Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Rahnama, Amirali
Master of Applied Science - MASc
University of British Columbia
The integration of Unmanned Aerial Vehicles (UAVs) in power line inspections significantly improves efficiency and safety compared to traditional methods. This thesis explores the comparison of UAV trajectories for detecting the corona effect, characterized by ultraviolet (UV) emissions due to high-voltage discharges, which indicate potential faults in power transmission systems. The study compares the effectiveness of straight and spiral UAV trajectories using advanced algorithms to determine the most efficient inspection paths. Simulations conducted in MATLAB reveal that straight trajectories offer an estimation error as low as 0.22%, demonstrating better accuracy and efficiency for large-scale inspections. In contrast, spiral trajectories, while providing detailed data for specific areas, exhibit higher estimation errors, up to 10.31%, and are more resource-intensive. This research uniquely incorporates a mobile receiver in UAV simulations, allowing a dynamic and realistic assessment of UAV performance. The findings contribute to enhancing the reliability and safety of power lines by comparing UAV trajectories for accurate fault detection, thereby improving maintenance protocols and helping to prevent power outages. This thesis provides significant insights into the practical applications of advanced UAV technology and trajectory planning in power line inspections.
Comparative Analysis of Straight and Arbitrary Unmanned Aerial Vehicle Trajectory Planning for Corona Effect Detection on Powerlines by Amirali Rahnama A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The College of Graduate Studies (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) October 2024 © Amirali Rahnama, 2024 ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis entitled: Comparative Analysis of Straight and Arbitrary Unmanned Aerial Vehicle Trajectory Planning for Corona Effect Detection on Powerlines submitted by Amirali Rahnama in partial fulfillment of the requirements for the degree of Master of Applied Science Dr. Julian Cheng, School of Engineering Supervisor Dr. Jahangir Hossain, School of Engineering Supervisory Committee Member Dr. Liwei Wang, School of Engineering Supervisory Committee Member Dr. Xiaoping Shi, Irving K. Barber Faculty of Science University Examiner iii Abstract The integration of Unmanned Aerial Vehicles (UAVs) in power line inspections significantly improves efficiency and safety compared to traditional methods. This thesis explores the comparison of UAV trajectories for detecting the corona effect, characterized by ultraviolet (UV) emissions due to high-voltage discharges, which indicate potential faults in power transmission systems. The study compares the effectiveness of straight and spiral UAV trajectories using advanced algorithms to determine the most efficient inspection paths. Simulations conducted in MATLAB reveal that straight trajectories offer an estimation error as low as 0.22%, demonstrating better accuracy and efficiency for large-scale inspections. In contrast, spiral trajectories, while providing detailed data for specific areas, exhibit higher estimation errors, up to 10.31%, and are more resource-intensive. This research uniquely incorporates a mobile receiver in UAV simulations, allowing a dynamic and realistic assessment of UAV performance. The findings contribute to enhancing the reliability and safety of power lines by comparing UAV trajectories for accurate fault detection, thereby improving maintenance protocols and helping to prevent power outages. This thesis provides significant insights into the practical applications of advanced UAV technology and trajectory planning in power line inspections. iv Lay Summary Traditional power line inspections are slow, costly, and challenging due to the remote or hard-to-reach locations of power lines. This research explores the use of Unmanned Aerial Vehicles (UAVs), to make these inspections faster and safer. A key aspect is detecting the corona effect, where high-voltage charges create sparks around power lines, emitting ultraviolet (UV) light. Detecting this effect is vital, as it often indicates faults that require repair. This study compares UAV flight paths, particularly straight and spiral trajectories. Straight paths allow for rapid coverage of long distances, while spiral paths provide detailed inspections of specific structures. By selecting the most suitable UAV flight path, this research aims to enhance the detection of power line faults, reducing outages and improving the reliability of electricity delivery. UAVs offer a modern, efficient solution for power line maintenance, contributing to public safety and energy security. v Preface This thesis is based on the research conducted under the supervision of Professor Julian Cheng, and the contents of the work have been reviewed by him. The research idea for this thesis was provided by Professor Cheng, while the identification, design, execution, and analysis of the research were carried out by the author. Generative AI tools were used for grammar corrections and improving the flow of information during the preparation of the thesis, as English is not the author’s first language. vi Table of Contents Abstract ......................................................................................................................................... iii Lay Summary ............................................................................................................................... iv Preface .............................................................................................................................................v Table of Contents ......................................................................................................................... vi List of Figures ............................................................................................................................... xi Acknowledgements ..................................................................................................................... xii 1 Introduction ................................................................................................................................1 2 Background ................................................................................................................................6 2.1 Understanding Corona Discharge on Power Lines .............................................................6 2.1.1 Physics of Corona Discharge ............................................................................................. 6 2.1.2 Impacts of Corona Discharge ............................................................................................. 7 2.2 UAVs in Power Line Inspection.............................................................................................8 2.2.1 Evolution of UAV Technology .......................................................................................... 8 2.2.2 Advantages of UAVs Over Traditional Methods .............................................................. 9 2.2.2.1 Safety and Cost-Effectiveness .................................................................................... 9 2.2.2.2 Increased Efficiency and Data Quality ..................................................................... 10 2.3 UAV Trajectory Planning ....................................................................................................10 2.3.1 Importance of Trajectory Planning .................................................................................. 10 2.3.2 Types of Trajectories ....................................................................................................... 11 2.3.3 Planning and Simulation .................................................................................................. 11 2.3.4 Algorithms and Optimization .......................................................................................... 11 vii 2.3.5 Straight vs. Spiral Trajectory Analysis ............................................................................ 12 2.3.5.1 Characteristics of Straight Trajectories ..................................................................... 12 2.3.5.2 Characteristics of Spiral Trajectories ........................................................................ 12 2.3.5.3 Comparative Analysis ............................................................................................... 12 2.4 UAV Sensors for Corona Detection .....................................................................................13 2.4.1 Types of Sensors Used ..................................................................................................... 13 2.4.2 Sensor Performance ......................................................................................................... 14 2.4.3 Comparative Sensor Analysis .......................................................................................... 14 2.4.4 Integration and Challenges .............................................................................................. 15 2.4.5 Future Developments ....................................................................................................... 15 2.5 Data Acquisition and Processing .........................................................................................16 2.5.1 Data Collection ................................................................................................................ 16 2.5.2 Data Analysis Techniques................................................................................................ 16 2.5.3 Integration of UV Communication Technology .............................................................. 17 2.5.4 Simulation and Modeling ................................................................................................. 17 2.5.5 Challenges and Future Directions .................................................................................... 17 2.6 Case Studies and Applications .............................................................................................18 2.6.1 Review of Relevant Studies ............................................................................................. 18 2.6.2 Application Scenarios ...................................................................................................... 18 2.7 Challenges and Future Directions .......................................................................................20 2.7.1 Current Limitations .......................................................................................................... 20 2.7.2 Future Research: Trajectory Planning ............................................................................. 20 2.7.3 Future Research: Sensor Integration ................................................................................ 21 viii 2.7.4 Addressing Data Processing Challenges .......................................................................... 21 2.7.5 Legal and Regulatory Advances ...................................................................................... 21 2.8 Review of Key Studies ..........................................................................................................22 2.8.1 Study 1 ............................................................................................................................. 22 2.8.2 Study 2 ............................................................................................................................. 23 2.8.3 Study 3 ............................................................................................................................. 24 2.8.4 Study 4 ............................................................................................................................. 25 2.8.5 Study 5 ............................................................................................................................. 26 2.8.6 Study 6 ............................................................................................................................. 27 3 Methodology .............................................................................................................................28 3.1 Introduction ...........................................................................................................................28 3.2 Research Design ....................................................................................................................28 3.3 Study Framework .................................................................................................................29 3.3.1 Conceptual Framework .................................................................................................... 29 3.4 Model Description .................................................................................................................30 3.5 Simulation Setup ...................................................................................................................31 3.6 Data Collection ......................................................................................................................32 3.7 Data Analysis .........................................................................................................................32 3.8 Formulas for UAV Trajectory Simulation .........................................................................34 3.8.1 Straight Path Trajectory ................................................................................................... 34 3.8.2 Spiral Path Trajectory ...................................................................................................... 36 3.8.3 Implementation in Simulation.......................................................................................... 38 3.8.4 Analytical Considerations ................................................................................................ 38 ix 3.9 Simulation Flowchart ...........................................................................................................38 3.9.1 Initialization of Simulation Parameters ........................................................................... 40 3.9.2 Photon Generation and Scattering Process ...................................................................... 40 3.9.3 Dynamic Updating of Receiver Location ........................................................................ 41 3.9.4 Probability Calculations and Decision-Making ............................................................... 41 3.9.5 Estimation of Transmitter Location ................................................................................. 41 3.9.6 Statistical Analysis and Visualization .............................................................................. 42 3.9.7 Iterative and Comparative Analysis ................................................................................. 42 3.10 Utilizing Poisson Distribution in Photon Emission Analysis...........................................43 3.10.1 Hypothesis Testing in Photon Emission Studies ........................................................... 43 3.10.2 Parameter Specification ................................................................................................. 44 3.10.3 Statistical Analysis and Threshold ................................................................................. 44 3.10.4 Establishing a Threshold Using Normal Approximation............................................... 45 3.10.5 Decision Rule Based on Statistical Thresholds ............................................................. 46 4 Simulation Results ...................................................................................................................47 4.1 Simulation Setup ...................................................................................................................48 4.2 Comparison of Trajectory Paths .........................................................................................48 4.2.1 Trajectory Comparison at Location (60,1,0) .................................................................... 49 4.2.2 Comparison at Location (130,1,0) ................................................................................... 50 4.2.3 Comparative Analysis at Location (210,1,0) ................................................................... 51 4.2.4 Analysis of Factors Contributing to Variability in Results Across Different Locations . 52 4.2.5 Comparison of normalized probabilities .......................................................................... 53 4.2.6 Importance of Estimation Errors ...................................................................................... 54 x 4.2.7 Aligning Expectations with Results ................................................................................. 54 4.3 Effect of UAV Velocity on Transmitter Location Estimation Accuracy .........................55 4.4 Implications of Trajectory Performance for UAV Localization Accuracy .....................56 5 Conclusion and Future Work .................................................................................................57 5.1 Conclusion .............................................................................................................................57 5.2 Future Work ..........................................................................................................................58 Bibliography .................................................................................................................................60 Appendices ....................................................................................................................................66 Appendix A ............................................................................................................................... 66 Appendix B ............................................................................................................................... 72 xi List of Figures Figure 3.1: Straight path trajectory ............................................................................................... 35 Figure 3.2: Spiral path trajectory (3D) .......................................................................................... 36 Figure 3.3: Spiral path trajectory .................................................................................................. 36 Figure 3.4: Geometric Calculation of UAV Coordinates ............................................................. 37 Figure 3.5: Flowchart of MATLAB Simulation Process .............................................................. 39 Figure 4.1: Probability Distribution at Location 1-straight path................................................... 49 Figure 4.2: Probability Distribution at Location 1-spiral path ...................................................... 49 Figure 4.3: Probability Distribution at Location 2- straight path.................................................. 50 Figure 4.4: Probability Distribution at Location 2-spiral path ...................................................... 50 Figure 4.5: Probability Distribution at Location 3-straight path................................................... 51 Figure 4.6: Probability Distribution at Location 3-spiral path ...................................................... 51 Figure 4.7: Effect of Velocity on Estimation Error ...................................................................... 56 xii Acknowledgements I am deeply grateful to my supervisor, Professor Julian Cheng, for his invaluable mentorship, guidance, and support throughout my research. Without his help and support, it would not have been possible to conduct this research. I would also like to express my heartfelt gratitude to my beloved family for their unwavering support and encouragement, which have been instrumental in the completion of my thesis. 1 Chapter 1 Introduction The integration of unmanned aerial vehicles (UAVs) into power line inspections represents a significant technological advancement in infrastructure management. UAVs offer a substantial improvement over traditional manual inspection methods, which are not only labor-intensive but also limited in their ability to provide comprehensive assessments, especially in high-altitude and challenging terrains. Traditional inspection methods heavily rely on handheld tools and visual inspections through binoculars, which fall short of delivering a detailed and comprehensive understanding of the overall condition of transmission towers, particularly in inaccessible areas [1]. The deployment of UAVs addresses these limitations by enhancing the efficiency and safety of power line inspections through advanced aerial capabilities. This study focuses on comparing two UAV trajectory paths for detecting the corona effect on power lines. The corona effect, characterized by ultraviolet (UV) emissions due to high-voltage electrical discharges, is a crucial indicator of potential faults in power transmission systems [2]. Accurate detection of this phenomenon is essential for maintaining the reliability and safety of power lines. The primary objective of this research is to compare the effectiveness of straight and arbitrary UAV trajectories in detecting the corona effect, utilizing advanced algorithms to determine the most efficient inspection paths. 2 Trajectory planning is a critical component in maximizing the performance of UAV-based inspections. Traditional algorithms, such as the Dijkstra algorithm, have been employed for UAV path optimization but are limited by their computational intensity and inability to find optimal paths in complex environments [1]. To overcome these limitations, this study focuses on simulating and comparing straight and arbitrary flight paths. By simulating these two distinct trajectories, we aim to determine which path provides better efficiency and accuracy in corona effect detection on powerlines. This approach is expected to enhance the overall effectiveness of UAV inspections, ensuring comprehensive coverage and precise identification of defects. The significance of this study lies in its potential to enhance the accuracy and efficiency of power line inspections. By selecting the most effective UAV trajectories, the research aims to improve the detection of the corona effect, thereby increasing the reliability of power transmission systems. The findings could inform the development of more effective inspection protocols, ensuring thorough and timely identification of potential faults. This is crucial for preventing power outages and maintaining the continuous delivery of electricity. Additionally, the study contributes to the broader field of infrastructure management by demonstrating the practical applications of advanced UAV technology. This research is distinct from previous studies in several ways. While existing research primarily focuses on the application of various algorithms for UAV trajectory planning, this study uniquely compares the effectiveness of straight and arbitrary flight paths for detecting the corona effect. The key difference in this research is the use of a mobile receiver in the UAV simulations, as opposed to the stable receivers used in other studies. This distinction allows for a more dynamic and realistic assessment of UAV performance in real-world inspection scenarios. 3 Furthermore, the study aims to provide a comprehensive analysis of the benefits and drawbacks of straight versus arbitrary flight paths in terms of accuracy and efficiency. By conducting detailed simulations and analyses, this research seeks to determine which trajectory planning method offers superior performance for corona detection. The major contributions of this thesis can be summarized as follows: • Detailed Examination of UAV Technology and Trajectory Planning: This thesis provides a comprehensive review of the current state of UAV technology and trajectory planning algorithms for power line inspections. • Comparative Analysis of Straight and Arbitrary Flight Paths: The research offers a novel approach by comparing the effectiveness of straight and arbitrary UAV flight paths in detecting the corona effect on power lines. This comparison is crucial for determining the most efficient and accurate trajectory planning method for UAV-based inspections. • Introduction of Mobile Receiver in UAV Simulations: Unlike previous studies that use stable receivers, this research incorporates a mobile receiver in the UAV simulations. This key difference allows for a more dynamic and realistic assessment of UAV performance in real-world inspection scenarios, enhancing the practical applicability of the findings. • Validation through Simulations: The thesis includes extensive simulations to validate the essential functions of the UAV-based inspection system. The implementation of advanced algorithms and the demonstration of their effectiveness in corona detection through these 4 simulations lay the foundation for future research and improvements in UAV technology for infrastructure inspections. The thesis is structured into five chapters, with summaries as follows: Chapter 1: Introduction, which provides an overview of the research background, significance, objectives, and contributions. Chapter 2: Literature Review, which details the current state of UAV-based inspection technologies, the corona effect, and trajectory planning methods. This chapter also compares existing studies and identifies gaps that this research aims to address. Chapter 3: Methodology, which discusses the simulation setup, including the design and implementation of straight and arbitrary trajectories, the mobile receiver integration, and the data collection process. Chapter 4: Results and Discussion, which presents the experimental results, compares the accuracy of the two trajectory planning methods, and discusses the implications of the findings. Chapter 5: Conclusion and Future Work, which summarizes the research contributions, highlights the key findings, and suggests directions for future research to further enhance UAV-based inspection systems. In summary, for the first time, this research presents a comprehensive analysis of different UAV trajectory planning methods for power line inspections. By comparing straight and arbitrary flight paths, the study aims to determine the most effective approach for detecting the corona effect. The results are expected to provide valuable insights for improving the performance of UAV-based inspection systems, promoting the efficiency and safety of power line maintenance. Through this detailed examination, the research offers significant 5 contributions to the advancement of UAV technology and its applications in critical infrastructure monitoring. In this study, two hypotheses were established to assess the presence of the corona effect on powerlines by analyzing photon emission rates. The null hypothesis (H₀) posits that all detected photon emissions are solely due to background noise, characterized by a defined rate, 𝜆b. Conversely, the alternative hypothesis (H₁) suggests that photon emissions exceed this background level due to the corona effect, implying an additional emission rate λs, which represents the rate at which photons are emitted directly as a result of the corona discharge, leading to a higher overall emission rate (λs + λb). Our analysis aims to establish a statistical threshold to distinguish between these scenarios, enhancing the accuracy and reliability of corona effect detection in electrical infrastructure monitoring. 6 Chapter 2 Background 2.1 Understanding Corona Discharge on Power Lines 2.1.1 Physics of Corona Discharge Corona discharge is defined as an electrical discharge which takes place when the electric field intensity around the conductors exceeds the breakdown voltage of the surrounding air. This in turn results in a local ionization of the air surrounding the conductor along with a visible and at times audible glow. Electric discharge commonly takes place where the electric field is most concentrated, for example at the sharp ends of the objects [3]. This event depends on the shape of the conductors, the weather, and the voltage level. The relative humidity of air, temperature, and pressure can be quite significant in determining the occurrence and intensity of corona discharges. For instance, humidity affects the electrical conductivity of the air because high humidity is more conductive and enhances the corona discharge. Also, it is concluded that the electric field intensity increases with the voltages or with the increase in the curvature of conductors, so the possibility of corona discharges and their intensity increases also [4]. 7 2.1.2 Impacts of Corona Discharge Corona discharge on power lines leads to significant power loss and operational issues, including the conversion of electrical energy into light, heat, and noise [5]. This loss is both inefficient and costly, necessitating the generation of additional power. The chemical by-products of corona discharge, such as ozone and nitrogen oxides, further degrade electrical components, requiring more frequent inspection and maintenance [5]. The factors influencing corona losses, including power line properties and environmental factors, are discussed in the literature [6]. Methods to minimize orona effects, such as the use of high-quality insulation and good service design for underground high transmission cables, are also explored [5]. The influence of wire insulation coating on power lines and the potential for reducing energy losses are considered [7]. Corona discharge from power transmission lines can lead to electromagnetic interference (EMI) that disrupts communication systems [8]. This interference can be mitigated by innovative mechanisms, such as the use of dielectric oil to create an isolated layer that protects the transmission line from high electrical fields [9]. EMI control measures are also crucial in minimizing the impact of corona discharge on telecommunication networks [10]. Advanced modeling techniques can be used to analyze the interference caused by high-voltage power lines on communication cables and to evaluate potential mitigative measures [11]. It is important to minimize these effects to maintain the functionality of both power transmission and communication. 8 Corona discharges from high-voltage power lines can have significant environmental and health impacts. These discharges can lead to the production of hazardous ozone and UV radiation, posing a risk to both maintenance personnel and the public, particularly those living near power lines [12]. The presence of volatile organic atmospheres and wire oxidation can further exacerbate corona and ozone production [13]. To address these issues, Balametov & Isaeva (2022) developed software for monitoring corona effects on overhead lines, which can help in understanding and mitigating these impacts [14]. Shah (1979) provided a comprehensive review of state and federal environmental regulations related to the electrical effects of overhead transmission lines, including corona-generated effects. These studies collectively highlight the need for effective monitoring and regulation of corona discharges to minimize their environmental and health impacts [15]. 2.2 UAVs in Power Line Inspection 2.2.1 Evolution of UAV Technology The development of UAVs for power line inspection has been significantly influenced by advancements in sensor technology and autonomous flying systems. Moore et al. (2018) highlighted the use of UV sensors for detecting corona discharges, a key indicator of power line issues [16]. They further emphasized the need for improved UAV position determination technology or increased sensor range for broader deployment. Kunz et al. (2022) underscored the importance of multi-sensor setups, including high-resolution aerial cameras, LiDAR, thermal cameras, and UV cameras, for enhanced power line monitoring [17]. These studies 9 collectively demonstrate the critical role of sensor technology in the evolution of UAVs for power line inspection. 2.2.2 Advantages of UAVs Over Traditional Methods UAVs have revolutionized power line inspection, offering increased safety, versatility, and cost-effectiveness [18]. These advantages stem from their ability to capture high-quality data without the need for physical contact with high-voltage areas, their capability to access remote or challenging locations, and their potential for automation. The use of UAVs in power line inspection is a growing trend, with companies responding to challenges such as safety, endurance, and automation [19]. The integration of advanced computation techniques, deep learning, and advanced sensors further enhances the potential of UAVs in this field [18]. They also offer higher versatility; it is possible to inspect the power lines in areas that are challenging to reach or are located in remote areas without much human interference. Also, UAVs can minimize the costs and time of power line inspections while providing better quality data for analysis. 2.2.2.1 Safety and Cost-Effectiveness The use of UAVs for power line inspections offers numerous benefits, including increased safety for human inspectors and reduced liability and insurance expenses for utility companies [20]. These drones can operate in any weather and reach inaccessible areas, making them a cost-effective solution for regular maintenance checks. The development of a powerline detection system using a deep learning algorithm further enhances the capabilities of UAVs in this context 10 [21]. However, challenges such as safety, endurance time, and automation need to be addressed to fully realize the potential of UAV power inspections [19]. 2.2.2.2 Increased Efficiency and Data Quality UAVs enhance power line inspections because they can survey large areas and gather detailed information that includes visual, thermal, and UV imaging. This data is also important in the early detection of problems such as wear, corrosion, and other related defects likely to cause failures. The high resolution and detailed information from UAVs help utility companies carry out accurate predictive maintenance to avoid outages and increase the durability of power structures [17, 19]. 2.3 UAV Trajectory Planning 2.3.1 Importance of Trajectory Planning Trajectory planning is critical for the operation of UAVs in power line inspections due to the impact it has on the operations’ efficiency and safety. Therefore, the well-defined trajectories of the UAVs cover all of the necessary areas systematically and reduce the risks and costs of UAV operations by minimizing the time UAVs spend in the air. This planning helps UAVs to avoid obstacles and also helps ensure that they are at the optimal distance from the power lines and towers which is crucial for early fault detection and maintenance planning [22,23]. 11 2.3.2 Types of Trajectories Depending on the inspection needs, UAVs use different flight paths. Straight and spiral patterns are often used in power line surveillance. Straight paths are desirable when surveying long distances of transmission lines at once, suitable for fixed-wing UAVs that need long, uninterrupted flights. Spiral trajectories, however, are more favorable for comprehensive inspections around specific structures such as towers, for which multi-rotor UAVs are ideal due to their ability to fly close to the structure and capture data from various angles [24]. 2.3.3 Planning and Simulation Due to the complexity of power line environments, there is a need to employ elaborate planning and simulation to ensure comprehensive coverage and operational safety. Simulation tools are used to model the UAV's flight paths around various obstacles. This modeling is required for adjusting the parameters of UAV’s flight in real-time to optimize data gathering and to meet the requirements of flight regulations [8,23]. 2.3.4 Algorithms and Optimization Some of the algorithms that are used in the optimization of the UAV flight paths include Genetic Algorithm (GA) and Simulated Annealing (SA), which address the multi-objective requirements of minimizing flight distance and maximizing data quality. These algorithms analyze different flight scenarios to define the best routes that would include all the inspection points of interest with no interference from the known obstacles and reduced probability of UAVs’ failures [1,25]. 12 2.3.5 Straight vs. Spiral Trajectory Analysis 2.3.5.1 Characteristics of Straight Trajectories Straight trajectories are mainly used for monitoring long lengths of power lines, aiming for the best coverage in terms of distance. This approach reduces the extent to which specific areas are inspected, which makes it useful in broad inspection. Straight paths are defined by fewer changes in the values of control inputs, which in turn means less stress on the components of the UAV and less energy used. These paths are most advantageous when the goal is to quickly survey large areas for significant problems such as physical damages or obstructions along the power line providing a bird’s eye view of the power infrastructure health [26]. 2.3.5.2 Characteristics of Spiral Trajectories Spiral trajectories are more specific for the examination of specific objects like power line towers or insulators. This method also allows the UAV to encircle the structure and its surroundings from different perspectives and heights, which is helpful in the assessment. Such detailed data collection is crucial for tasks that require a close-up view. Using spiral paths may provide a broader insight into certain faults, nevertheless, their application is generally more resource-intensive, requiring more energy and time [27]. 2.3.5.3 Comparative Analysis Studies show that straight trajectories are more accurate and are more desirable for longer power line inspections. They allow for faster coverage and are less complicated to organize and implement thereby cutting down on operational expenses. This trajectory type is particularly 13 beneficial for preliminary surveys or routine monitoring, where time and efficiency are critical [28]. On the other hand, spiral trajectories though they provide a more detailed picture of the target area are likely to add more operational challenges due to their planning requirements and are more energy and time consuming. While they are important for detailed diagnostics and are necessary when data are gathered from different directions, their application may be less effective compared to straight-line paths in large-scale surveys [29]. Every trajectory type serves a specific role and has to be applied based on the inspection’s objectives, the UAV’s characteristics, and the conditions. The choice of the trajectory type should correspond to the general goals of the inspection – if the area is to be checked quickly, or if some of the components are to be checked more thoroughly. Therefore, selecting the appropriate trajectory planning in line with these goals will greatly enhance the effectiveness of UAV inspections in the power industry [1]. 2.4 UAV Sensors for Corona Detection 2.4.1 Types of Sensors Used As for UAV-based Corona detection, different kinds of sensors are used to distinguish and measure high-voltage discharges correctly. UV sensors are specifically important because they can detect the UV emissions that are characteristic of Corona discharges. These sensors operate by detecting the UV rays produced when the electrical discharges in the surrounding air of high-voltage power lines become ionized. High-resolution cameras are employed for visioning and validating physical imperfections or irregularities that might trigger Corona discharges, with optical sensors. Thermal sensors can also be used to identify heat, which is often related to 14 electrical faults before they escalate to worse conditions. Each type of sensor offers unique advantages in terms of sensitivity, range, and the type of data they can provide for analysis [17,30]. 2.4.2 Sensor Performance These sensors’ performance depends on the trajectory patterns of the UAV. For example, straight trajectories ensure constant and swift data acquisition over a large area, which is useful in surveys to find potential hotspots. The practical efficiency of UV sensors can be poor on straight tracks if the UAV does not fly close enough to the source of Corona discharges. Spiral trajectories are beneficial as the UAV can approach a target from different directions and angles, and perform multiple scans. This is especially helpful in UV and thermal sensors [3, 30]. 2.4.3 Comparative Sensor Analysis When comparing the efficiency of these sensors, the UV sensors are more sensitive to the specific wavelengths related to Corona discharges but require closer proximity to the source for accurate detection. Optical sensors offer valuable visual context that helps in the diagnosis and documentation of the infrastructure's condition but they are not very efficient during the night or when obscured by environmental factors. Thermal sensors are useful for detecting overheating components that could be prone to Corona discharge in the future even if they are not currently creating Corona discharges. It can be concluded that the selection of the sensor and its incorporation into the UAV systems depends on the flight path of the UAV and the requirements of the power line inspection jobs [17,30]. 15 2.4.4 Integration and Challenges The incorporation of these sensors into UAV platforms requires consideration of factors such as the weight, and power consumption of the sensors as well as their data processing ability. Sophisticated UAV models with these sensors need to be capable of flying on their own while keeping the target structures’ coordinates aligned with the sensors’ field of view for accurate data capture. Stability and control of the payload are paramount to the UAV especially when flying along various trajectories and over difficult terrains in which wind or other atmospheric conditions may influence the results of the sensor’s readings [31,32]. 2.4.5 Future Developments Future development in the sensors and UAV systems is believed to offer better Corona detection solutions. For example, improvement of the UV sensors with higher sensitivity and range may decrease the number of proximity flights and increase safety. Also, advancements in data analysis and machine learning algorithms that are capable of analyzing the sensor data in real time will greatly enhance the effectiveness and precision of the UAV-based power line inspection [17,30]. 16 2.5 Data Acquisition and Processing 2.5.1 Data Collection The procedures of data acquisition in UAV-based power line inspections are quite different depending on whether the UAV follows a straight or spiral flight path which is more suitable for different types of inspection. In straight paths, UAVs usually move along the lateral of the power lines and can cover large stretches in a short time. This method is effective for general monitoring and evaluation as it gathers data in real time over large regions. Spiral movements entail the UAV flying around particular objects like transmission towers or the suspected areas of damage. [28,29]. 2.5.2 Data Analysis Techniques The processing of the data collected during UAV flights uses various methods that help in achieving the objective of detecting and estimating the Corona effect on power lines. Signal processing techniques are important, particularly in dealing with optical and thermal images. These algorithms assist in finding patterns and irregularities that signify the existence of Corona discharges. Sophisticated methods such as machine learning (ML) models are gradually being incorporated to perform classification and prediction of possible flaws from the data collected by the sensors. These methods improve with time as they adapt to new data, and therefore, make predictions more accurate and reliable in terms of identifying issues [33,34]. 17 2.5.3 Integration of UV Communication Technology Recent advancements in ultraviolet (UV) communication have been integrated into UAV systems for power line inspection. UV sensors, which can measure emissions of Corona discharges are essential for pinpointing the exact locations of faults. The non-line-of-sight communication capabilities of UV technology allow UAVs to transmit data effectively in complex environments, which increases the reliability of the data transfer between UAVs and ground stations, no matter the weather or environment [30]. 2.5.4 Simulation and Modeling To improve the efficiency of data acquisition and processing, simulation tools are utilized to model UAV flights and sensor performance. These simulations are useful in planning the flight path and sensor activities before the real flight; it is cheaper and safer than actual field exercises. The models can predict the actual outcome of the inspection missions under various flight conditions, sensor settings and environmental conditions, which helps in refining the UAV operations for better data quality and coverage [35]. 2.5.5 Challenges and Future Directions However, there is the issue of speed and efficiency of data processing considering the large amounts of data obtained from multiple trajectories. Future developments will probably be directed towards making onboard systems faster so that the data could be processed in real time and decisions made immediately. Moreover, the improvement of the integration of various kinds of sensors, for example, UV, optical, and thermal sensors, will probably be a trend for 18 offering more advanced analysis tools for Corona detection and power line condition evaluation [17,36]. 2.6 Case Studies and Applications 2.6.1 Review of Relevant Studies Several studies have explored the utilization of UAV trajectories for monitoring and inspecting powerlines, focusing on detecting phenomena like the Corona discharge. Vidan et al. (2015) emphasized the use of fixed-wing UAVs equipped with sensors capable of detecting Corona discharge by analyzing high voltage lines through images processed with MATLAB functions, specifically employing Radon transforms to detect line features from aerial imagery [37]. Similarly, Wu et al. (2023) utilized a simulated annealing algorithm in MATLAB to optimize UAV trajectory planning for power line inspections, aiming to enhance the efficiency of detecting faults on transmission towers [1]. These studies highlight the importance of trajectory planning in improving the detection capabilities and operational efficiency of UAVs in complex environments. 2.6.2 Application Scenarios In scenarios where extensive area coverage and efficient navigation through challenging landscapes are required, straight trajectories are often advantageous. For instance, the use of straight UAV trajectories can be instrumental in large-scale surveillance of powerlines spanning extensive and minimally obstructive landscapes. This trajectory allows for rapid, linear scans of infrastructure, minimizing time and energy consumption, as supported by He et al. (2019), 19 who discussed the efficiency of multi-rotor UAVs in performing detailed inspections of transmission lines using pre-planned straight route paths [38]. Conversely, spiral trajectories may be preferred in environments requiring detailed, close-range observations to identify defects effectively. The spiral pattern ensures thorough coverage of a localized area, allowing sensors to capture data from multiple angles and elevations, thus enhancing the detection accuracy of spatially and geometrically complex features. This trajectory type is particularly useful in environments where obstacles or varying topographies complicate the flight path, as detailed by Wu et al. (2023), who explored the adaptation of UAV paths to accommodate different operational constraints, optimizing the inspection process [1]. The selection of UAV trajectories can also be influenced by the precision requirements of the inspection tasks. For high precision needs, such as detailed component analysis on powerlines, spiral trajectories provide a closer and more detailed inspection capability. This approach is evident in the methodology described by Vidan et al. (2015), where intricate image analysis techniques were required to identify and diagnose the condition of high-voltage lines accurately [37]. In summary, the choice between straight and spiral trajectories for UAV operations over powerlines is heavily dictated by the specific requirements of the task at hand, including the need for efficiency, coverage, and precision. Both trajectories have distinct advantages that, when deployed under appropriate conditions, can significantly enhance the effectiveness of UAV-based powerline inspections. As UAV technology and image processing algorithms continue to evolve, the integration of these trajectories in different combinations may offer new pathways to improve both the detection capabilities and operational efficiencies of these aerial systems. 20 2.7 Challenges and Future Directions 2.7.1 Current Limitations The employment of UAVs for Corona detection on power lines has some technological and methodological drawbacks that restrain its usage. One issue is the endurance and payload capacity of the UAVs, which limit the hours of flight for a UAV and the kind of sensors that can be carried on a single flight. Moreover, Corona detection is also highly dependent on the quality and calibration of the sensors including UV, optical, and thermal imagers among others. Environmental factors, such as lighting conditions and weather, also highly impact sensor performance, possibly causing false readings or missed detections. Another issue is related to the control and stability of the UAV in difficult weather conditions, which may influence the quality of the data received during the power line inspection [30]. 2.7.2 Future Research: Trajectory Planning Future research in the trajectory planning field should therefore be directed towards the models that would enable real-time computation of the flight trajectories, considering the dynamic environmental changes and the operational feedback. This could include developing AI and machine learning algorithms that could be used in improving the UAV’s capability to self-adjust the flight plan for optimal data collection. Research could also explore advanced path planning techniques that consider different weather patterns, which highly affect the stability of a drone and its sensor alignment [39]. 21 2.7.3 Future Research: Sensor Integration Another important field is the enhancement of the integration of different sensors on UAVs. In future work, it is necessary to improve the integration of multi-sensor systems that include UV, optical, and thermal sensors to analyze the health status of power lines more comprehensively. This could include the design of lightweight sensors that are highly efficient with improved resolution and sensitivity. Research could also be done on the use of multiple swarming drones having different types of sensors to survey a larger region more effectively and to cross-verify data through multiple perspectives [17,40] 2.7.4 Addressing Data Processing Challenges There is a need for improvement in onboard data processing capabilities to deal with large volumes of data that UAVs capture. This involves not only the hardware aspects, such as more powerful, yet lightweight, processing units but also the software that can perform some basic analysis of the data in-flight. This would facilitate real-time decision-making, which in turn could facilitate the identification of critical issues during the UAV inspection process and adjust their inspection focus accordingly [41,42]. 2.7.5 Legal and Regulatory Advances Lastly, it is critical to focus on the legal and regulatory issues that are related to UAV operations in the public and commercial airspace domains. Future research should also look into the development of standardized procedures for UAV operations that would guarantee safety and privacy while at the same time not hampering the growth of innovative solutions. The 22 cooperation of various agencies, organizations, and universities will be vital in developing the necessary framework that will accommodate the use of UAVs in the national airspace [43]. 2.8 Review of Key Studies 2.8.1 Study 1 The study conducted by Mikhail et al., with the title "Review of Key Study on Autonomous UAV Trajectory Planning for Corona Discharge Inspection" aimed to develop a novel method for autonomous UAV trajectory planning specifically for inspecting Corona discharges in power lines using ultraviolet (UV) spectrum imaging. The researchers sought to address the critical issue of Corona discharges, which can lead to significant power system failures, by leveraging UAVs equipped with UV sensors for efficient and automated monitoring. The primary objective was to create a trajectory planning method that would consider the spatial and geometric features of power lines, the nature of potential damages, and the quality requirements for the collected data, ensuring a representative and standardized dataset. The methodology involved a detailed analysis of power line structures, defining quality requirements for data, simulating UAV flights in Blender software, and manually labeling the gathered UV images for segmentation. The researchers' findings were significant as they successfully simulated the UAV flights, which resulted in a collection of 200 UV images from one powerline tower, capturing 326 manually segmented Corona discharge areas. The study demonstrated that the proposed method could automate the data collection process effectively, enhancing the representativeness and quality of the gathered data. However, the research had limitations, including the simulation 23 environment's inability to perfectly replicate real-world conditions and the need to test the method on actual powerlines for validation. The accomplishment of creating a standardized method for UAV trajectory planning in UV imaging marks a critical step towards optimizing powerline inspections and ensuring energy security through timely detection and repair of damaged elements. The study sets the foundation for future work to expand this method to other types of powerline damages and real-world applications [3]. 2.8.2 Study 2 In the study "The Design and Implementation of a Hotline Tracking UAV" by Valipour et al., the researchers aimed to develop a vision-based system for power line inspection using quad-rotor UAVs. Their objective was to improve efficiency and reduce the costs of power line inspections, which are crucial for preventing power outages and assessing the condition of power lines. The methodology involved using a combination of Gaussian steerable filters and the probabilistic Hough line transform for power line detection, coupled with an unscented Kalman filter (UKF) to track the position of the power line relative to the UAV. The researchers simulated and implemented their approach, demonstrating its capability to follow power lines. They highlighted the use of computer vision as a key technology for both inspection and navigation, addressing challenges such as dynamic environmental parameters and sensor inaccuracies. The study's findings showed that the proposed vision-based system could effectively detect and track power lines, proving its practical applicability through both simulations and real-world tests. However, the researchers acknowledged several limitations, including the 24 computational overhead of the detection algorithms and the difficulties in handling unstructured environments. The accomplishment of integrating the UKF for stable tracking and designing a backstepping controller for the UAV marked significant advancements in autonomous power line inspection. Despite the robust detection and tracking capabilities, the researchers noted that further improvements could be made by incorporating distance measurements to the power lines and enhancing the system's robustness with additional sensor data such as GPS [44]. 2.8.3 Study 3 Yuan et al., in their 2020 study titled "Monte-Carlo Integration Models for Multiple Scattering Based Optical Wireless Communication," addressed the complex challenge of modelling optical wireless communication channels affected by multiple scattering [45]. Their primary goal was to develop Monte-Carlo integration (MCI) models that enhance computational efficiency and accuracy, crucial for applications like free-space non-line-of-sight (NLOS) ultraviolet communication and underwater optical wireless communication. To achieve this, they introduced three variations of the MCI model: MCI-US (uniform sampling), MCI-IS (importance sampling), and MCI-PIS (partial importance sampling). These models were rigorously tested through numerical simulations, and the MCI-PIS model, in particular, demonstrated significant improvements, achieving up to 12 times the efficiency of the original MCI model and 5.6 times that of the traditional Monte Carlo simulation (MCS) model [45]. The findings revealed that the MCI-PIS model could effectively balance simplicity and computational efficiency, making it highly suitable for practical applications. However, the study also highlighted limitations, particularly in handling higher-order scattering scenarios and the associated computational overhead. Despite these challenges, the research provided a robust 25 foundation for further refinement and optimization of these models. The advancements made by Yuan and colleagues represent a significant contribution to the field, offering a pathway toward more efficient and accurate modelling of optical communication channels, which is vital for improving the performance and reliability of advanced optical wireless communication systems [45]. 2.8.4 Study 4 In the paper "UV communications channel modeling incorporating multiple scattering interactions" by Drost et al., the researchers aimed to develop a robust Monte Carlo model for non-line-of-sight (NLOS) ultraviolet (UV) communication systems. Their primary objective was to accurately model the complex interactions of multiple scattering events, which are critical for predicting system performance in environments where direct line-of-sight is not possible. The methodology involved deriving a comprehensive Monte Carlo multiple-scattering UV channel model that could be used to study the contributions of different orders of scattering to path loss and impulse response functions. This model builds on previous research by incorporating detailed physics-based models of photon propagation and scattering, which significantly improve the robustness and convergence rate of the simulations [46]. The findings demonstrated that the proposed model could accurately predict the path loss and impulse response functions for various system geometries, providing insights into the relative contributions of different scattering orders. The simulation results validated the effectiveness of the model in both theoretical and practical scenarios, showcasing its potential for aiding in the design and optimization of UV communication systems. However, the study also identified limitations, such as the need for further refinement to handle complex 26 atmospheric conditions and the computational challenges associated with high-order scattering events. Despite these challenges, the accomplishments of Drost, Moore, and Sadler in developing a more accurate and efficient channel model mark a significant advancement in the field of UV communications, paving the way for more reliable and effective communication systems in challenging environments [46]. 2.8.5 Study 5 Cui et al. in their paper 'Path Planning Algorithms for Power Transmission Line Inspection Using Unmanned Aerial Vehicles," aimed to enhance the efficiency and effectiveness of UAV-based automated power transmission line inspections through optimized path planning. They addressed the specific requirements of tower monitoring and line corridor monitoring by employing multi-rotor and fixed-wing UAVs, respectively. For tower inspections, they utilized genetic algorithms (GA) to design safe and efficient paths, considering the UAV's proximity to the towers and camera specifications. For line corridor monitoring, they developed a mathematical model incorporating polar coordinate coding to handle constraints like maximum path deflection angle and minimum step length, using GA and genetic simulated annealing (GSA) algorithms to optimize the paths [23]. The study's findings revealed that the proposed GA and GSA algorithms could generate optimal inspection paths, significantly improving the operational efficiency of UAVs in both tower and corridor inspections. Simulation results demonstrated that the GA could quickly find optimal solutions for tower inspections, while the GSA approach effectively combined global and local search capabilities to avoid local optima in corridor inspections. However, the 27 researchers acknowledged limitations such as the UAVs' physical constraints and the need for further improvements in handling complex environmental conditions. Their accomplishments provide a robust foundation for the autonomous, safe, and reliable operation of UAVs in power transmission line inspections, contributing to the overall safety and stability of power lines [23]. 2.8.6 Study 6 Rymer et al., in their 2018 paper “Inexpensive, Lightweight Method of Detecting Coronas with UAVs," presented an innovative solution for enhancing the detection range of ultraviolet (UV) sensors used in UAV-based power line inspections. Recognizing the limitations of current UV cameras, which are heavy, expensive, and not ideal for UAV applications, the researchers aimed to develop a more practical and cost-effective method for Corona detection. Their primary objective was to use a shallow parabolic reflector to augment the range of a compact UV sensor, enabling safer and more efficient autonomous inspections of high-voltage structures [30]. The researchers conducted extensive testing and development, comparing the performance of UV sensors with both lens and parabolic dish augmentations. The study revealed that while lenses improved the sensor's range, they significantly narrowed the field of view (FOV), making them less suitable for flyby inspections. In contrast, the parabolic dish provided a balanced enhancement, increasing the sensor's range by approximately 39% without compromising the FOV. The optimal configuration was found with a 12.7 cm (5 inch) dish, which nearly doubled the sensor's range, allowing for a safer standoff distance from power lines. This improvement in sensor range and the lightweight, cost-effective design of the dish-augmented UV sensor demonstrated significant potential for autonomous high-voltage line inspections, addressing key challenges in the power industry's maintenance and safety protocols [30]. 28 Chapter 3 Methodology 3.1 Introduction This chapter explores the methodologies employed in the comparative analysis of straight and spiral UAV trajectory planning for detecting the Corona effect on powerlines under Non-Line-of-Sight (NLoS) conditions. By integrating advanced Monte Carlo simulations within a controlled computational environment, this study aims to systematically evaluate the UAV trajectories’ impact on detection capabilities and model the complex interactions between electromagnetic waves and atmospheric particles. The methodology underscores the theoretical underpinnings and practical applications of UAV technology in infrastructure monitoring, setting a precedent for future research in this domain. 3.2 Research Design The research employs a quantitative approach, utilizing computational simulations to address the research questions. This method was chosen for its ability to precisely model complex phenomena, such as photon scattering and absorption in NLoS conditions, which are challenging to replicate in experimental setups. 29 3.3 Study Framework 3.3.1 Conceptual Framework The conceptual framework for this study centers around hypotheses focused on the capability of UAVs to detect the presence or absence of a transmitting point (corona effect) on powerlines. This is framed through two specific hypotheses: H0 (Null Hypothesis): There is no emitting source (corona effect) detectable by the UAV along the powerlines, regardless of the UAV’s trajectory pattern (straight or spiral). H1 (Alternative Hypothesis): There is an emitting source (corona effect) detectable by the UAV, and the effectiveness of its detection varies with the UAV’s trajectory pattern (straight or spiral). Experimental Design: Simulations are conducted using MATLAB, leveraging its robust computational capabilities to model multiple scattering interactions typical of NLoS communication channels. Parameters such as the angular dispersion of photons, the UAV’s altitude and speed, and the environmental visibility conditions are methodically varied to simulate different real-world scenarios [47]. Monte Carlo Simulation: This technique is employed to model the random nature of scattering events and their impact on photon detection. It allows for the estimation of detection probabilities under various atmospheric conditions, providing a comprehensive understanding of how different UAV trajectories affect detection efficiency [48]. 30 Handling NLoS Conditions: The study specifically addresses the challenges posed by non-line-of-sight (NLoS) conditions, which occur when there is no direct visibility between the UAV and the transmitter due to obstacles like terrain or atmospheric interference. These conditions, which include but are not limited to, atmospheric obstructions, varying altitudes, and the inherent unpredictability of photon paths, are meticulously modeled to reflect real-world operational environments for UAVs in powerline inspections [49,50]. 3.4 Model Description The UV communication model employed in this thesis is specifically designed to address the challenges associated with NLoS conditions, a common scenario in UAV applications for infrastructure monitoring. This model is grounded in the fundamental principles of ultraviolet light behavior, particularly how UV light interacts with atmospheric particles, which significantly affects its travel and detection. The core of this model includes detailed characterizations of the UV source's emission patterns and the atmospheric medium’s reflective properties. Adaptations to the conventional UV communication models have been made to account for the mobility of UAVs. These adaptations are crucial as they influence both the range and resolution of detection, simulating how the UAV's movement can affect the detection capabilities of the onboard sensors. 31 3.5 Simulation Setup The simulation setup is meticulously designed to evaluate the UAV's detection and estimation performance under straight and arbitrary trajectories using MATLAB. This environment is chosen for its robust computational capabilities and extensive support for geospatial data integration, which allows for a realistic mapping of the area surrounding powerlines. In this setup, the UAV is programmed to execute two primary flight patterns: straight and spiral trajectories. Each of these patterns is tested at varying speeds to determine the optimal conditions for detecting UV emissions from powerlines. The parameters of each flight, such as the UAV's position, speed, and altitude changes, are continuously recorded, along with the rates of photon detection, to provide a comprehensive dataset for analysis. The complexity of the photon's travel through the atmosphere, influenced by variables such as air density and particulate matter, is simulated using the Monte Carlo method. This method is particularly well-suited for this research as it provides a probabilistic means to assess photon detection, accounting for random variations in atmospheric conditions and photon scattering behaviors. This probabilistic approach enables the simulation to produce a range of possible outcomes under similar conditions, offering a nuanced understanding of how different UAV trajectories and environmental factors influence the success rate of UV photon detection. This information is critical for selecting the most effective UAV flight paths for real-world applications in utility infrastructure monitoring. To implement the simulations and trajectory analyses, custom MATLAB codes were utilized. These codes were developed to accurately model and simulate the UAV trajectories and their interactions with environmental factors. The full MATLAB code listings used in this study are provided in Appendix A and B. 32 3.6 Data Collection Data collection for this thesis was conducted using simulated data generated within the MATLAB environment. This approach was chosen due to several practical constraints associated with conducting real-world experiments. Gathering experimental data would require significant financial resources, including the acquisition of UAVs, specialized sensors, and programming to control flight paths and record photon detection rates. Additionally, locating faulty powerlines in real-world environments presents challenges in terms of accessibility, safety regulations, and obtaining the necessary permissions from utility companies. Simulated data allows for controlled testing of various UAV trajectories under different conditions, providing insights into the probability of photon detection, location estimation error, and location estimation. The simulations can be adjusted to model different environmental scenarios and operational settings, making it an efficient and flexible method. While experimental data could provide additional validation, the complexity and cost involved in field testing made simulations the most feasible option for this research. 3.7 Data Analysis The data analysis process for this study was meticulously designed to evaluate the effectiveness of UAV trajectory planning in detecting the corona effect on powerlines using UV communication under non-line-of-sight conditions. The primary outcomes analyzed from the MATLAB simulations included the probability of photon detection at various locations along the powerline and the estimated location of the transmitter. These outcomes were crucial for assessing the precision of the UAV's detection capabilities in various simulated environments. To quantitatively 33 measure the accuracy of the transmitter location estimates, the study calculated the estimation error by comparing the actual transmitter location (XActual) with the simulated estimates (XEstimated). This error was computed by taking the absolute difference between the actual and estimated transmitter locations and then dividing that difference by the actual location., thus providing a relative measure of accuracy. Estimation Error (E) = |𝑋𝐴𝑐𝑡𝑢𝑎𝑙 − 𝑋𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑|𝑋𝐴𝑐𝑡𝑢𝑎𝑙 (3.1) For a comparative analysis, the UAV trajectories—straight and spiral—were evaluated to determine which trajectory resulted in lower estimation errors. This involved simulating each trajectory path ten times to mitigate any random anomalies and averaging the estimation errors from these runs. By analyzing these averaged errors, the study could identify the more effective trajectory for accurate transmitter location estimation under simulated conditions. Additionally, these simulations were conducted under varying UAV speeds and on different arbitrary paths, as well as at different transmitter locations, to ensure a comprehensive assessment across a range of operational scenarios. This approach allowed for a robust comparison of how each trajectory performed under diverse conditions, contributing to a nuanced understanding of UAV trajectory performance in monitoring systems. The analytical process also included a statistical evaluation of the photon detection probabilities, which were correlated with the UAV’s operational parameters to identify trends and optimal conditions for maximum detection efficiency. By systematically varying the speed of the UAV and the complexity of the flight paths, the analysis explored how these factors influenced the detection probabilities and estimation accuracies. This thorough statistical examination helped in formulating recommendations for UAV operational 34 strategies in utility infrastructure monitoring, ensuring that the findings are grounded in empirical evidence and are applicable to enhancing the accuracy and reliability of NLoS UV communication systems employed in real-world settings. 3.8 Formulas for UAV Trajectory Simulation The simulation of UAV trajectories incorporates specific mathematical models tailored for straight and spiral paths. These models are pivotal in determining how the UAV navigates through non-line-of-sight environments and how effectively it can detect photons emitted from powerlines. 3.8.1 Straight Path Trajectory The formula for the straight path trajectory simplifies the UAV's motion into a linear, constant-speed movement, which is crucial for maintaining a consistent approach relative to the powerline. Figure 3.1 illustrates this straight-path trajectory, highlighting the UAV’s alignment and movement strategy. 35 Figure 3.1: Straight path trajectory The UAV's position along the straight path is given by: Xn = Xn-1 + (Δt× V) (3.2) where: Xn is the UAV's position at time n, Xn-1 is the UAV's position at time n-1, V is the velocity of the UAV, Δt represents the incremental time step. This equation ensures that the UAV's progression along the powerline is calculated at each step, factoring in the velocity and the time elapsed. This straightforward computation allows for precise control over the UAV's flight path, which is essential for consistent photon detection along a defined route. 36 Figure 3.3: Spiral path trajectory (3D) Figure 3.2: Spiral path trajectory 3.8.2 Spiral Path Trajectory For the spiral path, the trajectory is more complex and is designed to enhance area coverage. The formulas governing the spiral path consider both radial expansion and angular progression, as visually represented in Figures 3.2 and 3.3. These figures illustrate the UAV’s expansive spiral movement, demonstrating the strategic increase in coverage area along the powerline. The UAV's position along the spiral path is given by: Xn = Xn-1 + (V × Δt) (3.3) Y = r – (r × Cos (θ)) (3.4) Z = r × Sin (θ) (3.5) θn = θn-1 +( (2𝜋𝑆𝑡𝑒𝑝 𝑠𝑖𝑧𝑒) × Δt) (3.6) where: θn and θn-1 are the angular positions at times n and n-1, Step size is the incremental change in the angular position, r is the distance between the receiver and transmitter, 37 Figure 3.4: Geometric Calculation of UAV Coordinates Y and Z are the Cartesian coordinates corresponding to the UAV's position in the vertical plane. These equations allow the UAV to spiral outward as it ascends or descends, effectively scanning a larger volume of space around the powerline. The combination of radial and angular adjustments provides a comprehensive sweep of the area. Figure 3.4 illustrates the geometric relationships used to calculate the coordinates Y and Z. This visual representation helps clarify how the UAV’s position is derived from its radial and angular parameters along the straight path. 38 3.8.3 Implementation in Simulation Implementing these equations in MATLAB involves setting initial values for Xn-1, θn-1, and r, and updating these values iteratively over the simulation duration. Each update represents a move to a new position based on the UAV's velocity and the chosen path—straight or spiral. The simulation tracks the UAV's position in real time, calculating photon detection probabilities based on its proximity to the theoretical emission sources. 3.8.4 Analytical Considerations The effectiveness of each trajectory in terms of detection accuracy is quantified by measuring the estimation error, which is the difference between the actual transmitter location and the estimated position derived from the simulation. By running multiple iterations and comparing the mean estimation errors for both trajectories under various operational conditions, the study robustly evaluates which path yields more reliable and accurate results. 3.9 Simulation Flowchart The MATLAB simulation of the UAV trajectory planning for photon detection is structured around a comprehensive flowchart, detailed in Figure 3.5. This flowchart serves as a visual guide to the sequence of operations and decision-making processes implemented in the simulation. It outlines the critical steps of the simulation, ensuring clarity and systematic progression through the simulation's complex algorithms. Each component of the flowchart is explained in detail in the sections that follow, providing a thorough understanding of how various factors influence photon detection and UAV performance under different atmospheric conditions. 39 Figure 3.5: Flowchart of MATLAB Simulation Process 40 3.9.1 Initialization of Simulation Parameters The simulation begins by setting critical parameters such as the distance between the transmitter and receiver (r), the directional angles of both the transmitter (Tdir) and the receiver (Rdir), the beam width of the transmitter (Tbw), the field of view of the receiver (Rfov), and the area of the receiver aperture (Sr). Additional parameters include the absorption (kabs), Rayleigh scattering (kray), and Mie scattering coefficients (kmie), along with atmospheric conditions (atm), the maximum number of scattering events (N), and the number of photons to simulate (its). These parameters establish the foundational conditions under which the simulation operates, crucial for accurate modeling of UV light behavior in atmospheric conditions. 3.9.2 Photon Generation and Scattering Process Upon initializing the positions of the transmitter and receiver, the simulation proceeds to generate photons one at a time, with each photon undergoing multiple scattering events. The MATLAB function increases the scattering order (i) with each event, calculating new positions for the photon based on the scattering distance (d), and the zenith and azimuth angles (θ and ϕ). These calculations consider complex interactions within the atmosphere, such as absorption and scattering by small and large particles (Rayleigh and Mie scattering), which are modeled using detailed phase function parameters stored in vectors K and C. 41 3.9.3 Dynamic Updating of Receiver Location As the simulation progresses, the location of the receiver is dynamically updated to simulate movement or changes in orientation. This step is crucial for testing different scenarios where the UAV (carrying the receiver) might change positions, affecting the detection probabilities of photons within its field of view (FOV). 3.9.4 Probability Calculations and Decision-Making For each photon, the simulation computes the probability of it being detected by the receiver, factoring in the current conditions and the photon's path. This involves detailed calculations using the phase function parameters, which influence how photons are scattered and absorbed by the atmosphere. The function checks if the current photon is the last in the sequence and if the accumulated photon count meets a predefined threshold for successful detection. If the threshold is met, the simulation estimates the transmitter's location based on the aggregated data from detected photons, calculating the position probability that provides insights into the effectiveness of the detection process under given conditions. 3.9.5 Estimation of Transmitter Location The transmitter location is estimated using a probabilistic approach based on the normalized probabilities of photon detection at different candidate locations along the powerline. The goal is to determine the most likely position of the transmitter based on these probabilistic measurements. 42 (3.7) The transmitter location is estimated using the following formula: ?̄? = ∑ Xi ∗ Pi𝑵𝒊=𝟏 Where: X̅: is the estimated transmitter location. Xi: represents the i-th candidate location on the powerline. Pi: denotes the normalized probability of photon detection at the i-th candidate location. N: is the total number of candidate locations considered. This formula computes the transmitter location as a weighted average of candidate locations. Each candidate location Xi is weighted by the normalized probability Pi of detecting photons at that location. The probability Pi is derived from the photon detection measurements, which reflect the likelihood of the transmitter being at that particular location. 3.9.6 Statistical Analysis and Visualization The simulation records each trial's outcomes, providing data that can be statistically analyzed to understand the effectiveness of different UAV trajectories and atmospheric conditions on photon detection. This includes plotting the probability distributions and comparing the detection efficiencies across different simulation runs. 3.9.7 Iterative and Comparative Analysis By repeatedly simulating the photon generation and scattering process under varied conditions (such as different UAV speeds, paths, and transmitter locations), the MATLAB function provides 43 a robust framework for systematically evaluating how different variables affect the NLoS communication efficacy. This iterative approach allows for a thorough analysis of how different UAV trajectories influence detection capabilities in real-world applications. 3.10 Utilizing Poisson Distribution in Photon Emission Analysis The Poisson distribution is integral to this study's approach to modeling discrete photon emissions from power lines. Due to its efficacy in scenarios where events occur independently at a constant average rate, the Poisson distribution is perfectly suited for our analysis. Specifically, the probability of observing an exact number of emissions k during a set interval is given by the Poisson probability mass function: P (X = k) = λ𝑘𝑒−λ𝑘! (3.8) where λ is the emission rate, e denotes the base of the natural logarithm, and k! is the factorial of k. This statistical tool allows us to quantify the likelihood of emission counts under normal operational conditions. 3.10.1 Hypothesis Testing in Photon Emission Studies To determine whether variations in photon counts are significant or merely random fluctuations, we employ hypothesis testing. The null hypothesis H0 posits that variations in photon counts are attributable solely to background environmental noise, represented by a baseline emission rate λb. Conversely, the alternative hypothesis (H₁) suggests that photon emissions exceed this background level due to the Corona effect, implying an additional emission rate 𝜆𝑠, which represents the rate at which photons are emitted directly as a result of the corona discharge, leading to a higher overall 44 emission rate (λs + λb.). This methodical testing framework enables us to assess observed photon counts systematically against established baseline levels. 3.10.2 Parameter Specification In the context of detecting the Corona effect, it is important to establish a baseline photon emission rate. Based on research by Chen et al. (2008), the background photon emission rate λb is determined to be 14,500 photons per second. This value represents the natural photon emissions that occur in the environment due to various sources, including solar radiation and other ambient factors. Establishing this baseline is essential as it helps differentiate between normal environmental emissions and those potentially caused by corona discharges, which are of particular interest in this study. The baseline serves as a reference point against which any observed increase in photon emissions can be measured to assess whether they are due to corona activity or simply background noise [51]. 3.10.3 Statistical Analysis and Threshold Calculation Confidence Level and Z-score: When analyzing the data to detect the Corona effect, we apply hypothesis testing to determine whether the observed photon emissions are likely due to corona discharges or just background noise. A 90% confidence level is chosen for this analysis, which corresponds to a 10% significance level (α). This means that we accept a 10% chance of incorrectly concluding that there is a corona effect when there isn't one. The Z-score, which is 1.645 for a 90% confidence level, is a statistical 45 tool used to calculate the threshold. This threshold is the point at which the probability of observing a higher photon count under the null hypothesis (H₀) is less than 10%. Essentially, if the observed photon count exceeds this threshold, it suggests that the emissions are not just due to background noise but likely include contributions from a corona discharge. 3.10.4 Establishing a Threshold Using Normal Approximation Given the relatively high background emission rate (λ𝑏 = 14,500 photons per second), the Poisson distribution, which models the probability of a given number of events happening in a fixed interval of time, can be approximated by a normal distribution for easier computation. This approximation is valid because the mean photon count (14,500) is large enough for the distribution to resemble a normal curve. The normal distribution is characterized by its mean, which in this case is λ𝑏 , and its standard deviation, √λ𝑏. To calculate the threshold k at which we distinguish between normal background emissions and potential corona discharges, we apply the Z-score to the normal distribution. To find the threshold k using the Z-score, we use: k = λb + Z × √λ𝑏 (3.9) Substituting the values: K = 14,500 + 1.645 × 120.416 ≈ 14,698 This threshold of approximately 14,698 photons per second is the critical value. If the observed photon count exceeds this number, it suggests the presence of additional photon emissions that could be attributed to corona discharges. 46 3.10.5 Decision Rule Based on Statistical Thresholds The decision rule for determining whether the observed photon emissions indicate a corona effect is based on the calculated threshold: - If X (number of photons detected by the receiver) ≤ 14,698: The observed photon count is considered to be within the range of normal background noise, so we accept the null hypothesis (H₀). No corona effect is detected. - If X (number of photons detected by the receiver) > 14,698: The observed photon count exceeds what would be expected from background noise alone, so we reject the null hypothesis (H₀) in favor of the alternative hypothesis (H₁). This suggests that the additional photons are likely due to a corona discharge. This decision-making process ensures that the detection of the corona effect is both accurate and reliable, minimizing the chances of false positives while being sensitive enough to detect true instances of corona activity. 47 Chapter 4 Simulation Results Introduction This chapter presents the comparative analysis of simulation results for UAV trajectories—specifically, the straight and spiral paths. The objective is to evaluate the accuracy of receiver location estimation under varying conditions. For this purpose, the transmitter was placed at three different predefined locations: (60,1,0), (130,1,0), and (210,1,0). By conducting multiple iterations of the simulation for both trajectory types, this analysis measures the estimated location accuracy of the receiver and the corresponding probability distributions for these estimations. The estimation error for each scenario is calculated to compare the accuracy between the two UAV flight paths. The simulation process involved setting the transmitter at the specified locations and running separate simulations for each trajectory type. Each simulation iteration aimed to estimate the transmitter's location based on the UAV's received data. The key outputs from these simulations included the estimated transmitter location and the probability of the transmitter being at each estimated location. The estimation error was calculated as the difference between the actual transmitter position and the estimated position provided by the simulation. This error metric served as the primary criterion for assessing the effectiveness and accuracy of each UAV trajectory method. 48 4.1 Simulation Setup The simulations were set up to ensure consistent conditions for both trajectory types. Parameters such as the UAV's speed, the number of photons simulated, the number of scatterings, and the atmospheric parameters were held constant to isolate the impact of the trajectory type on estimation accuracy. 4.2 Comparison of Trajectory Paths In this section, we will conduct a detailed comparison of the straight and spiral trajectory paths utilized in our UAV simulations. Each trajectory has been evaluated at three distinct transmitter locations: (60,1,0), (130,1,0), and (210,1,0), to determine their efficacy in accurately estimating the transmitter's location under varying conditions. This comparison is critical to understanding how the path of the UAV affects its ability to perform accurate location estimations, which is essential for improving UAV operations in real-world applications. By analyzing the estimated locations and the corresponding probabilities of detection, we aim to highlight the strengths and limitations of each trajectory type. The outcomes from these simulations will provide insights into the potential applications of each trajectory, informing future UAV path planning strategies to enhance accuracy and reliability in photon detection tasks. Figures will be presented side by side for each location, allowing for a visual and analytical evaluation of the performance differences between the straight and spiral paths. This comparative analysis seeks to answer key questions regarding trajectory efficiency and error margins, ultimately guiding the choice of trajectory in practical scenarios. 49 Estimated Transmitter Location: 60.1929 Initial Transmitter Location: 60 Estimation error: %0.32 Estimated Transmitter Location: 62.6193 Initial Transmitter Location:60 Estimation error: %4.36 Figure 4.1.: Probability Distribution at Location 1-straight path Figure 4.2: Probability Distribution at Location 1-spiral path 4.2.1 Trajectory Comparison at Location (60,1,0) The analysis of the UAV trajectories at (60,1,0), shown in Figures 4.1 and 4.2, reveals significant differences in performance. The straight path trajectory (Figure 4.1) demonstrates much lower estimation errors and a higher probability of accurate location detection compared to the spiral path (Figure 4.2). This indicates that the straight path is more effective for precise transmitter location estimation, highlighting the trade-offs between trajectory simplicity and area coverage. 50 Estimated Transmitter Location: 129.706 Initial Transmitter Location:130 Estimation error: %0.22 Estimated Transmitter Location: 126.847 Initial Transmitter Location:130 Estimation error: %2.42 Figure 4.3: Probability Distribution at Location 2- straight path Figure 4.4: Probability Distribution at Location 2-spiral path 4.2.2 Comparison at Location (130,1,0) Figures 4.3 (straight path) and 4.4 (spiral path) demonstrate clear differences in performance at the second transmitter location. The straight path records a minimal estimation error of 0.22%, indicating high precision, whereas the spiral path has a significantly larger error of 2.42%. This comparison underscores the straight path’s superiority for accurate location detection, contrasting sharply with the less precise spiral trajectory. 51 Estimated Transmitter Location: 209.306 Initial Transmitter Location:210 Estimation error: %0.33 Estimated Transmitter Location: 188.340 Initial Transmitter Location:210 Estimation error: %10.31 Figure 4.5: Probability Distribution at Location 3-straight path Figure 4.6: Probability Distribution at Location 3-spiral path 4.2.3 Comparative Analysis at Location (210,1,0) Figures 4.5 and 4.6 reveal significant differences in UAV trajectory performance at (210,1,0). The straight path, shown in Figure 4.5, maintains high accuracy with an estimation error of just 0.33%. In contrast, the spiral path, presented in Figure 4.6, records a much higher error of 10.31%, indicating its limitations in precision. This comparison highlights the superiority of the straight path for precise transmitter localization in complex simulation scenarios. 52 4.2.4 Analysis of Factors Contributing to Variability in Results Across Different Locations The differences in estimation errors and probabilities of detection across various locations are primarily influenced by the scattering effects and the symmetry of the transmitter’s position relative to the UAV’s trajectory. Scattering effects, particularly Rayleigh and Mie scattering, cause variations in how the photons emitted by the transmitter interact with the atmosphere. As the UAV travers’s different locations, changes in atmospheric conditions such as air density, humidity, and the presence of particulate matter can alter the path of these photons, leading to differing levels of signal reception at each location. This variability in signal propagation due to scattering directly impacts the outcomes observed at different locations. Symmetry also plays a crucial role in the observed differences in results. When the transmitter is symmetrically positioned relative to the UAV’s flight path, the signals tend to be more evenly distributed, leading to more consistent results. In contrast, asymmetrical positions disrupt this uniformity, causing variations in how the signals are detected at different points along the trajectory. For instance, when the corona effect is located at 210 meters, the UAV can only scan a short distance beyond this point, leading to asymmetry and higher estimation errors. In contrast, at 130 meters, the UAV’s scan is more balanced, covering significant distances both before and after the corona effect, resulting in lower estimation errors. These differences in symmetry can lead to fluctuations in the results across various locations, as the UAV’s ability to capture and process the signals is influenced by the relative positioning of the transmitter. 53 4.2.5 Comparison of normalized probabilities The figures illustrating the straight and spiral path trajectories show that the straight path consistently achieves a higher normalized probability compared to the spiral path. This higher probability indicates a greater certainty in locating the transmitting point when using the straight trajectory. For example, In Figure 4.5 where the UAV follows a straight path, the probability distribution around the transmitting point is more concentrated, indicating a higher certainty in detecting the transmitter's location. This concentration suggests that the straight path allows the UAV to maintain consistent and direct data collection, reducing the complexity of variables that could introduce errors. The linear and predictable nature of the straight trajectory minimizes disturbances and ensures that the signals received by the UAV are more uniform, leading to lower estimation errors and higher detection accuracy. Conversely, in Figure 4.6, where the UAV follows a spiral trajectory, the probability distribution around the transmitting point is much more dispersed, with higher noise levels. The increased noise and lower certainty in detecting the transmitting point can be attributed to several factors. The spiral path involves continuous changes in the UAV's position and orientation, which can introduce variations in the signal strength and directionality. This trajectory adds complexity to the environmental interaction, leading to fluctuations in the received signals and increased susceptibility to interference. As the UAV moves in a spiral, it covers varying distances from the transmitter at different angles, which can cause inconsistencies in the data and result in a less precise localization of the transmitter. These factors collectively contribute to the higher noise levels and lower certainty in detecting the transmitting point in Figure 4.6. 54 4.2.6 Importance of Estimation Errors The varying levels of estimation errors in UAV trajectory planning offer crucial insights into the precision and effectiveness of the inspection process. Lower estimation errors suggest that the UAV's flight path is well-suited for the specific operational environment, enabling accurate localization of the transmitter and, consequently, precise detection of phenomena like the Corona effect. This indicates that the selected trajectory and speed are ideally suited to the given conditions, ensuring reliable and efficient data collection. Such accuracy is vital for minimizing errors in the identification of potential issues along powerlines, thereby enhancing the overall effectiveness of the UAV's inspection mission. On the other hand, higher estimation errors expose challenges in the UAV's ability to accurately pinpoint the transmitter's location, which can stem from complex environmental factors, suboptimal trajectory planning, or unfavorable operational conditions. These errors highlight areas where the UAV's operational parameters may require adjustment to improve accuracy and data reliability. High estimation errors can lead to significant issues, such as incorrect localization of the problem area, resulting in inefficient maintenance operations, wasted resources, and extended system downtimes. Therefore, minimizing these errors is essential for improving UAV performance and ensuring the success of powerline inspections. 4.2.7Aligning Expectations with Results The results obtained from the simulations align with the expectations grounded in the practical understanding of UAV trajectory planning and its interaction with environmental factors. Straight trajectories, known for their efficiency in covering long distances with minimal energy usage and reduced stress on UAV components, consistently resulted in lower estimation errors. This outcome is anticipated due to the simplicity and directness of straight paths, which allow for more accurate 55 and efficient data collection over extensive stretches of powerlines. On the other hand, the higher errors observed in spiral trajectories were expected, given the complexity of these paths [26]. 4.3 Effect of UAV Velocity on Transmitter Location Estimation Accuracy This section of the thesis presents a detailed analysis of the impact of UAV velocity on the accuracy of transmitter location estimation. The study evaluates how different velocities influence the UAV’s ability to pinpoint the exact location of a transmitter initially positioned at coordinates (130,1,0). The velocities tested were 5, 6, 8, 10, 12, 14, and 15 m/s. Each simulation was repeated ten times to ensure the reliability of the results by averaging the outcomes, thereby reducing the impact of any anomalies or outliers that might skew the accuracy of the data. The compiled results, illustrated in Figure 4.7, show varying levels of relative estimation errors across the tested velocities. A critical observation from the data is that the UAV achieves the lowest relative estimation error at a velocity of 10 m/s, suggesting an optimal balance between speed and accurate localization capability. Velocities below and above this value were associated with higher estimation errors, indicating diminished accuracy. This pattern highlights the nonlinear relationship between UAV velocity and transmitter location estimation accuracy, emphasizing the necessity to calibrate UAV speed carefully to improve performance. 56 Figure 4.7: Effect of Velocity on Estimation Error (%) (m/s) Figure 4.7, which shows varying estimation errors at different UAV speeds, highlights the significant impact of velocity on the accuracy of transmitter localization. At lower speeds, the UAV has more time to gather data from a given location, which could theoretically improve accuracy. However, excessively slow speeds might introduce noise due to prolonged exposure to environmental disturbances, leading to higher estimation errors. Conversely, at very high speeds, the UAV might not have sufficient time to collect enough data, resulting in lower detection probabilities and higher estimation errors. 4.4 Implications of Trajectory Performance for UAV Localization Accuracy The data captured from these simulations underline the straight path's advantage in precision. This pattern was consistent across all tested locations, reinforcing the conclusion that the straight path not only offers better consistency but also excels in accuracy. 57 Chapter 5 Conclusion and Future Work 5.1 Conclusion This thesis conducted a comparative analysis of straight and spiral UAV flight paths for detecting and locating the Corona effect on powerlines. The results showed that the straight path consistently achieved higher accuracy in both the detection and localization of the Corona effect, compared to the spiral path. Under the conditions modeled in this study, the straight trajectory produced lower estimation errors and was more efficient in covering larger distances, making it the optimal path for Corona detection in the context of these simulations. While the spiral path allowed for more detailed inspections of specific areas, it exhibited higher estimation errors and was more resource-intensive. The findings of this thesis have important implications for powerline inspection and maintenance. Utility companies can apply straight flight paths for broad, rapid scans of powerline networks, which would allow for early detection of potential faults that might lead to power outages. This method enhances operational efficiency by covering large areas in a short amount of time, helping to reduce the costs and manpower traditionally needed for inspections. Moreover, by using UAVs to perform these tasks, utility companies can improve safety by minimizing the need for human workers in hazardous or hard-to-reach locations. 58 In addition to powerline inspections, the comparative approach used in this study can be applied to other infrastructure monitoring tasks. The ability to quickly and accurately assess structural integrity using UAVs with appropriate sensors could extend to bridges, railways, and pipelines, where regular maintenance is crucial for preventing failures. The straight path, in particular, could be adopted for scenarios that require efficient coverage of large areas, ensuring timely detection of faults and reducing the risk of major disruptions or accidents in various infrastructure sectors. 5.2 Future Work 1. Improving Computational Efficiency: The simulations in this study were constrained by the computational demands of modeling a high number of photon scatterings and interactions. Future work should focus on enhancing algorithms for faster processing, possibly by using parallel computing or cloud-based solutions. 2. Extending Simulation Scenarios: The current research explored a limited set of UAV trajectories and conditions. Future studies should expand the scope by testing additional configurations, such as different weather conditions or varying altitudes, to better understand the impact of these factors on photon detection accuracy. This would provide a broader dataset to refine trajectory planning and improve UAV inspection performance. 3. Real-World Validation: While this research relied on simulated data, future work should involve real-world testing to validate the accuracy of the probabilistic models used in the 59 simulations. 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Optics express 16, no. 19 (2008): 15059-15068. 66 Appendices Appendix A Straight Path Trajectory Planning Codes function [S,P,Pk] = mcs(r,Tdir,Tbw,Rdir,Rfov,Sr,kabs,kray,kmie,atm,N,its) % Multiple Scattering Ultraviolet Non-Line-of-Sight Simulation % P = Probability of a successful photon transmission % Pk = Vector [P1,P2,...,Pk,...] of probabilities of successful photon % transmission after exactly (k-1) scattering events % Pik = Raw probabilities computed during simulation % r = Distance from transmitter to receiver % Tdir = [Ttheta,Tphi], the directional angles of the transmitter % Note that Ttheta is the inclination angle, not the elevation angle % Tbw = Angular transmitter beamwidth (full width) % Rdir = [Rtheta,Rphi], the directional angles of the receiver % Rfov = Angular field of view of receiver (full width) % Sr = Area of receiver aperture % kabs = Absorption coefficient % kray = Rayleigh scattering coefficient % kmie = Mie scattering coefficient % atm = [gamma;f;g], vector of atmospheric parameters % N = Maximum number of scattering events to consider % its = Number of photons to simulate % tol = Tolerance for numerical solution of scattering angle (Defaults to % 1e-12) % Note on input coordinate system: linear distance units must be % consistent, angles are in radians, and angular directions are measured tol = 1e-12; %This means that the numerical algorithm used to compute the scattering %angle will terminate when the solution is within this tolerance, %ensuring a high level of precision in the computed scattering angles. Rrec = sqrt(Sr/pi); % Radius of spherical Rx with cross-sectional area Sr for Reciever FOV kscat = kray+kmie; % Total scattering coefficient- ray:small particles, mie:large particles K = zeros(6,1); % Parameters for computation of the integral of the phase function K(1) = 3*kray/kscat/(8*(1+2*atm(1)))*(1+3*atm(1))-1/2*kmie/kscat*(1-atm(3)^2)*atm(2)/(2*(1+atm(3)^2+0.845*atm(3))^(5/2)); K(2) = 3*kray/kscat/(8*(1+2*atm(1)))*(1/3)*(1-atm(1))+1/2*kmie/kscat*(1-atm(3)^2)*atm(2)/(2*(1+atm(3)^2+0.845*atm(3))^(5/2)); K(3) = 1/2*kmie/kscat*(1-atm(3)^2)/atm(3); K(4) = 1+atm(3)^2; 67 K(5) = 2*atm(3); K(6) = 1/2*kray/kscat-1/2*kmie/kscat*(1-atm(3)^2)/atm(3)*(1+atm(3)^2+2*atm(3))^(-1/2); C = zeros(5,1); % Parameters for computation of the phase function C(1) = kray/kscat*3*(1+3*atm(1))/(16*pi*(1+2*atm(1)))-kmie/kscat*(1-atm(3)^2)/(8*pi)*atm(2)/(1+atm(3)^2+0.845*atm(3))^(5/2); C(2) = kray/kscat*3*(1-atm(1))/(16*pi*(1+2*atm(1)))+kmie/kscat*3*(1-atm(3)^2)/(8*pi)*atm(2)/(1+atm(3)^2+0.845*atm(3))^(5/2); C(3) = kmie/kscat*(1-atm(3)^2)/(4*pi); C(4) = 1+atm(3)^2; C(5) = 2*atm(3); ktot = kscat+kabs; % Extinction coefficient (scattering and absorption) % Direction cosines for Tx pointing direction Tdircos = [cos(Tdir(2))*sin(Tdir(1)); sin(Tdir(2))*sin(Tdir(1)); cos(Tdir(1))]; experimentlength = 24; plot_x = zeros(1,experimentlength); plot_y = zeros(1,experimentlength); for time = 1:experimentlength; Pik = zeros(its,N); distances = zeros(its,N); for photon = 1:its % showing the probability of detection when the R is facing each location % on the powerline pos = [60;r;0]; % Transmitter Actual location on the powerline pos(1) = pos(1)-time*10; Rpos = [0,0,0]; Rpos(1) = Rpos(1)+time*10; Rpos(2) = r; % Progress update if ~rem(photon,1000) photon; %#ok<NOPRT> end Pcur = 1; % "weight" of the current photon Prec = 0; % Probability of reception after current number of scatterers dcur = 0; % Distance traveled by simulated photon to current scatterer for scatter = 1:N % First "scattering" (at Tx) has Tx beam function around Tdircos if scatter == 1 diroff = [acos(1-rand*(1-cos(Tbw/2))); 2*pi*rand]; % Offset angles % calculates the offset angles for the direction of the photon after the first scattering event. dircos = dircosup(Tdircos,diroff); % Photon direction cosine update else % Subsequent scatterings obey general phase function diroff = [scatang(K,tol);2*pi*rand]; % Offset angles dircos = dircosup(dircos,diroff); % Photon direction cosine update end %diroff(1): The elevation angle , represents the angle between the photon's new direction and the original direction. 68 %diroff(2): The azimuthal angle, represents the rotation around the original direction. %Tdircos is the initial direction cosine of the photon. %dircos is the updated direction cosine of the photon after each scattering event. %dircosup represents the new direction in which the photon is traveling after the scattering event. ds = -log(rand)/kscat; % Distance to next scatterer dcur = dcur + ds; % Update distance traveled by photon newpos = pos + ds*dircos; % Photon position update % Weight update accounting for probability of photon reception at % Rx from previous scatterer and for probability of absorption % along simulated path from previous scatterer to current scatterer Pcur = Pcur*(1-Prec)*exp(-kabs*ds);%*exp(-kabs*ds); %current probability that the photon has not been absorbed or scattered before reaching the current scatterer % End simulation of photon whose simulated path actually arrives at Rx. Dnear = norm(cross(pos,newpos))/norm(newpos-pos); % Distance photon could come to Rx if Dnear <= Rrec % Photon's extended path passes through sperical Rx % Direction of photon (toward or away from Rx) tmp = -dot(pos,newpos-pos)/norm(newpos-pos)^2; % Photon travels toward Rx and travels far enough to reach Rx if tmp > 0 && ds >= norm(pos) break; end end % Probability photon is received at Rx from current scatterer Prec = exp(-ktot*norm(newpos))*Pang2rec(newpos,dircos,Rdir,Rfov,Sr,C); % Store probability photon is received at Rx Pik(photon,scatter) = Prec*Pcur; % Store distance received photon would have traversed if received % at Rx from current scatterer distances(photon,scatter) = dcur + norm(newpos); % Set position for next time through loop pos = newpos; end end % Probability of photon reception after k scatterings Pk = sum(Pik)'/its; % Total probability of photon reception 69 P = sum(Pk); outputString = sprintf('POD: %.2e', P); disp(outputString); S= P*10^20; outputString1 = sprintf('NOPD: %.2e', S); disp(outputString1); outputString3 = sprintf('RFL: (%d,%d,%d)\n', Rpos(1),Rpos(2),Rpos(3)); disp(outputString3); %Visualization of pdf of transmitter location plot_x(1,time) = Rpos(1); plot_y(1,time) = P; % end pdf = plot_y/sum(plot_y); figure(1) stem(plot_x,pdf,'Marker', 'd', 'MarkerEdgeColor', 'r') xlabel('X-axis Position') ylabel('Normalized Probability') figure(2) stem(plot_x,plot_y,'Marker', 'd', 'MarkerEdgeColor', 'r') xlabel('X-axis Position') ylabel('Reception Probability') Expected_transmitter_loc = sum(pdf.*plot_x) function [dircosup] = dircosup(dircos,diroff) % Updates direction cosines given offset angles % % dircosup = Direction cosines of new path % dircos: cosine values of the angles between the current direction and each of the three coordinate axes % dircos = Direction cosines of current path % diroff = [alpha, psi], where alpha is the angle between current direction and the % new direction, and psi is the azimuth angle of the new direction dircosup = zeros(3,1); sintheta = sqrt(1-dircos(3)^2); % Updated direction cosines computed using equations from Ding, 2009 dircosup(3) = -sin(diroff(1))*cos(diroff(2))*sintheta+dircos(3)*cos(diroff(1)); if sintheta > 1e-4 70 dircosup(1) = sin(diroff(1))/sintheta*(dircos(1)*dircos(3)*cos(diroff(2))-dircos(2)*sin(diroff(2)))+dircos(1)*cos(diroff(1)); dircosup(2) = sin(diroff(1))/sintheta*(dircos(2)*dircos(3)*cos(diroff(2))+dircos(1)*sin(diroff(2)))+dircos(2)*cos(diroff(1)); % An attempt to avoid numerical issue when sintheta is close to 0. % Probability of this event is small, probably has no effect on simulation % results. Not clear if all potential numerical issues are avoided, but at % least the extreme cases are handled. else if sintheta == 0 % Avoid division by 0 cosphi = 1; % Arbitararily set phi = 0 sinphi = 0; % Arbitararily set phi = 0 else cosphi = dircos(1)/sintheta; sinphi = dircos(2)/sintheta; %checks if either cosphi or sinphi has resulted in an infinite value. % Check if sintheta is so small that above computation overflowed if isinf(cosphi) || isinf(sinphi) cosphi = 1; % Arbitrarily set phi = 0 sinphi = 0; % Arbitrarily set phi = 0; end end % Rearranged equation from Ding, 2009, to remove division dircosup(1) = sin(diroff(1))*(cosphi*dircos(3)*cos(diroff(2))-sinphi*sin(diroff(2)))+dircos(1)*cos(diroff(1)); dircosup(2) = sin(diroff(1))*(sinphi*dircos(3)*cos(diroff(2))+cosphi*sin(diroff(2)))+dircos(2)*cos(diroff(1)); end function alpha = scatang(K,tol) % Outcome is alpha which is the angle between old and new path of photon after scattering % It calculates the Scattering angle % Produce an appropriately distributed scattering angle via bisection search % alpha = Angle between new path and current path % Using bisection search to find the scattering angle % K = Parameters of integrated phase function % tol = Tolerance of numerical solution: algorithm terminates when mu is % within tol of exact answer x = rand; % Uniform random variable to be transformed- Generates a random number between 0 and 1 mu = 0; % Initial guess for the cosine of the scattering angle step = 1; % Initial step size and maximum error- Initial step size for the search while step >= tol % Continue until maximum error is less than tol step = step/2; % Bisection search %phase function describes how likely a photon is to scatter at different angles 71 % If phase function evaluated at mu is too small, need to increase mu if K(1)*mu+K(2)*mu^3+K(3)*(K(4)-K(5)*mu)^(-1/2)+K(6) <= x mu = mu + step; else % Else mu is too large mu = mu - step; end end alpha = acos(mu); % Scattering angle function Pang = Pang2rec(pos,dircos,Rdir,Rfov,Sr,C) % Probabilty that current photon will be scattered in a direction such that % it could be received at the Rx % Pang = Probability that photon is in FOV of Rx and is scattered toward Rx % pos = Location of current scatterer (where origin is at Rx) % dircos = Direction of photon prior to scattering % Rdir = Pointing direction of receiver % Rfov = Angular field of view of receiver (full width) % Sr = Area of receiver aperture % C = Parameters phase function theta = acos(pos(3)/norm(pos)); % Inclination angle of photon position if pos(1)==0 phi = pi/2; else phi = atan2(pos(2),pos(1)); % Azimuth angle of photon position end % angle between the receiver's pointing direction and the line from the receiver to the photon's position % Angular offset from Rx pointing direction to line from Rx to photon angfromR = acos(cos(theta)*cos(Rdir(1))+sin(theta)*sin(Rdir(1))*cos(phi-Rdir(2))); if angfromR > Rfov/2 Pang = 0; % Photon not in Rx FOV else d1 = norm(pos); % Distance from Rx to photon d2 = norm(pos+dircos); % Distance from Rx to a point at unit distance along path of photon mu = 1/(2*d1)*(1+d1^2-d2^2); % Scattering angle from scatterer to Rx (Law of Cosines) % Approximate probability the photon scatters toward the Rx area Pang = Sr/d1^2*(C(1)+C(2)*mu^2+C(3)*(C(4)-C(5)*mu)^(-3/2)); end 72 Appendix B Spiral Path Trajectory Planning Codes function [S,P,Pk] = mcs(r,Tdir,Tbw,Rdir,Rfov,Sr,kabs,kray,kmie,atm,N,its) % Multiple Scattering Ultraviolet Non-Line-of-Sight Simulation % P = Probability of a successful photon transmission % Pk = Vector [P1,P2,...,Pk,...] of probabilities of successful photon % transmission after exactly (k-1) scattering events % Pik = Raw probabilities computed during simulation % r = Distance from transmitter to receiver % Tdir = [Ttheta,Tphi], the directional angles of the transmitter % Note that Ttheta is the inclination angle, not the elevation angle % Tbw = Angular transmitter beamwidth (full width) % Rdir = [Rtheta,Rphi], the directional angles of the receiver % Rfov = Angular field of view of receiver (full width) % Sr = Area of receiver aperture % kabs = Absorption coefficient % kray = Rayleigh scattering coefficient % kmie = Mie scattering coefficient % atm = [gamma;f;g], vector of atmospheric parameters % N = Maximum number of scattering events to consider % its = Number of photons to simulate % tol = Tolerance for numerical solution of scattering angle (Defaults to % 1e-12) % Note on input coordinate system: linear distance units must be % consistent, angles are in radians, and angular directions are measured tol = 1e-12; %This means that the numerical algorithm used to compute the scattering %angle will terminate when the solution is within this tolerance, %ensuring a high level of precision in the computed scattering angles. Rrec = sqrt(Sr/pi); % Radius of spherical Rx with cross-sectional area Sr kscat = kray+kmie; % Total scattering coefficient K = zeros(6,1); % Parameters for computation of the integral of the phase function K(1) = 3*kray/kscat/(8*(1+2*atm(1)))*(1+3*atm(1))-1/2*kmie/kscat*(1-atm(3)^2)*atm(2)/(2*(1+atm(3)^2+0.845*atm(3))^(5/2)); K(2) = 3*kray/kscat/(8*(1+2*atm(1)))*(1/3)*(1-atm(1))+1/2*kmie/kscat*(1-atm(3)^2)*atm(2)/(2*(1+atm(3)^2+0.845*atm(3))^(5/2)); K(3) = 1/2*kmie/kscat*(1-atm(3)^2)/atm(3); K(4) = 1+atm(3)^2; K(5) = 2*atm(3); K(6) = 1/2*kray/kscat-1/2*kmie/kscat*(1-atm(3)^2)/atm(3)*(1+atm(3)^2+2*atm(3))^(-1/2); C = zeros(5,1); % Parameters for computation of the phase function C(1) = kray/kscat*3*(1+3*atm(1))/(16*pi*(1+2*atm(1)))-kmie/kscat*(1-atm(3)^2)/(8*pi)*atm(2)/(1+atm(3)^2+0.845*atm(3))^(5/2); C(2) = kray/kscat*3*(1-atm(1))/(16*pi*(1+2*atm(1)))+kmie/kscat*3*(1-atm(3)^2)/(8*pi)*atm(2)/(1+atm(3)^2+0.845*atm(3))^(5/2); 73 C(3) = kmie/kscat*(1-atm(3)^2)/(4*pi); C(4) = 1+atm(3)^2; C(5) = 2*atm(3); ktot = kscat+kabs; % Extinction coefficient (scattering and absorption) % Direction cosines for Tx pointing direction Tdircos = [cos(Tdir(2))*sin(Tdir(1)); sin(Tdir(2))*sin(Tdir(1)); cos(Tdir(1))]; theta = Rdir(1); experimentlength = 24; stepsize = 4; plot_x = zeros(1,experimentlength); plot_y = zeros(1,experimentlength); for time = 1:experimentlength Pik = zeros(its,N); distances = zeros(its,N); for photon = 1:its pos = [60;r;0]; % Transmitter Actual location on the powerline pos(1) = pos(1)-(time-1)*10; pos(2) = r*cos((2*pi/stepsize)*(time-1)); pos(3) = r*sin((2*pi/stepsize)*(time-1)); Rdir(1) = theta + (2*pi/stepsize*(time-1)); Rpos = [0,0,0]; Rpos(1) = Rpos(1)+(time-1)*10; Rpos(2) = r; Rpos(3) = 0; % Progress update if ~rem(photon,1000) photon; %#ok<NOPRT> end Pcur = 1; % "weight" of the current photon Prec = 0; % Probability of reception after current number of scatterers dcur = 0; % Distance traveled by simulated photon to current scatterer for scatter = 1:N % First "scattering" (at Tx) has Tx beam function around Tdircos if scatter == 1 diroff = [acos(1-rand*(1-cos(Tbw/2))); 2*pi*rand]; % Offset angles dircos = dircosup(Tdircos,diroff); % Photon direction cosine update else % Subsequent scatterings obey general phase function diroff = [scatang(K,tol);2*pi*rand]; % Offset angles dircos = dircosup(dircos,diroff); % Photon direction cosine update end %diroff(1): The elevation angle , represents the angle between the photon's new direction and the original direction. %diroff(2): The azimuthal angle, represents the rotation around the original direction. %Tdircos is the initial direction cosine of the photon. 74 %dircos is the updated direction cosine of the photon after each scattering event. %dircosup represents the new direction in which the photon is traveling after the scattering event. ds = -log(rand)/kscat; % Distance to next scatterer dcur = dcur + ds; % Update distance traveled by photon newpos = pos + ds*dircos; % Photon position update % Weight update accounting for probability of photon reception at % Rx from previous scatterer and for probability of absorption % along simulated path from previous scatterer to current scatterer Pcur = Pcur*(1-Prec)*exp(-kabs*ds);%*exp(-kabs*ds); % End simulation of photon whose simulated path actually arrives at % Rx. This is a technicality, since the probability of actual % arrival is so small that it likely has no effect on simulation results. Dnear = norm(cross(pos,newpos))/norm(newpos-pos); % Distance photon could come to Rx if Dnear <= Rrec % Photon's extended path passes through sperical Rx % Direction of photon (toward or away from Rx) tmp = -dot(pos,newpos-pos)/norm(newpos-pos)^2; % Photon travels toward Rx and travels far enough to reach Rx if tmp > 0 && ds >= norm(pos) break; end end % Probability photon is received at Rx from current scatterer Prec = exp(-ktot*norm(newpos))*Pang2rec(newpos,dircos,Rdir,Rfov,Sr,C); % Store probability photon is received at Rx Pik(photon,scatter) = Prec*Pcur; % Store distance received photon would have traversed if received % at Rx from current scatterer distances(photon,scatter) = dcur + norm(newpos); % Set position for next time through loop pos = newpos; end end % Probability of photon reception after k scatterings Pk = sum(Pik)'/its; % Total probability of photon reception P = sum(Pk); outputString = sprintf('POD: %.2e', P); disp(outputString); S= P*10^20; 75 outputString1 = sprintf('NOPD: %.2e', S); disp(outputString1); outputString3 = sprintf('RFL: (%d,%d,%d)\n', Rpos(1),Rpos(2),Rpos(3)); disp(outputString3); %Visualization of pdf of transmitter location plot_x(1,time) = Rpos(1); plot_y(1,time) = P; % end pdf = plot_y/sum(plot_y); figure(1) stem(plot_x,pdf,'Marker', 'd', 'MarkerEdgeColor', 'r') xlabel('X-axis Position') ylabel('Normalized Probability') figure(2) stem(plot_x,plot_y,'Marker', 'd', 'MarkerEdgeColor', 'r') xlabel('X-axis Position') ylabel('Reception Probability') Expected_transmitter_loc = sum(pdf.*plot_x) function [dircosup] = dircosup(dircos,diroff) % Updates direction cosines given offset angles % % dircosup = Direction cosines of new path % % dircos = Direction cosines of current path % diroff = [alpha, psi], where alpha is the angle between dircos and the % new direction, and psi is the azimuth angle of the new direction dircosup = zeros(3,1); sintheta = sqrt(1-dircos(3)^2); % Updated direction cosines computed using equations from Ding, 2009 dircosup(3) = -sin(diroff(1))*cos(diroff(2))*sintheta+dircos(3)*cos(diroff(1)); if sintheta > 1e-4 dircosup(1) = sin(diroff(1))/sintheta*(dircos(1)*dircos(3)*cos(diroff(2))-dircos(2)*sin(diroff(2)))+dircos(1)*cos(diroff(1)); dircosup(2) = sin(diroff(1))/sintheta*(dircos(2)*dircos(3)*cos(diroff(2))+dircos(1)*sin(diroff(2)))+dircos(2)*cos(diroff(1)); % An attempt to avoid numerical issue when sintheta is close to 0. 76 % Probability of this event is small, probably has no effect on simulation % results. Not clear if all potential numerical issues are avoided, but at % least the extreme cases are handled. else if sintheta == 0 % Avoid division by 0 cosphi = 1; % Arbitararily set phi = 0 sinphi = 0; % Arbitararily set phi = 0 else cosphi = dircos(1)/sintheta; sinphi = dircos(2)/sintheta; % Check if sintheta is so small that above computation overflowed if isinf(cosphi) || isinf(sinphi) cosphi = 1; % Arbitrarily set phi = 0 sinphi = 0; % Arbitrarily set phi = 0; end end % Rearranged equation from Ding, 2009, to remove division dircosup(1) = sin(diroff(1))*(cosphi*dircos(3)*cos(diroff(2))-sinphi*sin(diroff(2)))+dircos(1)*cos(diroff(1)); dircosup(2) = sin(diroff(1))*(sinphi*dircos(3)*cos(diroff(2))+cosphi*sin(diroff(2)))+dircos(2)*cos(diroff(1)); end function alpha = scatang(K,tol) % Produce an appropriately distributed scattering angle via bisection search % % alpha = Angle between new path and current path % K = Parameters of integrated phase function % tol = Tolerance of numerical solution: algorithm terminates when mu is % within tol of exact answer x = rand; % Uniform random variable to be transformed mu = 0; % Initial guess step = 1; % Initial step size and maximum error while step >= tol % Continue until maximum error is less than tol step = step/2; % Bisection search % If phase function evaluated at mu is too small, need to increase mu if K(1)*mu+K(2)*mu^3+K(3)*(K(4)-K(5)*mu)^(-1/2)+K(6) <= x mu = mu + step; else % Else mu is too large mu = mu - step; end end alpha = acos(mu); % Scattering angle function Pang = Pang2rec(pos,dircos,Rdir,Rfov,Sr,C) 77 % Probabilty that current photon will be scattered in a direction such that % it could be received at the Rx % Pang = Probability that photon is in FOV of Rx and is scattered toward Rx % pos = Location of current scatterer (where origin is at Rx) % dircos = Direction of photon prior to scattering % Rdir = Pointing direction of receiver % Rfov = Angular field of view of receiver (full width) % Sr = Area of receiver aperture % C = Parameters phase function theta = acos(pos(3)/norm(pos)); % Inclination angle of photon position if pos(1)==0 phi = pi/2; else phi = atan2(pos(2),pos(1)); % Azimuth angle of photon position end % Angular offset from Rx pointing direction to line from Rx to photon angfromR = acos(cos(theta)*cos(Rdir(1))+sin(theta)*sin(Rdir(1))*cos(phi-Rdir(2))); if angfromR > Rfov/2 Pang = 0; % Photon not in Rx FOV else d1 = norm(pos); % Distance from Rx to photon d2 = norm(pos+dircos); % Distance from Rx to a point at unit distance along path of photon mu = 1/(2*d1)*(1+d1^2-d2^2); % Scattering angle from scatterer to Rx (Law of Cosines) % Approximate probability the photon scatters toward the Rx area Pang = Sr/d1^2*(C(1)+C(2)*mu^2+C(3)*(C(4)-C(5)*mu)^(-3/2)); end
Thesis/Dissertation
eng
Civil Engineering
Vancouver : University of British Columbia Library
Attribution-NoDerivatives 4.0 International
Graduate
Cheng, Julian
Comparative analysis of straight and arbitrary unmanned aerial vehicle trajectory planning for corona effect detection on powerlines
Text