, are evaluated using bias, MAE, and correlation. See Appendix E for a summary of verifica-tion metrics.Bias, MAE and correlation were calculated for each thermal rating method at each forecasthour over the 20-month data period. The distributions of those metrics over the 47-hour forecasthorizon are shown in Figures 4.10, 4.13, and 4.15, respectively.434.2.1 BiasAll of the bias-corrected DTR forecast methods have essentially no systematic error (Fig. 4.10).Biases that are approximately +/- 20 A, relative to DTR forecasts that range from approximately800 A to 2500 A, are order ∼1%, and essentially negligible. The biases of EDW DTR forecasts,QSTRs, and STRs are much larger, and generally vary in magnitude more (bias distributions havelarger spread except at Agassiz).Figure 4.10: Boxplots of the biases of each forecast method at each weather station. Boxplotsare color-coded by location. Biases closer to zero are better.The conservative nature of QSTRs and STRs leads to large negative biases. At each weatherstation, the bias trends of both the STR and QSTR are almost exactly the same with notable diurnalcycles, except that the QSTR always has smaller biases than the STR. The QSTRs and STRs havetheir largest biases at Agassiz, and their lowest biases at D’arcy. Because there is no daily variationin the QSTR or STR, increasingly negative biases during the day mean ampacity is increasing,making the QSTR and STR increasingly conservative. This increase in ampacity indicates a generalincrease in wind speed during the day, seen at all stations except Pemberton Base (Fig. 4.11).The raw EDW forecasts have significantly larger positive biases than all bias-corrected forecastconfigurations at each location, showing that bias correction was successful (Figs. 4.10 and 4.11).At Agassiz, EDW has the lowest bias during the day and a short peak of larger bias at night (Fig.44l l ll l l ll ll lllll l lllll lll l l llll ll ll lllll ll lll llll l l l l l l l ll l ll l l l l l l l l l l l l l l l l ll l l ll l l l l l l l ll ll l llll l l llllll l l lll l lllll l l lllll l lllll l l lll l lllll l l lllll l lll ll llllll lll llll ll l ll l l lll l l l l lll l ll lll l l l l l ll ll lllllll ll l ll l ll l l ll l ll ll llllllll l l lll l lllll l l lllll l lll ll l l lll l lllll l l lllll l ll−400−350−300−250−200−150−100−500501001502001 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47Forecast hourBias (amperes)lllllllllEDWBDEBWBEBDWBEDBWDEBWEBDWEDBWQSTRSTRAgassizFigure 4.11: Forecast bias of each thermal rating method at each forecast hour for Agassiz.Forecast hours start at 0100 PST. Biases closer to zero are better.4.11), while at the other locations, EDW has the highest bias at night, and lowest during the day(not shown). Positive biases indicate overforecasting of DTRs from overforecasted wind speeds, aswas shown in Figures 4.1c and 4.1d. Agassiz has the lowest positive wind speed bias (Figs. 4.1a- 4.1d), which can be seen in the smallest overforecasting bias of DTRs (4.10). Bias-correction ofwind speeds and of DTRs calculated from raw wind speeds has very similar success, reducing DTRbias at all locations (Figure 4.1e, 4.1f, 4.10, 4.11).BDEBW, BEDBW, and BEBDW have similar, essentially negligible, biases. Their largest bi-ases are overnight at Agassiz − approximately -15 A (Fig. 4.11). DEBW, EDBW, and EBDW havelarger bias, larger diurnal cycles of bias, and larger hour-to-hour variability of bias than BDEBW,BEDBW, and BEBDW (Figs. 4.10 - 4.12). This is most notable at Agassiz, when their negativebiases become larger (up to -60 A) overnight. This indicates that the forecast configurations thatemployed bias-correction only once − either to weather forecasts or to DTR forecasts calculatedfrom raw weather forecasts − over-corrected the positive EDW bias, yielding negatively biasedDTR forecasts that assume less environmental line-cooling at night than really occurs. This over-correction, however, is only an issue at Agassiz, and even then, a small one.Biases of all bias-corrected DTR forecast configurations have a diurnal cycle, but BDEBW,BEBDW, and BEDBW have less variation by offset (forecast hour) than the other forecast con-figurations at each location (e.g. Fig. 4.12). The reduction in bias magnitude and variability is45llllllllllllll lllllllllllllllllllllllllll lllllllllll lll llll llllll l ll l llllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllll ll l llllllllllll l lll ll ll l llllllllllllll lllll lllllll llllllll llllllllllll ll ll lllll lllll lllllllllllllll lllll lllll llll l−20−15−10−5051015201 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47Forecast hourBias (amperes) llllllBDEBWBEBDWBEDBWDEBWEBDWEDBWD'arcyFigure 4.12: Timeseries of the bias-corrected DTR forecast methods at each forecast hour forD’arcy. Biases closer to zero are better.clear between DEBW, EBDW, and EDBW and their DTR configuration counterparts that have hada second round of bias correction applied (BDEBW, BEBDW, and BEDBW).The reason there is any remaining bias in the bias-corrected DTR forecasts is that the analysesin this thesis mimic an operational forecast setting where the future bias is not known yet. Hence,all bias corrections are based on the biases over the past 30 days, as was explained in Chapter 3.A Tukey honest significant difference (HSD) test found that there are no statistically significantdifferences between the bias distributions of any of the bias-corrected DTR forecast methods. How-ever, while statistically no one bias-correction method is superior to all others, visual inspectionreveals that BDEBW, BEBDW, and BEDBW have smaller biases than DEBW, EBDW, and EDBW.This indicates bias correction methods that perform a second bias-correction of the final ensembleaverage of DTR forecasts are more successful, essentially eliminating systematic error.4.2.2 MAEAll bias-corrected DTR forecasts and QSTRs have significantly higher MAEs at Agassiz thanat the other locations, all of which behave similarly (Fig. 4.13). At Big Silver 2, Pemberton Baseand D’arcy, EDBW, BEDBW, DEBW, and BDEBW have lower MAEs than EBDW and BEBDW,which have higher MAEs but smaller MAE distributions. EDBW has the lowest MAE of all thermal46rating methods throughout the forecast horizon (Fig. 4.14). A Tukey HSD test performed on theMAE distributions found that MAEs of EDBW, DEBW, BEDBW, and BDEBW are not statisticallydifferent from each other, but are statistically better than the MAEs of EBDW and BEBDW. Thehigher ensemble mean errors of BEBDW and EBDW are consistent with their wider probabilisticspread (section 4.1).Figure 4.13: Boxplots of the MAEs of all forecast methods. Smaller MAE is better.The MAEs of all thermal rating methods have clear diurnal cycles at each weather station (Fig.4.14− only Big Silver 2 and D’arcy shown). The success of MAEs between thermal rating methodsstays approximately the same offset (i.e. forecast hour) to offset.47ll l l llll ll lllllllll l ll llll l l lll lll lll llllll l l l lll l llllll lllllll l llll l lll lll lll lllll l l l lll l l lll lllllllll llll l l lll lllllll l l l llllll l lll l lll lllllllllllllll lll ll l lll lllllllllllllll llll l l llll ll l lllll ll l ll llll l l lll lll llll ll l l l lll l llll l lll llll l l lllll l ll llll ll l l l llll l ll l ll l lll llll lll l llllll l ll l llllll llll ll l l lllll ll l llll lll llllllll lll lll ll l llll lll llllllll ll1001502002503003504001 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47Forecast hourMAE (amperes)lllllllllEDWBDEBWBEBDWBEDBWDEBWEBDWEDBWQSTRSTRBig Silver 2(a)ll l l lllllll ll ll l lllllll ll ll l llll lll l lll lllll l lll l l l llll llllllll ll l l l llll lllll l lllll l l lllllll l lll lllll l ll ll ll l lllllll l lll llll l l llll l lllllllllllllll lllll ll l l lll l lll llllll ll llll l l l l l lll l lll llllll ll llll l l l llll lll ll llllllll l llll l l llll lll ll lllllll l l lllll lllll l l l llllll ll lllllll lll ll llllllllllllll l l l l ll ll llllllllllllllll l ll l l l llll l lllllll lll ll l l l llll llllll lll1001502002503003504001 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47Forecast hourMAE (amperes)lllllllllEDWBDEBWBEBDWBEDBWDEBWEBDWEDBWQSTRSTRD'arcy(b)Figure 4.14: Mean absolute error of all the DTR forecast methods at each forecast hour at (a) Big Silver 2, and (b) D’arcy.Smaller MAEs are better.48At Big Silver 2, Pemberton Base, and D’arcy, the MAEs of all bias-corrected DTR forecasts,QSTR, and STR increase by approximately 100 A to peak around midday, and are at their lowestovernight. However, at Agassiz, all DTR forecast MAEs are very similar throughout the forecasthorizon, being highest in the early morning, and lowest in the evening. EDW MAE diurnal cyclesare different than those of the bias-corrected forecast methods at all locations except Agassiz. Thisindicates that for those three stations, overnight errors are more systematic in nature and can beremoved via bias correction, whereas midday errors are more random in nature. MAEs of DTRforecasts were either greatly reduced by bias-correction, or at the very least, did not increase (Agas-siz).The STRs, QSTRs, and EDW DTRs have the largest MAEs, although QSTRs do the bestwith MAEs in the same range as those of the bias-corrected DTR forecasts at Pemberton Base andD’arcy. STRs have the largest range of MAEs over the forecast horizon, while EDW forecasts havethe highest MAEs.4.2.3 CorrelationThe correlation coefficient distributions for all DTR forecast methods are similar at all loca-tions, except Big Silver 2, that has noticeably higher and more variable correlations (Fig. 4.15).Raw EDW forecast correlations are not substantially different than the bias-corrected methods, butQSTRs and STRs do exhibit a much larger range of correlations than the DTR forecast methods. ATukey HSD test found no statistical difference between the correlations of the various DTR fore-cast methods, and that all methods have statistically better correlations than the QSTRs and STRs.This shows that forecast association was not worsened by the bias correction. The correlations ofall methods are, unfortunately, low. There is no location where any thermal rating method is wellcorrelated with ampacity.Correlations of all methods reach daily minima in the morning through early afternoon, andpeak overnight (Fig. 4.16). At no forecast hour does any thermal rating method become well corre-lated with ampacity. DTR forecasts have smaller diurnal variations in correlation than QSTRs andSTRs, and hover near correlations of 0.35. QSTRs and STRs have similar correlations overnight.The significant drop in correlation of QSTRs and STRs during the day at Agassiz, Big Silver 2, andD’arcy, indicates there is greater ampacity variability that those methods are not able to follow. Thisis most likely due to greater wind speed variability during the day than at night, indicating windspeeds are the dominant factor influencing ampacity during the day, whereas less variable temper-atures have more of an influence overnight, when wind speeds are also less variable. This impliesthat 0.35 is the approximate correlation value that results from the thermal ratings roughly followingthe annual temperature cycle. The fact that DTR forecast correlations do not drop as significantly49Figure 4.15: Boxplots of correlation coefficients of each forecast method at each location.Correlation values closer to +1 are better, and 0 indicates no skill.during the day as those of QSTRs and STRs indicates that, although their correlations are low,NWP-based DTR forecasts do have some skill (more so than monthly and seasonal averages, i.e.climatology) at forecasting increased ampacity variability during the day, as well as the transitionfrom low to high ampacity variability and vice versa.An examination of input variables indicates that poor correlation between DTR forecasts andampacity is proportional to the poor correlation between wind speed forecasts and observations, aswind speed is proportional to thermal ratings (Fig. 4.1; section 2.1, equations 1,3,4, and 6). Rawand bias-corrected wind speed forecasts are not well correlated with wind speed observations atany forecast hour or weather station, whereas temperature correlations are quite good (e.g. greaterthan 0.9; Fig. 4.17). Wind speed forecasts have correlations between 0.20 and 0.30, similar toDTR forecast correlation values. All other weather stations have similar magnitudes of correlationcoefficients for temperature and wind speed forecasts (not shown).The fact that wind speed forecasts are not well correlated with wind speed observations islikely due to multiple reasons, including physical parameterization schemes that are not correctlyrepresenting small scale turbulence, and horizontal grid resolution that is not fine enough to resolvewind speed flow in mountainous terrain.50l llllll lllll lllllllllllllllll lllll lllll llllllllll ll ll llllllllllll lllllllllllllllll lllll ll ll l l lll llllllllllllll llllllllllll llllllllllll llllllllllll llll l llll lllll llllllllll lllllllllllllll lllllllll l llllllllllll l lll l lllll lllllll lll l ll llllll lllllllll lllllllllllll llllllllll lllllllll− 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47Forecast hourCorrelation CoefficientlllllllllEDWBDEBWBEBDWBEDBWDEBWEBDWEDBWQSTRSTRAgassizFigure 4.16: Timeseries of correlation coefficients of all thermal rating methods at each fore-cast hour at Agassiz. Correlation values closer to +1 are better, and 0 indicates no skill.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23Temperature Correlation at AgassizForecast hourCorrelation Coefficient0. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23Wind Speed Correlation at AgassizForecast hourCorrelation Coefficient0.−corrected(b)Figure 4.17: Correlations of 24-hour (a) air temperature and (b) wind speed forecasts withobservations at Agassiz.514.2.4 SummaryDTR forecast methods cannot be meaningfully differentiated via correlation, as there are in-significant differences between correlations of DTR forecast methods, and all correlations are small.Therefore, bias and MAE results are examined to determine, by process of elimination, the mostsuccessful thermal rating method.QSTRs, STRs and EDW forecasts have the largest biases and MAEs out of all thermal ratingmethods. QSTRs have similar MAEs to those of the bias-corrected DTR forecasts, but much largerbiases. BEBDW and EBDW have larger MAEs than the other bias-corrected DTR forecast methodsat three out of the four weather stations. DEBW, EDBW, BDEBW and BEDBW are all among thelowest MAEs. The median and variation in biases of DEBW, EBDW, and EDBW are larger (worse)than BDEBW, BEBDW, and BEDBW.Therefore, BDEBW and BEDBW appear to be equally the most skillful DTR forecast configu-rations, and thus the most skillful thermal rating methods.4.3 Probabilistic forecast verificationIn this section, probabilistic raw and bias-corrected DTR forecasts are evaluated using theCRPS, calibration deviation ratios (CDRs), and PIT histograms to assess forecast accuracy andcalibration. See Appendix E for a summary of verification metrics. CRPS and calibration deviationratios were calculated for each forecast hour over the 20-month data period. The distributions ofthose metrics over the 47-hour forecast horizon are shown in Figures 4.20 and PIT histogramsA reliable, or calibrated, probabilistic forecast is one where the observed frequencies corre-spond to the forecasted probabilities, e.g., ampacity verifies below the 20th percentile of a DTRforecast 20% of the time. Calibration can be assessed using PIT histograms that show the frequencywith which observed values verify in different forecast probability bins, over a given time. PIThistograms with a pyramid or U-shape indicate the spread of the probabilistic forecasts is eithertoo wide or too narrow, respectively; the majority of observations are verifying in the mid-range orextreme forecast probabilities. If a PIT histogram is skewed to either the left or right, the forecastshave a under- or over-forecasting bias, respectively, and the observations verify too often in thelow or high forecast probabilities. A flat PIT histogram is best, because it means observations areverifying equally in all forecast probabilities and indicates a calibrated forecast. PIT histograms foreach DTR forecast configuration, forecast hour, and location were examined. Four PIT histogramsare shown here to illustrate the effectiveness of calibration.520.0 0.2 0.4 0.6 0.8 1.0Cumulative probability05101520Frequency (%)station=679, offset=6, config=raw(a)0.0 0.2 0.4 0.6 0.8 1.0Cumulative probability05101520Frequency (%)station=185, offset=6, config=raw(b)Figure 4.18: Raw EDW forecast PIT histograms for (a) Agassiz and (b) Big Silver 2. A flatterdistribution is better.At each location and all offsets, raw EDW forecast PIT histograms are greatly skewed to theright, indicating that this configuration overpredicts DTRs significantly, and are therefore uncali-brated (Fig. 4.18). The PIT histogram for Agassiz shows a mix of overforecasting and underdisper-sion. The overforecasting bias is smaller at Agassiz than at the other stations, which is consistentwith findings in section 4.2.The PIT histograms of all DTR forecast configurations to which calibration was applied aremuch closer to being perfectly flat (Fig. 4.19). There is a large improvement in reliability betweenFigures 4.18 and CDRDue to sampling error, the PIT histogram of a calibrated forecast will never be completelyflat. In Figure 4.19, PIT histograms are almost flat, with some variation among bins. Calibrationdeviation ratios (CDRs) ([actual deviation] / [expected deviation]; Appendix E) show whether aprobabilistic forecast is acceptably calibrated (i.e. its PIT histogram is flat enough) or not. The ex-pected deviation from a completely flat histogram is a function of the number of forecast probabilitybins and the sample size. The actual deviation is a function of the forecast probability bin counts.CDR values equal or less than 1 are ideal, as this means the deviation from perfect is equal or lessthan the expected deviation given the sample size and number of bins.The expected deviations at Big Silver 2, Pemberton Base, and D’arcy are nearly the same atall offsets: 0.0134, 0.0133, and 0.0133, respectively. The expected deviations at Agassiz vary by a530.0 0.2 0.4 0.6 0.8 1.0Cumulative probability05101520Frequency (%)station=679, offset=6, config=meanFactor2cal(a)0.0 0.2 0.4 0.6 0.8 1.0Cumulative probability05101520Frequency (%)station=185, offset=6, config=meanFactor2cal(b)Figure 4.19: Calibrated BEDBW forecast PIT histograms for (a) Agassiz and (b) Big Silver 2.A flatter distribution is better.factor of 0.0001 offset to offset, but are approximately 0.0124.The CDRs of all bias-corrected, calibrated DTR forecast methods demonstrate the effectivenessof probabilistic calibration in producing reliable forecasts at each weather station (Fig. 4.20). Thedifferences in calibration between each method are negligible. However, EDW forecasts are notcalibrated at any location.Further, all bias-corrected DTR forecast methods are calibrated at all offsets (forecast hours;not shown). There is no diurnal cycle and no significant variation in deviation ratios over the fore-cast period for any of the bias-corrected, calibrated forecast methods. The deviation ratios of theuncalibrated EDW forecasts by contrast do have a diurnal cycle and significant variation at eachlocation.4.3.3 CRPSThe CRPS is a measure of the probabilistic forecast errors, integrated over all forecast proba-bilities. Smaller values are better, meaning there is less discrepancy between the cumulative distri-bution function of the forecasts and the step function of the observations (Appendix E). The CRPSof a probabilistic forecast reduces to the MAE for the equivalent deterministic forecast (Hersbach,2000). It can be broken down into reliability, resolution, and uncertainty components.The non-calibrated, non-bias-corrected EDW forecasts have much higher CRPSs than any ofthe bias-corrected DTR forecast methods (Fig. 4.21). The median and distribution of CRPSs formost calibrated, bias-corrected methods (EDBW, DEBW, BEDBW, and BDEBW) are very similar.54Figure 4.20: Boxplots of the CDRs (dimensionless) of all DTR forecast methods at each loca-tion. Smaller values (equal or less than 1) are better.As with MAE scoring of the deterministic forecasts, accuracy for probabilistic forecasts is worst atAgassiz. At all stations except Agassiz, EBDW and BEBDW, have noticeably larger (worse) scoresthan the other bias-corrected DTR configurations. At Agassiz, all bias-corrected DTR forecastmethods have CRPSs that are very similar.The diurnal cycles of CRPS for the bias-corrected DTR forecasts are large (Fig. 4.22). CRPS islargest (worst) during the afternoon at all stations except for Agassiz. There, CRPS is lowest in theafternoon and highest in the morning. BEBDW and EBDW have the worst CRPSs at the majority offorecast hours throughout the forecast horizon at all weather stations except Agassiz, while EDBWhas the best.55Figure 4.21: Boxplots showing the CRPSs for each DTR forecast method at each location. Asmaller CRPS is better.56(a) (b)Figure 4.22: Timeseries of CRPS of all DTR forecast methods at each forecast hour at (a) Agassiz and (b) Pemberton Base.Smaller CRPS is better.57Because the uncertainty component of CRPS is the same for all methods (function of the ob-servations only), and reliability is roughly the same for all bias corrected, calibrated methods, thelarger CRPSs of BEBDW and EBDW is an indication that they have worse resolution than the othermethods. Resolution refers to the ability of the probabilistic forecasts to skillfully differentiate whenand by how much the forecast distribution departs from the climatological distribution.A Tukey HSD revealed significant differences between the best four methods (EDBW, DEBW,BEDBW, and BDEBW) and the worst three (EBDW, BEBDW, and EDW), but no significant differ-ences amongst the best four. For the deterministic forecasts, MAE showed similar results.4.3.4 SummaryThe results of the CRPS and CDRs for each DTR forecast method show that the probabilisticbias-corrected DTR forecasts are calibrated and significantly more accurate than the EDW forecasts.The CRPS of all thermal rating methods shows the same results as the MAEs − BEBDW andEBDW have the largest CRPSs out of all bias-corrected forecast configurations, while all otherbias-corrected configurations have equally the lowest CRPS. All DTR forecasts, except EDW, haveCDRs very close to 1, indicating the bias-corrected probabilistic DTR forecasts are calibrated. ThePIT histograms help visualize this result. Further, a check revealed that calibration did not worsenthe MAE of any bias-corrected DTR forecast configuration.The bias-corrected DTR forecast configurations cannot be differentiated by calibration (be-cause they are all calibrated), but the CRPS results show EDBW, DEBW, BEDBW, and BDEBWgive the most accurate probabilistic DTR forecasts. This conclusion, combined with the results atthe end of section 4.2, suggest that BEDBW and BDEBW are recommended for both deterministicand probabilistic DTR forecasts.4.4 Assessing DTR forecast safety and efficiencyIn this section, probabilistic DTR forecast percentiles are evaluated in terms of their miss andextended hit rates, to determine their suitability for safe operational decision making relative toconventional thermal rating methods.As stated in section 3.1, the miss rate is defined as the percentage of forecast-observation pairswhere the DTR forecast exceeds the observed ampacity. This is important because it quantifies therisk of the DTR forecast. Namely, the miss rate gives the percentage of forecasts that could causeunsafe current to be put through the line.Forecast percentiles should directly correspond to the percent risk of calibrated (reliable) prob-abilistic forecasts violating ampacity. For example, observed ampacities would fall below a 10th58percentile DTR forecast, if perfectly reliable, 10% of the time. This makes calibrated probabilisticDTR forecasts extremely useful for utilities. Contrast this with QSTRs and STRs, which do notchange with the weather, and hence have an average, constant level of risk.Figure 4.23 shows the miss rates (i.e., percent risk) of each DTR forecast method for eachprobabilistic forecast distribution percentile, compared to the miss rates of the QSTR and the STR(which are constant), at Big Silver 2. A perfect (reliable) probabilistic forecast would have themiss rates equal to the forecast percentiles, i.e, aligned along the 1:1 line. While the bias-corrected,calibrated DTR forecasts have near-perfect miss rates, the EDW DTR forecast percentiles have missrates that exceed forecast percentiles by up to∼30%. This means that EDW forecasts too frequentlyexceed the observed ampacity of the line. For example, the 30th percentile DTR forecast, whichshould have a miss (or risk) rate of 30%, actually exceeded the observed ampacity over 50% of thetime. Uncalibrated forecast percentiles, produced from empirical ensemble member distributionsor other methods, cannot be trusted. Calibration is an essential post-processing step to ensure safeDTR forecasts.lllllllllllllllllllllllll l l01020304050607080901000 10 20 30 40 50 60 70 80 90 100Forecast PercentilesMiss Rate (%)lllllllEDWBDEBWBEBDWBEDBWDEBWEBDWEDBWQSTRSTRBig Silver 2Figure 4.23: The miss rate for each DTR forecast configuration for all probabilistic forecastdistribution percentiles, the QSTR and the STR at Big Silver 2. Values closer to thestraight line segment are better.Among the different bias-corrected, calibrated DTR forecast configurations, the miss rates59are almost identical at any individual forecast percentile. Miss rates below the 50th percentile forthe bias-corrected DTR forecast methods are almost exactly equal to their corresponding forecastpercentile. Beginning at the 80th percentile, the miss rates become noticeably lower than theircorresponding forecast percentile. At all locations, the miss rates of the bias-corrected forecastmethods for 80th percentiles and above are ∼5% lower than they should be. This indicates that, forexample, the 90th percentile forecast, only exceeded observed ampacity 85% of the time. However,these differences are small, especially since DTR forecast percentiles at this end of the forecastdistribution, with more than an 80% risk, are very unlikely to be used in operations.The constant STR and QSTR miss rates are more safe than most calibrated DTR forecastpercentiles, but less safe than the lowest DTR forecast percentiles − the ones likely to be used inoperations. At all locations, the miss rates of the 1st, 5th, and 10th bias-corrected and calibratedforecast percentiles are lower than (i.e.,safer than) those of the QSTRs and STRs. At PembertonBase, this is also true of the miss rates of the 20th percentile DTR forecasts, and is also true of thoseof the 20th and 30th percentiles at D’arcy. The 1st percentile calibrated DTR forecasts are saferthan the STRs and QSTRs throughout the 20-month study period at all locations, and are congruentwith the typical 1% risk tolerance chosen for DTR use, as reported by others (Zhang et al., 2008;Heckenbergerova et al., 2013).Focusing on just calibrated, bias-corrected DTR configurations, the miss rate - forecast per-centile differences are small (< 10%), and become increasingly negative with higher forecast per-centiles (Fig. 4.24). 20th percentile forecasts and above have miss rates that are too low, meaningthose DTR forecast percentiles are too low (overly conservative). At most locations and for manypercentiles, EDBW and DEBW have larger negative differences than the other configurations, indi-cating they are more conservative (fewer ampacities fall below the forecasts). At D’arcy, BEDBWhas the smallest differences out of all configurations for the 1st to 60th percentiles. There is verylittle difference between methods at Agassiz.Focusing on the three lowest DTR forecast percentiles (1st, 5th, 10th), there is a small positivedifference between the miss rates and their corresponding percentiles (Fig. 4.25). This indicates thatthe DTRs are less safe (more ampacities fall below the forecasts) than they should be by only about1% − a small error. The results in section 4.3, and the miss rates calculated in this section, haveproven that the calibrated probabilistic DTR forecasts are within the expected range of reliability,and the deviation from perfect calibration is very small. Therefore, if the 1st, 5th, or 10th forecastpercentiles were chosen as the risk tolerance in an operational DTR forecast scenario, it would beexpected that the actual risk would be within about 1% of those percentiles.The reliability of lower range forecast percentiles is the most crucial to thermal ratings be-cause these are the percentiles that give the least risk and are therefore most likely to be used as60lllllllllllllllllllllllllllllllll lllll lllllll llll−6−5−4−3−2−10121st 5th 10th 20th 30th 40th 50th 60th 70th 80th 90th 95th 99thForecast PercentilesMiss Rate (%) − Percent (%)llllllBDEBWBEBDWBEDBWDEBWEBDWEDBWBig Silver 2Figure 4.24: The difference between the miss rate and its corresponding DTR forecast per-centile, for each calibrated, bias-corrected DTR forecast configuration, at Big Silver 2.Values closer to zero are operational thermal rating. A reliability error of 1% when a planner is using the 5th percentileforecast because they can only tolerate a 5% risk, could be considered substantial. Therefore it’simportant to verify the results of a calibrated forecast system, as has been done here. Knowing this,the engineer may choose a 4th percentile DTR forecast to ensure no greater than 5% risk. The keyis that the engineer knows the level of risk they are taking, and can choose their level of risk. In anSTR or QSTR system, the risk is likely greater because (a) the average risk of the STR and QSTRratings calculated here using standard methods is fairly high; and (b) the rating is fairly static, whiletrue ampacity varies substantially with the weather, so the risk on a given day is unknown to usersof STRs and QSTRs.The degree of reliability amongst the calibrated, bias-corrected DTR forecast configurationsvaries between the 1st, 5th, and 10th percentile, as well as by location. BEBDW has the smallestdifference between miss rate and 1st forecast percentile at each location except Agassiz. DEBWand EDBW have the smallest differences between miss rate and the 5th and 10th percentiles ateach weather station except D’arcy. No one bias-corrected DTR forecast configuration is the mostreliable at all percentiles and locations. This has been a consistent result throughout Chapter 4.61llllllllllllllllll0121st 5th 10thForecast PercentilesMiss Rate (%) − Percent (%)llllllBDEBWBEBDWBEDBWDEBWEBDWEDBWBig Silver 2Figure 4.25: The difference between the miss rate and the DTR forecast percentile it wascalculated with, for each bias-corrected DTR forecast configuration, for the 1st, 5th,and 10th percentiles, at Big Silver 2. Values closer to zero are better.As stated in section 3.1, the extended hit rate shows how often a thermal rating is the highest(ideal for maximizing system optimization), while also being at or below the observed ampacity(ideal for safe system operations), relative to two conventional thermal rating methods. Extended hitrates were calculated for each forecast percentile of each DTR forecast configuration relative to theQSTR and the STR, and vice versa (e.g., shown for BEDBW in Fig. 4.26). Comparisons betweenall other bias-corrected, calibrated forecast methods and the QSTR and STR are very similar, soonly those of BEDBW forecasts will be discussed here. Larger values mean that thermal ratingmethod is more efficient, while still being safe, more often than the other methods. The percentilesof greatest interest are the 1st, 5th, and 10th, as they are the lowest risk and most likely to be usedoperationally.It is important to note that QSTRs and STRs have fixed average risk ratings of ∼19-46% and∼12-35%, respectively. Technically, then, QSTRs should only be compared to∼20 - 50th percentileDTR forecasts, because they have a similar risk; the same can be said for STRs only being compared62llll l lllllllllllll l lllllllll l l l l l ll ll l l02040601st 5th 10th 20th 30th 40th 50th 60th 70th 80th 90th 95th 99thForecast PercentilesExtended Hit Rate (%)lllBEDBWQSTRSTRStation=185, config=BEDBWFigure 4.26: The extended hit rates of BEDBW forecasts, the QSTR, and the STR for eachforecast percentile, at Big Silver 2. Larger values are ∼10 - 40th percentile DTRs. For example, comparing 5th percentile DTR forecasts to STRs isunfair, because they are effectively held to different standards for safety. That said, a full comparisonis conducted here to contrast probabilistic DTR forecasts with current industry practice (QSTRs andSTRs).The extended hit rates of the BEDBW forecasts, relative to the QSTR and the STR, increasethen decrease with increasing forecast percentiles. It is the opposite for the extended hit rates of theQSTR and the STR, relative to the BEDBW forecasts. This is because the probabilistic forecastswill be greater than observed ampacity more often (riskier) with increasing percentiles by definition.Low DTR forecast percentiles give lower ratings and therefore, are very safe and conservative.QSTRs and STRs have an average risk, as they have been determined in this thesis, are less safe,and thus are often more efficient, than the lowest DTR percentiles. As DTR forecasts increase withincreasing forecast percentiles they can more often be more efficient than the QSTR and STR whilestill being safe (20th to 50th percentile, Fig. 4.26). Soon, however, they become less frequently safeand efficient than the QSTR, and eventually the STR (60th to 99th percentile, Fig. 4.26). The STRsalways have lower ratings than the QSTRs, except in March and October. Therefore if they wereboth less than the observed ampacity, the QSTR would almost always be more efficient.The extended hit rates of BEDBW forecasts exceed those of the QSTR and STR for the 30th,40th, and 50th percentiles at Agassiz; the 20th, 30th, 40th, and 50th percentiles at Big Silver 2;63and the 30th, 40th, and 50th percentiles at Pemberton Base. At D’arcy, BEDBW forecasts are themost successful, having higher extended hit rates at the 5th, 10th, 20th, 30th, 40th, 50th, and 60thpercentiles. At most locations, the 1st, 5th, and 10th percentiles of the probabilistic BEDBW fore-casts have lower extended hit rates than the QSTR, because they are lower risk, and so ampacitiesfall below them less often. The QSTR has higher average risk (18 - 46%), and therefore higher ex-tended hit rates, because it allows higher currents. At D’arcy, the extended hit rates of the BEDBWforecasts at the 1st and 5th percentiles are comparable to those of the QSTRs, but larger than theQSTR at the 10th percentile. At all other locations the QSTR is more often the most efficient, whilebeing safe, for the three lowest percentiles by large amounts. For all locations, the STR (12 - 35%average risk) has lower extended hit rates than BEDBW from the 1st to the 80th percentiles. TheSTR always has a lower extended hit rate than the QSTR. This means it is the least efficient thermalrating method, while still being safe, when compared to the QSTR and the BEDBW forecasts.Probabilistic DTR forecasts are calibrated such that ampacities will only fall below the proba-bilistic forecast the correct percentage of time. This means the lowest DTR forecast percentiles arethe safest thermal rating methods. However, safety comes at the expense of efficiency. If the 1st,5th, or 10th percentile DTR forecast was used operationally at Agassiz, Big Silver 2, and PembertonBase, it would be more conservative, and not more efficient, than the QSTR. Conversely, if the 20th,30th, 40th, or 50th percentile DTR forecast was used operationally, it would be safe while beingmore efficient more often than the QSTR and the STR. However, it is unlikely that a power utilitywould implement a new thermal rating method that is unsafe 30% or 40% of the time, despite thefact that their current method, perhaps unknowingly, is unsafe 30 or 40% of the time.Figure 4.27 shows the extended hit rates at each forecast percentile for each DTR forecastconfiguration relative to the QSTR and the STR. EDW has similar extended hit rates as the bias-corrected DTR forecast configurations for the 1st, 5th, and 10th percentiles, as well as 20th and30th at Agassiz, but above those percentiles, they drop significantly compared to the other forecastconfigurations. The extended hit rates are very similar between bias-corrected forecast methods,meaning DTR forecasts are similarly efficient over the QSTR and STR. Extended hit rates are lowerfor BEBDW and EBDW at Big Silver 2 and Pemberton Base for the 20th - 80th percentiles, aswell as for the 1st - 30th percentiles at D’arcy. This means BEBDW and EBDW are less efficientthan the other bias-corrected, calibrated methods, but there is no clear winner among those superiormethods. There is no significant difference between the bias-corrected configurations at Agassiz.Probabilistic bias-corrected DTR forecasts are more efficient, while being safe, more oftenthan QSTRs and STRs, but only more efficient for engineers willing to take risks of 20 - 50%.As mentioned near the beginning of this section, however, when considering safety alone, the 1st,5th, and 10th percentile calibrated, bias-corrected DTR forecasts are safer than QSTRs and STRs.64llll l lllllllllll l llllllllll llllll llllll llllllll l l0102030401st 5th 10th 20th 30th 40th 50th 60th 70th 80th 90th 95th 99thForecast PercentilesExtended Hit Rates (%)lllllllBDEBWBEBDWBEDBWDEBWEBDWEDBWEDWStation=185Figure 4.27: The extended hit rates for each DTR forecast configuration percentile relative tothe QSTR and the STR, at Big Silver 2. Larger values are better.In the lowest forecast percentiles, all bias-corrected, calibrated DTR forecast methods show similarextended hit rates, indicating that in an operational environment, all forecast methods would performequally well. Miss rates and extended hit rates show that BEBDW and EBDW, as well as EDW, areless safe and less efficient, than the other bias-corrected DTR forecast configurations.BEDBW, and BDEBW are among the most successful DTR forecast methods, which is con-sistent with results from previous sections.65Chapter 5DTR forecast evolution mappingIn this chapter, the spatial and temporal variations of raw EDW DTR forecasts along the case-study powerline are analyzed. An overview of the synoptic and regional weather is given to showthe direct effect of the environmental conditions on DTR forecasts. Gridded wind speed and tem-perature forecast maps from the GFS-initialised MM5 model, run at 4-km horizontal grid spacing(Table 3.2) are presented here to analyze the environmental conditions used to calculate EDW DTRforecasts. This ensemble member was chosen because, on average, it gave the lowest (most conser-vative) DTR forecasts along the line. January 18 and August 19 have weather scenarios typical ofwinter and summer, and are examined next to illustrate DTR variations along the powerline.5.1 Winter exampleOn January 18, 2016, an upper level trough was just off the west coast of BC, bringing south-westerly flow across the south coast and case-study region throughout the day (Fig. 5.1). A low-pressure system arrived at the South Coast in the afternoon bringing windy, cloudy, and rainy condi-tions overnight. Wind speeds were forecasted to generally increase in the region late in the day frombelow 1-10 km/h to 1-20 km/h (Figs. 5.2 - 5.6). The wind speed range is due to local variations alongthe powerline. Wind direction was kept constant in the DTR forecasts, as was discussed in Chapter3 and Appendix D, therefore wind direction forecast vectors are not discussed. Temperatures alongHarrison Lake and surrounding Pemberton were forecasted to be above zero (approximately 2◦C),and below zero everywhere else along the line (approximately -2◦C). A small increase in tempera-tures was forecasted for midday, but for the most part temperatures stay consistent throughout theforecast period. The largest variability of environmental conditions comes from the wind speedforecasts.66Figure 5.1: NCEP reanalysis of mean sea level pressure, atmospheric thickness, and 500hPageopotential heights valid on 1000 PST on January 18, 2016.67Figure 5.2: Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 0100 PST on January 18,2016.68Figure 5.3: Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 0600 PST on January 18,2016.69Figure 5.4: Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 1200 PST on January 18,2016.70Figure 5.5: Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 1800 PST on January 18,2016.71Figure 5.6: Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 2300 PST on January 18,2016.Figures 5.7 - 5.11 show EDW DTR forecasts for each segment of the powerline at 0100, 0600,1200, 1800, and 2300 PST on January 18, 2016. Lower DTR forecasts, shown by shades of red,result from environmental conditions that are less conducive to line cooling (lower wind, higher airtemperature and solar radiation); whereas higher DTR forecasts, shown by shades of blue, resultfrom environmental conditions that are more conducive to line cooling (higher winds, lower airtemperature and solar radiation).EDW DTR forecasts are lowest along Harrison Lake and near Pemberton throughout the fore-cast period, coincident with simultaneous lower wind speeds (less than 7.5 km/hr) and above zerotemperature forecasts (Figs. 5.2 - 5.6). DTR forecasts are lowest early in the day between HarrisonLake and Pemberton, and increase with increasing wind speed forecasts (less than 7.5 km/h at 0100,0600 and 1200 PST, up to 10 km/h at 1800 PST, and up to 15 km/h at 2300 PST). North of Pember-ton, DTRs are predominantly high throughout the period, coincident with below-zero temperaturesand relatively high wind speeds (up to 10 km/h for most of the day, up to 20 km/h at 2300 PST).This case demonstrates that wind speed is the dominant factor in determining DTR magnitude.72Higher DTRs correspond with higher wind speed forecasts, and vice versa, despite variations inforecasted temperature. For example, at 1800 PST, wind speeds (Fig. 5.5) and DTRs (Fig. 5.10)are lower at the northern end of the line, and higher closer to Pemberton. Colder temperatures atthe northern end of the line did not make for higher DTR forecasts, because wind speed effectsdominated. Similarly, at 0600 PST, when wind speed forecasts are uniformly low (less than 5 km/h)along the line (Fig. 5.3), DTR forecasts are also uniformly low along the line (Fig. 5.8) despitetemperature forecast variations. Operation planners using this type of tool (Figs. 5.7 - 5.11) needto know the lowest DTR forecast on the line at one time. The transmission capacity (or ampacity)of a powerline connecting two substations is limited by the transmission capacity of its lowest-rated (most at risk) segment. This planview forecast map would allow planners to see variations inthermal rating along the line, as well as the lowest DTR forecast along the powerline (labeled bythe magenta diamond with a black outline).Figure 5.7: DTR forecasts (amperes) along the case-study powerline for 0100 PST initializedon January 18, 2016. The magenta diamond with the black outline shows the locationof of the lowest DTR forecast along the transmission line. This DTR forecast limits theamount of current allowed for the whole line.73Figure 5.8: DTR forecasts (amperes) along the case-study powerline for 0600 PST initializedon January 18, 2016.74Figure 5.9: DTR forecasts (amperes) along the case-study powerline for 1200 PST initializedon January 18, 2016.75Figure 5.10: DTR forecasts (amperes) along the case-study powerline for 1800 PST initializedon January 18, 2016.76Figure 5.11: DTR forecasts (amperes) along the case-study powerline for 2300 PST initializedon January 18, 2016.5.2 Summer exampleOn August 19, 2016, an upper-level high pressure ridge and surface high pressure were inplace over the southern BC, bringing clear, hot, calm and dry conditions to the case study regionthroughout the day (Fig. 5.12). Wind speeds were forecasted to decrease in the region from 5-30km/h at 0100 PST, down to less than 5 km/h at 1200 PST, where they stayed for the rest of the day(Figs. 5.13 - 5.17). Spatial temperature forecast patterns are similar to those on January 18, withwarmer temperatures surrounding Harrison Lake and the Pemberton area, and cooler temperaturesbetween Harrison Lake and Pemberton, as well as north of Pemberton. There is a strong diurnalcycle of temperature − forecasts are approximately 5◦C higher at midday than in the morning andovernight.77Figure 5.12: NCEP reanalysis of mean sea level pressure, atmospheric thickness, and 500hPageopotential heights valid on 1000 PST on August 19, 2016.78Figure 5.13: Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 0100 PST on August 19,2016.79Figure 5.14: Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 0600 PST on August 19,2016.80Figure 5.15: Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 1200 PST on August 19,2016.81Figure 5.16: Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 1800 PST on August 19,2016.82Figure 5.17: Wind and air temperature forecasts from the MM5 model, with 4 km horizontalresolution, initialized with GFS initial conditions, valid for 2300 PST on August 19,2016.Figures 5.18 - 5.22 show EDW DTR forecasts for each segment of the powerline at 0100, 0600,1200, 1800, and 2300 PST on August 19, 2016. EDW forecasts are highest at 0100 PST, when windspeed forecasts were above 30 km/h along Harrison Lake, up to 20 km/h near Pemberton, and 10km/h or below everywhere else. At 0600 PST, wind speeds were forecasted to decrease (below25 km/h south of Harrison Lake, approximately 5 km/h everywhere else), and thus so did DTRs.For the rest of the day, DTRs are forecasted to be uniformly, extremely low along the powerline,coincident with uniformly low wind speeds (less than 5 km/h).As in January, wind speeds are the dominate EDW DTR variations in August. For example,at 1800 PST, despite temperature forecasts being among the hottest along the line, DTR forecastsare relatively higher at the southern end of the line, coincident with higher wind speed forecasts (upto 10 km/h) in that area. Conversely, at 1200 PST, DTR forecasts are the lowest possible along thepowerline, coincident with low wind speed forecasts (less than 5 km/h) that allow high temperatureforecasts (approximately 26◦C) to lower DTRs further beyond the effects of low winds speeds alone.83Figure 5.18: DTR forecasts (amperes) along the case-study powerline for 0100PST, initializedon August 19, 2016.84Figure 5.19: DTR forecasts (amperes) along the case-study powerline for 0600PST, initializedon August 19, 2016.85Figure 5.20: DTR forecasts (amperes) along the case-study powerline for 1200PST, initializedon August 19, 2016.86Figure 5.21: DTR forecasts (amperes) along the case-study powerline for 1800PST, initializedon August 19, 2016.87Figure 5.22: DTR forecasts (amperes) along the case-study powerline for 2300PST, initializedon August 19, 2016.5.3 DiscussionOverall, EDW DTR forecasts increase and decrease substantially hour-to-hour (by a few hun-dred amperes), as well as spatially between adjacent powerline segments at a given forecast hour.Similarly large day-to-day DTR variability is seen throughout the months of January and August,2016 (not shown). Most DTR variability along the line is due to wind speed and temperatureforecasts, however, fine scale variability is most likely due to varying powerline elevation anglesinfluencing the solar heating of the line.As expected, thermal ratings decrease more during the day in August than in January due toenvironmental conditions that are much less conducive to cooling of the line (low wind, high airtemperatures). However, DTRs increased just as much in August as in January (Fig. 5.18), furtherindicating that high wind speed is the dominant factor cooling the line at any time of year.885.4 RecommendationsThe southernmost end of the line often sees both the highest temperature and highest windspeed forecasts along the line. This is associated with strong outflow and inflow winds exiting andentering the Interior, respectively, via the Fraser Valley gap in the Coast Mountains. However, windspeed forecasts can also be relatively low in this region, especially farther north along Harrison Lakefrom the Fraser Valley gap. Lower wind speeds in conjunction with higher temperature forecastsat that location led to the minimum DTR forecast along the line most often being in the Harri-son Lake region on January 18 and August 19. The combination of high variability and frequentline-minimum thermal ratings indicate this portion of the line is most important to instrument andmonitor.Thermal rating methods that do not account for ever-changing environmental conditions, andinstead assume local weather averages or a flawed “worst-case” weather scenario (QSTR or STR,respectively), by definition cannot accurately represent the high spatial and temporal variability ofampacity. Traditional thermal ratings will almost inevitably be higher than the lowest point ampacityalong an entire stretch of powerline.DTR forecast products such as Figures 5.7 - 5.11, and 5.18 - 5.22 can improve the ability ofa power utility to understand the variation in ampacity along a stretch of powerline under varyingweather scenarios throughout the day and year. In the long term, these products are useful to deter-mine which sections of the line are regularly at risk of thermal overload − which is crucial. Whenat-risk sections are identified, further investigation can be done as to whether conductor upgradesshould be made to add capacity to that section of line. Identification of the exact section that is regu-larly at risk will reduce upgrade costs because no extra material will be used to upgrade sections thatare not in need of it. Alternatively, engineers can use this information to determine where to placeampacity sensors on the line, for monitoring real-time ampacity and calibrating DTR forecasts.Further, DTR forecast maps are useful in studying what sections of the line are vulnerable underparticular weather patterns. This will help with system planning when certain weather patterns areforecast for the powerline; planning engineers will know if they can transmit more or less amountsof power depending on the forecasted weather pattern. These DTR forecast products can also helpengineers develop a sense for their confidence in raw planview DTR forecasts− they may recognizecertain patterns associated with a high or low bias in the DTR forecasts, and can adjust and actaccordingly. These products bring further insight into why and how the grid functions under variousenvironmental conditions. Ultimately, having spatial, hourly DTR forecast products along the linewill help power managers be more informed, resulting in a more efficient power grid operation anddecreased upgrade and maintenance costs.895.5 AnalysisWhile there is great variability in EDW DTR forecasts along the powerline at one time (Figs.5.7 - 5.11, and 5.18 - 5.22), the minimum DTR forecast along the line at each forecast hour is mostimportant to consider, as it represents the line segment feeling the least environmental cooling, andlimits the maximum current possible in the line at that forecast hour. Figure 5.23 shows the differingdistributions of line-minimum DTR forecasts between August and January.Figure 5.23: Boxplot of the lowest DTR forecast along the powerline at each forecast hour, oneach forecast initialization day in August 2016 and January 2016.Regional environmental conditions are much different in August than in January. Air tem-peratures are lower in January, and low pressure systems bringing strong winds and rain are morecommon in January than in August in southern BC. This means January has a smaller diurnal cycle,as conditions are regularly overcast throughout the day and the sun angle is low. These wintertimeenvironmental conditions help maintain cooler line temperatures, keeping minimum DTR forecastshigher. In the summer, the jet stream weakens and shifts north, so there are much fewer low-pressuresystems. High pressure ridging is more common in August, bringing quiescent, clear conditions,and thus stronger diurnal temperature cycles and generally lighter to calm winds. This means envi-90ronmental conditions less conducive to cooling the line are more frequent in the summer, resultingin lower minimum DTR forecasts.The complex terrain where the case-study powerline is located alters the summertime diurnalcycle. The sun sets earlier and rises later for powerline sections in the mountain valleys.Sectionsof powerline that are on mountain ridges or flatter terrain would have more time to heat up duringAugust days, potentially resulting in even lower daily minimum DTR forecasts.The line-minimum EDW DTR forecast spreads of August and January support that there isa larger range of summertime environmental conditions that allow for more extreme high and lowDTR forecasts, whereas wintertime environmental conditions allow for more consistently high DTRforecasts (Fig. 5.23). The higher upper bound of the August spread than the January spread indi-cates that it is possible to have strong environmental cooling of the line in August, similar to andgreater than that in January (as was seen in Figs. 5.7 - 5.11, and 5.18 - 5.22). In general, however,environmental conditions are much less conducive to cooling in August than in January, causing thelower bound and median minimum DTR forecast to be much lower in August.The winter STR (1242 A) and QSTR for January (1298 A) are higher by approximately 200A than the hourly line-minimum DTR forecasts throughout January (Fig. 5.24a). This means thatthe winter STR and the QSTR for January assume environmental cooling of the line that is muchstronger than the NWP-based DTR forecasts produce. Therefore, in January, daily minimum DTRforecasts are safer and more conservative than the STR and QSTR methods.Figure 5.24b shows that the STR and QSTR are not consistently the most conservative thermalrating in August. There are only a few forecast hours overnight where the QSTR (1063 A) is lessthan the line-minimum DTR forecast. The summer STR (959 A) is the lowest thermal rating in theovernight hours, but exceeds the line-minimum DTR forecasts during the day. This means the DTRforecast would put the powerline at greater risk of thermal overload at the time of day when thermaloverload is most likely. The STR and QSTR are closer in magnitude to the DTR forecasts in Augustthan in January. This indicates that the environmental conditions assumed in the summer STR andthe August QSTR are closer to the environmental conditions of the August weather forecasts, thanthe corresponding values for the January weather forecasts.For the two cases in Figure 5.24, neither the STR nor the QSTR is the safest thermal rating(lowest and most conservative) to be applied to the powerline. Ampacity observations are not avail-able, therefore it is impossible to know if any of the thermal ratings are actually too high for thecase-study powerline. However, hourly line-minimum DTR forecasts have a clear diurnal cycle insummer and winter, which is a much more realistic daily variation in ampacity in time and space,that the STRs and QSTRs completely fail to represent. Assuming or averaging historical weatherconditions is not conservative enough, nor does it sufficiently represent diurnal variations in envi-91(a)(b)Figure 5.24: Hourly minimum DTR forecasts along the powerline, at each forecast hour oneach forecast initialization day in (a) January and (b) August 2016. The winter STRand QSTR for January is plotted for January 2016 (a), and the summer STR and QSTRfor August is plotted for August 2016 (b).92ronmental cooling, making for less safe and efficient electricity transmission. This is especially truein complex terrain where there are many and varied microclimates along a transmission line. Thesafest and most efficient thermal rating method should represent the seasonal, daily, subdaily, andspatial variations of environmental conditions along a line.93Chapter 6Conclusion6.1 DTR forecastsThis study sought to show that ensemble NWP-based DTR forecasts would allow for increasedrealisation of transmission capacity, and safer operations, compared to traditional thermal ratingmethods. QSTRs and STRs do not account for any submonthly and subseasonal environmentalvariations, respectively, often leading to rating violations and unrealized transmission capacity. Anincrease in transmission capacity rating leads to greater system flexibility, and safe thermal operationleads to lowered maintenance costs. Power utilities must operate the grid under multiple constraints,therefore greater flexibility leads to better system optimization, and reduced operational costs.On-transmission-line weather and/or ampacity observations are needed to post-process DTRforecasts, but neither were available for the case-study powerline. Therefore, ensemble NWP fore-casts were interpolated to four weather stations near the powerline (Fig. 3.1), and fed into sevendifferent post-processing configurations to create DTR forecasts (Fig. 3.2) from which the bestconfigurations could be found. “Observed ampacity”, calculated from weather observations, wasused to both post-process and evaluate all DTR forecast configurations. Raw DTR forecasts werealso produced spatially along the line, but could not be post-processed or verified due to a lack ofon-transmission-line weather observations.Raw (EDW) DTR point forecasts, which were neither bias-corrected nor calibrated, had largesystematic and random errors, and were probabilistically unreliable. Bias-corrected and calibratedconfigurations did much better. BEBDW and EBDW had negligible bias, but larger MAE andCRPS than other DTR methods, resulting in wider probability distributions. EDBW and DEBWhad among the lowest MAE and CRPS, and were calibrated, but had relatively larger and morevariable biases. BEDBW and BDEBW, which had among the smallest and most stable bias, the94lowest MAE and CRPS, and were reliable. QSTRs and STRs had large underforecasting biases.STRs had much larger MAE than all bias-corrected, calibrated DTR forecasts, while QSTRs hadMAE similar to BEDBW and BDEBW.Correlations were unfortunately low for all DTR configurations. All thermal rating methodshad comparable correlations at night, however, the daytime DTR forecast correlations were higherthan those of the STR and the QSTR, which drop significantly. This indicates increased ampacityvariability during the day, that DTR forecasts better represent than QSTRs and STRs.Due to higher wind speeds, deterministic DTR forecast performance was consistently the worstat Agassiz. DTR forecasts and observed ampacities were consistently higher there than the otherthree locations in this study. Located in an area where very strong outflow and inflow winds exitand enter the Interior via the Fraser Valley gap in the Coast Mountains (Figs. 5.7-5.11, and 5.18-5.22), Agassiz has wind speed observations that are much higher than the other weather stations.Higher wind speeds are the most notable difference between Agassiz and the other weather stations.Less skillful DTR forecasts at Agassiz (larger bias, MAE, and CRPS) indicates the models havea harder time forecasting high wind speeds. However, the variation of deterministic wind speedsand ampacity were well represented by wind speed and DTR forecasts at all weather stations. Thisinstills confidence that the bias-corrected, calibrated DTR forecast configurations are capable ofskillfully representing the variation of ampacity at different locations with different environmentalconditions.Probabilistic DTR forecasts provide more information and flexibility than deterministic DTRforecasts (Fig. 4.8). The raw forecast distributions are unreliable, but they became reliable afterprobabilistic calibration was applied. Ensemble NWP-based, calibrated DTR forecasts account fordaily and hourly changes in both ampacity and uncertainty, whereas QSTRs and STRs do not.Another very valuable advantage is that probabilistic DTRs allow the user to select their ownrisk tolerance depending on their own unique needs for balancing safety and transmission systemefficiency. In a reliable probabilistic DTR forecast, forecast percentiles are equivalent to user risktolerance (e.g, the 5th forecast percentile corresponds to a 5% risk of thermal violation). If a lowerDTR forecast percentile is chosen, the DTR forecasts will be safer over the long term, but allowless electricity to be transmitted. Higher risk tolerances, from higher DTR forecast percentiles,have correspondingly higher risk of thermal overload but allow more current in the lines. Figure4.26 shows that tradeoff for DTR forecasts. By providing probabilistic DTR forecasts, the useris presented with the best, most accurate and complete information to make an informed decisionspecific to their needs.The miss rate, or risk, of a thermal rating is the percentage of thermal rating - ampacity pairs forwhich the thermal rating is higher than the observed ampacity (i.e., a potential thermal violation).95All bias-corrected, calibrated DTR forecast configurations had very similar miss rates. Overall,DEBW and EDBW often have the worst (highest) miss rates. There is no one DTR configurationthat has the lowest miss rates overall for the 1st, 5th, and 10th percentiles − the risk tolerancesthat are most likely to be used in an operational environment. QSTRs and STRs have constantaverage miss rates that are worse than the 1st, 5th, and 10th percentile DTR forecasts. In this studyQSTRs and STRs were found to have fairly high (10-50%) average miss rates. Outside of this study,however, the miss rates of QSTRs and STRs are not typically known, meaning they could be evenless safe.The extended hit rate was defined as the percentage of thermal rating − ampacity pairs forwhich a thermal rating method is both safe and allows the highest transmission capacity of thethree thermal rating methods. BEBDW and EBDW had the lowest (worst) extended hit rates outof all bias-corrected, calibrated DTR configurations. DTR forecasts have higher extended hit ratesthan STRs for the lowest forecast percentiles (risk tolerances; 1st to 80th percentiles). Because oftheir higher risk, QSTRs have higher extended hit rates than the 1st, 5th, and 10th percentile DTRforecasts. For DTR forecast percentiles (risk tolerances) closer to the risks of QSTRs (20th - 50th),DTR extended hit rates are the same or better than QSTRs. Thus, if power utilities were to adoptDTR forecasts with the same risk as QSTRs (as calculated in this study), the DTR forecasts wouldallow for greater transmission capacity. Although the 1st, 5th, and 10th DTR forecast percentiles donot allow for as much transmission capacity as QSTRs, their risk and skill are superior, thus moreuseful.Deterministic and probabilistic bias-corrected, calibrated DTR forecasts are more skilled andsafer than both the monthly QSTRs and seasonal STRs by all measures. For many risk tolerancesDTR forecasts also allow for the most power transmission. The most skillful DTR forecasts arethose that were calculated from bias-corrected ensemble NWP forecasts, in particular, those DTRforecast methods that applied a second bias correction to the final ensemble average: BDEBW andBEDBW.Given the accuracy of DTR forecasts in this study, power utilities looking to adopt these DTRforecasts operationally could choose to either: 1) adopt low risk DTR forecasts (1st - 10th per-centiles) that will always be safer than, but reduce transmission capacity compared to, conventionalmethods; or 2) maintain the higher level of risk associated with conventional methods, and slightlyincrease transmission capacity (by using 20th - 50th DTR forecast percentiles). In either case, DTRforecasts are a better choice, offering cost savings over conventional QSTR and STR methods.EDW DTR forecast maps such as Figures 5.7-5.11 and 5.18-5.22 would give power utilitiesmore information about how ampacity reacts to varying environmental conditions along the line atone time, and throughout the year. This type of DTR forecast product would help identify at-risk96sections of the powerline under particular weather scenarios, reduce costs, and give more informa-tion for a more efficient and optimized power grid.6.2 How to implement DTR forecasts operationallyHourly, probabilistic DTR forecasts can be very useful for electrical power operations. Tomake operational DTR forecasts, the methods described in chapters 4 and 5 would be combined inorder to be used by power utilities. It is important to note that all forecast products presented in thisthesis were produced in a real-time, operational sense − that is, forecasts initialized on a given dateonly used forecast/observation data from before that date.The most skillful DTR forecast configurations (BDEBW and BEDBW) bias-correct weathervariable forecasts before calculating DTRs. Therefore, the first step to make operational probabilis-tic DTR forecasts for powerlines, is to install weather stations at powerlines so that transmission-lineweather data is available for post-processing and verification. Further, observed ampacity, calcu-lated from weather-observation-based ampacity sensors, is needed to obtain the most skillful DTRforecasts.It is, however, impossible to have weather observations at every point along the line. For agiven line (where line is defined as the transmission line between two substations), an assessmentof the variation of weather along that line should be performed in order to determine the key points− locations where the lowest NWP-forecasted DTRs most often occur. Weather and DTR forecastmaps such as those in Chapter 5 are well suited for this environmental conditions assessment.Once the utility installs at-transmission-line weather stations (to observe wind and temperature)at key points, gridded NWP wind speed and temperature forecasts from each ensemble memberwould be interpolated to and bias-corrected at those key points. Either BEDBW or BDEBW cali-brated probabilistic DTR forecasts would be generated. The current-limiting point − the key pointwith the lowest bias-corrected, calibrated DTR forecast along the line − is identified. A risk toler-ance is chosen by the utility, and the corresponding DTR forecast percentile for that current-limitinglocation is applied as the operational thermal rating to the entire line at that time.The operational DTRs for all lines would be displayed on a forecast map of the entire powergrid. Power operations managers would be able to easily see how much current can pass througheach line. This information would be fed into the method or software used for system optimization(i.e., determining the best operations plan for the entire power grid).976.3 Limitations and future workThe most limiting factor of this study is the lack of weather observations on the powerline,at powerline height. Without observations, the environmental conditions on the line cannot beanalyzed, and NWP and DTR forecasts cannot be post-processed and verified to improve forecastskill. Line DTR forecasts could neither be post-processed nor verified, and their skill is not known.There were only four weather stations nearby whose observations could be used to calculatea proxy to ampacity. There are large periods of observations missing from three out of the fourstations. Wind speed observations are recorded at 10-m and temperature at 2-m AGL, which aremuch lower than the height of a powerline conductor. Observed ampacity, and thus DTR forecasts,could only be calculated from wind speed and temperature at these heights, which may not berepresentative of the environmental conditions at the conductor. Unless the observed ampacity is arealistic estimate of the true ampacity, bias-correction and calibration of DTR forecasts is not useful.In addition to wind speed and temperature at powerline height, varying wind direction relativeto the line, cloud cover, and rain are needed to realistically represent conditions on the line. TheIEEE 738 assumes no cloud cover and no rain, and calculates solar heat gain from the location ofthe sun depending on the time of year, all of which are conservative, but unrealistic, assumptions.The wind component relative to the transmission line is the most influential factor on ampacity.It is proportional to current (thermal rating eqns. 2.3, 2.4, 2.6). Further, the convective coolingterm, which is a function of wind speed and direction, is the largest term in the energy balanceequation of a powerline (equation 2.1). Wind speed forecasts are vitally important to the successand usefulness of DTR forecasts. Spatial wind speed forecast differences were responsible forthe most and largest EDW DTR forecast variability along the powerline. Low correlation of allDTR forecast configurations, at all weather stations, is most likely due to low wind speed forecastcorrelations. Including improved wind direction forecasts could improve DTR forecast skill. Windspeed and direction forecast accuracy and association at powerline height is the NWP componentof DTR forecasts with the largest room for improvement.Wind speeds are predominantly light (< 10 km/hr) meaning they can be affected by smallerscale processes and terrain features. NWP models skillfully represent larger-scale winds, but poorlyforecast light winds. To improve wind speed forecasts in this case, higher grid resolution anddifferent boundary layer and/or microphysical schemes could be tested in NWP models. Windspeeds increase with height. Therefore, perhaps wind speeds at powerline height may be moreskillfully represented than the 10-m wind speed forecasts used in this study. For this, weatherstations at powerline height would be needed. Powerline conductor height varies due to its catenaryshape and undulating terrain. Multiple stations at different heights could be deployed in a temporary98field study to decide the height most representative of powerline-height winds.The bias-correction and calibration used in this study are calculated based on all DTR fore-casts of the past 30 days, for each forecast hour. Additive and multiplicative bias-correction tech-niques were applied to temperature, and wind speed and DTR forecasts, respectively. More involvedpost-processing techniques such as Kalman-filtering or neural networks could be applied instead tofurther improve forecasts. Different calibration schemes could be tested to see which is the mostsuccessful. Further, both bias-correction and calibration could be applied to forecasts based on fore-cast value. For example, if low wind speeds are overforecast, and high wind speeds underforecast,bias determined from all wind speeds will cancel out the different biases for low and high windspeeds, resulting in less effective bias-correction. Conversely, the distribution of low wind speedscould be skewed left, and the distribution of high wind speeds skewed right, meaning calibrationof these forecast subsets could be more effective than calibration of the entire forecast distribution.Therefore, different biases and distributions between subsets of NWP and DTR forecast valuesshould be identified, so more focused bias-correction and calibration can be applied. In turn, thiswill improve DTR forecast skill and reliability.One of the primary goals of this study was to prove DTR forecasts could allow higher trans-mission capacity than traditional methods. Unfortunately, extended hit rates showed the safest DTRforecast percentiles (1st, 5th, and 10th) did not allow increased transmission capacity more oftenthan QSTRs, but those of comparable risk to QSTRs (20th - 50th percentiles) did. The actual ampac-ity gains and losses of DTR forecasts over lower-risk QSTRs, and vice versa, should be determinedto further understand the benefits of DTR forecasts versus QSTRs. Heckenbergerova et al. (2013)suggests one way to quantify these gains/losses is to sum ampacities over all instances when therating is higher than ampacity (more transmittable current) and vice versa (less transmittable cur-rent). This method does not take into account rating violations, but does show which thermal ratingallows higher transmission capacity, and by how much.DTR forecast percentiles of similar risk to QSTRs (∼20th - 50th percentiles) allow the same orincrease transmission capacity relative to QSTRs, while not violating ampacity. Power utilities aremore likely to want lower risk thermal ratings (1% - 10%), such as the 1st, 5th, and 10th DTR per-centiles, which are safe and reliable. QSTRs with risks of approximately 1, 5, and 10% would needto be recalculated to determine if the 1st, 5th, and 10th DTR forecast percentiles are more efficientthan these more conservative QSTRs. The higher the maximum daily temperature percentiles usedto calculate QSTRs (section 3.3), the lower the QSTR, and the less risk it will have. Temperaturepercentiles that yield QSTRs with ∼1%, 5%, and 10% risk would need to be identified, and theresulting QSTRs compared with the 1st, 5th, and 10th DTR forecast percentiles.Other potential research directions that could unlock further deterministic and probabilistic99DTR forecast skill are:1. Smoothing hourly DTR forecasts could better represent the hourly variability of ampacity.2. Verifying thermal rating methods on shorter timescales, e.g., seasonally, monthly, time of day,could reveal further insight into DTR forecast performance, especially relative to seasonalSTRs and monthly QSTRs.3. 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Conductor surface temperatures areless in the winter when the air temperature is colder.108901001102015−01−012015−03−012015−05−012015−07−012015−09−012015−11−012016−01−012016−03−012016−05−012016−07−012016−09−01timeTemperature (degC)TcoreTsfcPemberton Base bias−corrected ensavg day 1 forecastsFigure A.1: Timeseries of conductor core and surface temperature (◦C) over the 20-monthstudy period at Pemberton Base.109Appendix BCatenary ApproximationThe smooth catenary curve of a transmission line is approximated with three rigid straightsegments of equal length s. Assume the cable attachment points (A, D in Figure B.1) on the twoadjacent towers are separated by horizontal distance L and vertical distance h. The straight-linedistance between attachment points is d =√h2+L2, and ε = sin−1( hd ) is the slope angle betweenthose points. The angles α,β , γ , and ε are defined as positive as drawn in Figure B.1. The equationspresented below are also valid for negative angles, such as for a taut cable where C is higher than D,or when tower D is on higher terrain than tower A. Let the cable length (3s) be greater than d, anddefine their ratio as:Fd = 3s (B.1)where F is the cable-length factor as a function of the straight line distance between two towers.The cable length between two towers is longer than d by 1% (Lu, 2016), making F = 1.01.Solving the force and torque balances at points A, B, C, and D five the following three coupledequations for the three unknown angles α,β , γ:Ls= cosα+ cosβ + cosγ (B.2)hs= sinα+ sinβ − sinγ (B.3)0 = tanα−2tanβ − tanγ (B.4)For the special case where points A and D are at the same height, with d = L, the resulting110ABLαβ γεhdsc6c4c2c3c5s sC DFigure B.1: Schematic and notation of 3-segment catenary.three-segment catenary is symmetric (subscript s), meaning α = γ = αs and ε = β = 0.For this symmetric case, geometry allows us to find L as a function of αs and s:L = 2scos(αs)+ s (B.5)Substituting equation B.1 into equation B.5, and rearranging, gives the simple solution:αs = cos−1(12( 3F−1))(B.6)The exact general solution to equations (B.2 - B.4) is:α = αs+ ε,β = ε,γ = αs− ε (B.7)The angles from equation (B.7) can be used to improve the calculation of wind and solarradiation effects on the thermal line rating.In Chapter 5, α,β , γ , and ε are defined as positive, and are calculated for each tower pair ofthe case-study powerline. First, L and h are determined from the latitude, longitude coordinates andelevations (m ASL) of the transmission towers, from which d and ε are found. Once d is known, sand αs are found with equations B.1 and B.6, respectively. Finally, α , β , and γ are determined fromequation B.7.111Appendix CConductor Thermal ExpansionSensitivity StudyThe heights above ground and the elevation angles of a conductor will change as the conductorundergoes thermal expansion. Namely, the transmission line sags more when warmer. To test thethermal expansion of a Drake ACSR conductor, two transmission towers with equal terrain elevationand heights of 40 meters are conceptually placed 300 meters apart. The equation for linear thermalexpansion (Halliday et al., 2001) is:dL = LoαdT (C.1)where dL is the increase in conductor length (m), Lo is the original length of the conductor(m), α is the linear thermal expansion coefficient for a 26/7 Drake ACSR conductor, 18.8 ×10−6 1◦C(Loudon et al., 2017), and dT is the increase in conductor temperature. The new length of theconductor, L f , after an increase in conductor temperature is:L f = Lo(1+αdT ) (C.2)The length of a conductor, L, is:L = Fd (C.3)where F is the ratio of conductor length L to the straight-line distance d between the tops ofthe transmission towers.Combining equations C.2 and C.3:112Ff = Fo(1+αdT ) (C.4)where Ff is F at the final increased conductor temperature, and Fo is the F at the originalconductor temperature. The increase in conductor temperature, dT, is:dT = Tf −To (C.5)where Tf is the final conductor temperature (Kelvins), and To is the original conductor tem-perature (Kelvins). The conductor will expand or contract depending on the difference between theconductor temperature and the air temperature. The true F at a given air temperature can be foundwith:Fa = F(1+α(Ta−T )) when Ta > TFa = F(1+α(T −Ta))−1 when Ta < TFa = F when Ta = T(C.6)where Fa is for the ambient air temperature, Ta, and a reference conductor temperature, T, forwhich F is known. The conductor is assumed to be 1% longer than the distance between the towertops (F = 1.01; Lu (2016)) when the conductor temperature, T, is 20◦C. Were the conductor to heatup to it’s maximum temperature (100◦C) under a given air temperature, the maximum value of F is:Fmax = Fa(1+α(Tmax−Ta)) (C.7)The Fa,Fmax, the elevation angles, and the heights above ground of the conductor are calculatedfor air temperatures from -35◦C to 35◦C to analyze the extent of conductor thermal expansion at itsmaximum allowed temperature.Table C.1: The difference between each variable when calculated at 35◦C and -35◦C.F max F a α γ L c2, c6 c3, c4, c51.9171×10−6 1.3281×10−3 8.7500×10−4 8.7500×10−4 5.7513×10−4 -7.7682×10−4 -1.5536×10−3Table C.1 shows the difference between elevation angles, Fmax,Fa, and heights of each segmentabove ground for a range of conductor temperature, Ta, -35◦C to 35◦C. The differences betweenelevation angles, Fmax,Fa, and conductor segment heights between air temperatures of 35◦C and -35◦C are negligible. Therefore, it is acceptable to assume zero thermal expansion of the conductor.113Appendix DWind Direction Forecast AnalysisThere is no strong relationship between wind direction observations and the raw NWP ensemble-average point wind direction forecasts at D’arcy (Figure D.1), therefore, little to no skill in thosewind-direction forecasts. Scatterplots for the other locations also showed no association betweenwind direction observations and raw ensemble-average forecasts (not shown). The grouping of winddirection forecasts around 200◦ (Big Silver 2, Pemberton Base, and D’arcy) and 50◦ (Big Silver 2and D’arcy) is most likely due to large scale wind direction over the entire forecast grid, and notdue to location-specific terrain-induced variation in wind direction. Wind direction observationsat Agassiz are recorded in tens of degrees, i.e. 9 means 90◦, while weather forecasts two decimalplaces, making any relationship or pattern between wind direction forecasts and observations atAgassiz difficult to identify. Scatterplots of wind direction observations and raw individual modelforecasts were also made for each weather station, showing the same patterns as the raw ensembleaverage forecasts. Wind direction forecasts and observations were also split into seasons, hours,morning, afternoon and night, as well as converted to u and v components, with no correlationfound. Therefore, using wind direction forecasts as time varying input in thermal rating calcula-tions will not add skill to DTR forecasts, and will most likely decrease DTR forecast skill becausethe wind direction forecasts are so bad.114Figure D.1: Scatter plot of the ensemble average of hourly raw wind direction forecasts versushourly wind direction observations at D’arcy for the 20-month study period.115Appendix EVerification metricsBias, or mean error, expresses if the forecasts have a tendency to under- or over-predict theobservations:Bias =1nn∑i=1(xˆi− xi) (E.1)where xˆi and xi are ith forecast and observation, respectively, out of n forecast-observation pairs(Wilks, 2006).The mean absolute error (MAE) represents the forecast error magnitude over all forecast-observation pairs evaluated (Wilks, 2006):MAE =1nn∑i=1|xˆi− xi| (E.2)The Pearson correlation coefficient was calculated to assess association between DTR forecastsand ampacity:rxy =Cov(x,y)sxsy(E.3)where Cov indicates covariance between the two variables x and y, s is the standard deviation,and r is the correlation coefficient between -1 and 1 (Wilks, 2006). Values closer to 1 are ideal, asthey indicate strong positive relationships between the tested variables.The continuous ranked probability score (CRPS) is a measure of the probabilistic forecast errorover a continuous distribution of forecast probabilities:116CRPS =∞∫−∞[F(y)−Fo(y)]2dy (E.4)Fo(y) =0, y < observed value1, y≥ observed valuewhere F(y) is the cumulative distribution function of the forecast variable y, and Fo(y) is the stepfunction that increases from 0 to 1 when y equals the observation. The CRPS is the the differencebetween the cumulative distribution function of the probabilistic forecasts and the step function ofthe observed value (E.1; (Wilks, 2006)).Figure E.1: Cumulative distribution functions of three forecasts (labelled 1 - 3) and the ob-served step-function (thick black line) of variable y (Wilks, 2006)To visualize calibration, histograms are created from PIT values Nipen and Stull (2011):p =1||τ||∑t∈τH(p− pt) (E.5)where pt are the PIT values of the cumulative distribution function of the observed variable attime t out of τ times, and the PIT values p are the corresponding forecast probabilities. H(x) is theHeaviside (or step) function defined by:117H(x) =0, x < 01, x≥ 0Calibration deviation ratios (CDR) are calculated to evaluate if the probabilistic DTR forecastsare calibrated within an acceptable deviation from perfect. To quantify the number of times theobservations verify in each bin, the calibration deviation, D, is computed:D =√√√√ 1BB∑i=1(bi||τ|| −1B)2(E.6)where bi is the observed value count in bin i out of B forecast probability bins, which is arbitrary(Nipen and Stull, 2011). The squared term within brackets measures reliability - if the forecastedprobabilities ( 1B ) are equal to the observed frequencies (bi||τ|| ). Low values are better. A calibrationdeviation of perfectly calibrated probabilistic forecasts is equal to the variance of the observations(Nipen and Stull, 2011):E[Dper f ect ] =√1−B−1||τ||B (E.7)The ratio of E.6 and E.7 is calculated in COMPS:CDR =DE[Dper f ect ](E.8)Values of CDR closest to or less than 1 are best.118


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